Patrick Ballard
email: ballard@lms.polytechnique.fr

DYNAMICS OF RIGID BODIES SYSTEMS WITH UNILATERAL OR FRIC TIONAL CONSTRAIN TS Formulation And Well-Posedness

Keywords: Analytical Dynamics, Non-smooth Mechanics, Impact, Friction

HAL is

Introduction

The point of departure of any mechanical theory is a geometric description of the system under study and all its possible (or, more exactly, admissible) evolutions. This is always a schematization. Linear forms on the space of admissible (virtual) velocities define on turn the most general representation of inte::Ċal and external forces which is consistent with the geometric description. Naturally, obtaining their precise expression for a given system remains a part of the modelling process. The mass ::stribution leads to the definition of the kinetic energy of the system which is a positive definite quadratic form on the space of velocities. Taking a time derivative, we obtain the expression of the virtual power of inertia fo::āes (or acceleration) in any virtual velocity. The Fundamental Principle of Classical Mechanics asserts that the virtual power of inertia forces should equal the virtual power of external and internal forces in any admissible virtual velocity. As a consequence, we derive the equation of motion. For some class of geometric descriptions, the equation of motion, associated with some initial conditions, determines completely the subsequent motion of the system. We shall say that the evolution problem associated with the dynamics is well-posed. On the opposite, there are many examples of mechanical theories in which initial conditions and equation of motion are not enough to determine the subsequent motion of the system. This is generally a::Ĭbuted to the excess of schematization of the geometric description. The missing physical information is added through a constitutive law. Actually, well-posedness of the resulting evolution problem serves generally implicitly as a guideline to identify the gene::ÿl form of the constitutive law, although some thermodynamical considerations can also play an important part.

In this paper, we are concerned with the dynamics of rigid bodies systems. Speaking of rigid bodies systems is, actually, the geometric description of the system. It could be said that this is the most simple geometric description of solids. Working in the framework of rigid bodies system means that we are not interested in the prediction of the deformation of the bodies. It does not mean that we do not consider physical situations in which bodies deformability play a role. Let us illustrate this by examining the impact of two billiard balls. Billiard balls are always deformable. But, generally we are not interested in the deformations of the balls but only on their 'global' motion. Thus, we shall use a geometric description based on the rigidity assumption. However, we know that impacts are governed by deformation wave propagation in each of the balls. So, we can not expect the simple theory based on the geometric assumption of rigidity to be able to predict the outcome of an impact experiment. We must expect that some indetermination will remain. To get well-posedness of the theo::ē (this is necessary to make predictions which is the final aim of any me chanical theory), we are led to add to the theory an impact constitutive equation. This is nothing but injecting back in the theory the outcome of the impact, since the physical phenomena which governs the impact have been eliminated. Of course, in practical situations, we have to identify the impact constitutive equa tion. The choices are, either to make experiments or to use a refined theory (the elasticity theory which is based on a refined geometric assumption) in order to g _ et the outcome of each situation of impacts. In some situations, identifying the impact constitutive equation can represent a huge amount of work. In such a case, depending on the desired precision of the predictions of the theory, one may be led to question the relevance of the simple geometric assumption that has been adopted. The use of one geometric description or another to model a given real situation is always a compromise between the desired precision of the predictions, the amount of computation which is possible and the physical informations on the system which are available.

Since in this case, no constitutive law has to be identified, the main field of application of rigid bodies dynamics has been for a long time, celestial mechan ics where remarkable precision of the predictions was reached. Recently, some new fields of application of rigid bodies dynamics have emerged: robotics, granular dynamics, virtual reality, ... All these fields have in common that de termining the deformation in the bodies is of no interest. Nevertheless, in these applications, impacts are possible events that have to be incorporated in the theory. Very often, precision of the predictions is not so important and one may accept very approximate impact constitutive equations. Hence, the need has emerged to enrich the well-established theory of rigid bodies dynamics with the modelling of more complicated phenomena like impacts or friction, some of them relying physically on the deformation of the bodies. This new field is o::¤en called, after Jean Jacques Moreau, Non-smooth mechanics.

Actually, those more complicated phenomena are taken into account through constraints. A constraint is a kinematical specification of the motion with which some forces are associated: the reaction forces. In general, the kinematical specification in itself is not enough to determine the reaction force: a constitutive law of the constraint has to be added. It conveys some physical assumption on the way the constraint acts.

At the time being, it seems that only the rigid bodies dynamics with perfect holonomic bilateral constraints has firm mathematical foundations in the sense that the theory ensures the well-posedness of the evolution problem describing the dynamics. In this paper, we are concerned by the systematic formulation and well-posedness of the evolution problem describing the dynamics of sys tems involving more general constraints such as unilateral or frictional ones. As seen above, this program will necessarily involve the discussion of some constitutive law. Our aim will not be to try to identify any realistic one but just to characterize the general forms of constitutive laws that are compatible with the well-posedness of the theory. My opinion is that well-posedness should be considered as a requirement of any theory in classical dynamics. ::ath this idea in mind, the discussion of well-posedness is intimately connected with the discussion of constitutive laws. Actually, we shall consider well-posedness as the final aim of the theory. After having written the Fundamental Principle of Classical Dynamics, we shall look for the supplementary hypotheses that are necessa::Ĕ to get well-posedness. Each time an hypothesis will be made, we shall try to motivate it by a counter-example. These hypotheses will be classified into two categories. Those which convey physical assumption will be called 'constitutive' hypotheses and the other one whose aim is to prevent from mathematical pathologies will be called 'regularity' hypotheses. Since one aim is to obtain gene::Āl forms of constitutive laws, one has to make su::ă that the constitutive laws do not depend on any particular parametrization of the system. For this reason, we are going to try to obtain intrinsic formulations of dynamics, that is, formulations which do not rely on a particular choice for the parametrization of the system. This necessarily requires the use of the language of differential geometry. But, only the most elementary level of differential ge ometry is required.

The major enhancement of mathematical consistency which seems to be de sired at the time being concerns the modelling of impacts and that of f::ć ction. These two subjects are the major concerns in this paper and I believe that a mathematically satisfactory theory is obtained on both points-of-view of gen eral formulation as well as well-posedness. However, the task is far from being achieved. In this paper, we examine the cases of impacts and friction sepa rately. There remains to mix the two theories to discuss, for example, frictional unilateral constraints, which is not done here. The result would be a general theory of the evolution of mechanisms consisting of rigid bodies.

Section 1 recalls briefly the basics of intrinsic formulation and well-posedness of the dynamics of rigid bodies systems. The aim of this section is to provide precise description of the framework and notations. Section 2 contains also only well-known material. It shows that superimposing perfect holonomic bi lateral constraints does not modify the structure of the theory. In Section 3, perfect unilateral constraints are discussed. The general form for the impact constitutive equation is provided and the general formulation for the evolution problem is derived. Well-posedness is fully discussed. In Section 4, the case of general perfect non-holonomic bilateral constraints is examined. Actually, this type of constraint is a particular case of non-firm constraints which are the concern of Section 5. A complete theory of non-firm constraints is derived, including systematic formulation and well-posedness. In Section 6, the formal ism of non-firm constraints is applied to the description of frictional bilateral constraints. The underlying idea is that friction should be considered as a dis sipation mechanism obeying the ::Unciple of Maximal Dissipation. In some cases (for example, systems of punctual particles), we recover standard dry friction laws such as Coulomb fric::Īon and, in some cases, we do not. Section 7 provides a brief description of the situations that are not contained in the above theories and the extensions of the content of the paper that could be done later on.

1.

The dynamics of rigid bodies systems

1.1

The geometric assumption: rigidity Classical mechanics postulates the existence of a three-dimensional oriented affine Euclidean space£, sometimes called the (Galilean) real world, and an absolute chronology represented (after the choice of an origin) by a real number, generally denoted by t. The vector space associated with £ will be denoted by

E.

A solid is represented by its real world reference configuration which is nothing but a possible geometric locus of all the material points of the solid in £. The geometric assumption of rigidity can be stated as follows: the only real world configuration of that solid which can be observed are obtained ::£om the real world reference configuration by direct isometries. Therefore, once the real world reference configuration has been fixed, any real world conguration of the solid is represented by a direct isometry q. Considering a material point of the solid identified by its location M E £ in the real world reference configuration, the current location of that material point in the configuration defined by q is: m( M, q) = q(M) .

(1.1) Since any direct isometry on £ can be split into a translation and a rotation, the set of all direct isometries can be identified toE x §00 (where §03 denotes the set of all direct orthogonal endomorphisms on E, endowed with its standard manifold structure). It is said that E x §00 is the (abstract) configuration manifold of the rigid solid. Since its dimension is 6, we say that the rigid solid has 6 degrees of freedom (do::ħ. Any (local) chart on the configuration manifold is called a (local) parametrization. The configuration manifold is generally denoted by Q and a configuration (an element of the configuration manifold), by q. A local chart (parametrization) will be denoted generally by '1/J. Thus, for a rigid solid, the symbol '1/J(q) denotes an element of JR6.

Other idealizations of rigid solids can appear: the infinitely thin rigid bar whose configuration manifold is Ex §2 ( §2 denotes the two-dimensional sphere equipped with its standard manifold structure) and the punctual particle whose configuration manifold is simply E.

A motion of a rigid solid is a curve on its configuration manifold (a mapping from a time interval I into Q). The derivative of the motion at instant t is denoted by q(t). It is called the (abstract or sometimes, generalized) velocity. It is an element of the tangent bundle TQ of the configuration manifold. One often encounters the name 'state space' for TQ, in which case q(t) is also called a state of the system. Since the mapping m defined by formula (1.1) is obviously smooth, the material velocities are expressed in terms of the (abstract) velocity by:

m = oqm(M, q) • q, (1.2)
where oqm (M, q) is a linear operator ::¢om the tangent space TqQ into Tm£ = E. The mass distribution in the rigid solid is specified on the real world reference configuration. It is a bounded positive measure on £. It is denoted by p, .

Considering an arbitrary motion (I , q ( t)) of the rigid solid, the kinetic energy K at instant t is by definition: K = � le l l ml l 1 d p,(M ) .

(1.3) Combining formulae (1.2) and (1.3), we obtain easily the expression of the kinetic energy in terms of the (abstract) velocity. Then, it is easily noticed that the kinetic energy defines a nonnegative quadratic form on each tangent space TqQ of the configuration manifold. The mass distribution is said to be consistent with the geometric description if this quadratic form is positive definite. The following are easily proved:

• A mass distribution 11in the three-dimensional solid Ex §(()) 3 is consistent if and only if its support Supp 11contains at least three non-aligned points.

•

A mass distribution 11in the infinitely thin barE x §2 is consistent if and only if Supp 11contains at least two distinct points.

• A mass distribution 11in the punctual particle E is consistent if and only if Supp 11is non-void. >From now on, we shall assume that the mass distribution is always consistent with the geometric description. As a result, the kinetic energy defines a scalar product on each tangent space of Q, endowing the configuration manifold with a Riemannian structure. This Riemannian metric is naturally called the kinetic metric. From now on, whenever we spe::o of a configuration manifold, it will always be supposed to be equipped with its Riemannian structure.

A rigid bodies system is a finite collection of rigid bodies. The configuration manifold of a rigid bodies system is the cross-product

Q 1 x Q 2 x • • • x Q n of the individual configuration manifold Q i of each rigid body of the system.
The fundamental idea which is behind these definitions is that the config uration manifold conveys all the necessary information on the system and no more. For example, we should keep aware that the kinetic metric conveys all the relevant information about the mass distribution but, one can not, generally, recover the mass distribution from the kinetic metric.

Remark 1. The reader who is not familiar with elementary differential ge ometry could have the feeling that we have expressed very simple (and well known) things in a complicated way. Such a reader would probably prefer a presentation where the parametrization of the system is introduced at first and each definition (the abstract configuration, the kinetic metric, ... ) is made in terms of real matrices. Such a presentation should then precise what are the effects on these matrices of a change of parametrization. This leads to heavy and boring formulae and is often left aside, but this is not the main reason why I have chosen the above presentation. The possibility of defining every concept without any reference to a given parametrization ensures that all what has been defined is intrinsic (that is, does not depend on the particular parametrization under consideration). This fact is p::scularly crucial when one deals with con stitutive equations and introducing constraints necessarily involves constitutive equations. In the end, I believe that the intrinsic presentation, making appar ent the structure of the theory, provides deeper understanding. However, the reader who feels more comfortable with it, might consider that the configura tion manifold Q is an open subset of JR d equipped with a 'variable' symmetric positive definite matrix (9 ij ( q)), which is nothing but considering a particular parametrization of the system. The following convention notations are made on that purpose.

Notations. For Q being a smooth Riemannian manifold of dimension d, we shall denote by:

• TQ and T* Q, the tangent and cotangent bundles,

• Ilq and IIq, the natural projection mappings of TQ and T* Q,

• ( • , • ) q • the local duality product between tangent space T q Q and cotangent space T; Q,

• (•, •) q and ll• l l q ' the local scalar product and norm on T q Q (a* will be added when refe::Čng to the scalar product and norm on T* Q),

• 11 (and U = t>-1 , its inverse), the isomorphism of vector bundles from TQ onto T* Q naturally associated with the Riemannian metric of Q.

For q(t ) being a curve on Q, we have decided above to denote the derivative at t by q( t) E TQ. In order to be consistent with the suggestion made in remark 1; we shall alternatively use the notation (q(t ), q(t)) as often as it will not be too heavy or confusing. This is clearly a redundant notation since the base-point q = Ilq (q) is contained in the derivative, but I believe that this notation may help the understanding. More generally, an element v of TQ will also be denoted by (q , v) with q being the base-point of v. For 1/J being a local chart on Q, '1/J(q) is an element of JRd that we denote by (q 1 , q 2, •.. , q d). Still to be consistent with the suggestion of remark 1, we shall sometimes keep the notation q to refer to 1/J ( q) . Thus, for q being an abstract configuration, we might write q = ( q 1 , q 2, .•. , q d) . More generally, each time it will not be confusing, we will keep the same notation for an object and its representative in a chart. As usual, the natural basis of T q Q (resp. T; Q) naturally associated with the chart 1/J is denoted by ( e! (q ), e 2 (q) , ... , e d (q)) (resp. ( e 1 ( q), e2 (q) , ... , ed (q) ) ). For (q , v) belonging to TQ, we denote by v i (i = 1, 2, . . . d) its components in the natural basis ::qd we shall write: v = v i ei (q ). Einstein's summation convention will always apply unless explicitly stated. For q(t) being a curve, we shall write: q( t) = q i (t) ei (q(t)) , and q i (t) is the derivative at timet of the real-valued function q i (t) . As usual, Y ij ( q) will be the covariant components of the metric in the considered chart and g ij ( q) its contravariant components; q k ( q) will be the associated Christoffel symbols:

r i . ( ) = ! ih ( ) ( o gh k ( ) + 8 gjh ( ) _ 8 gjk ( ))
J k q 2 g q aq j q oq k q oq h q • For q(t) being a curve on Q and v a vector field on that curve, the covariant derivative of v along q( t) is denoted by: � v (t) = ( : t v i (t) + r] k (q(t))v j (t)q k (t)) e i (q(t)).

1.2

Formulation of the dynamics Consider a rigid bodies system of configuration manifold Q and a motion q(t) of that system. The power of inertial fo rces at instant t is, by de::¡nition, the time derivative at t of the kinetic energy: d . dt K (q, q)

� : t (q(t), q(t)) q ( t) ' (�q(t), q( t )) ' q(t )

= I 'p d D q(t), q(t)) .

\ t q( t)

Hence, it is seen that the power of inertial forces at time t defines the cotangent vector 'rJ Dq(t)jdt E r ;(t)Q. An arbitrary element TqQ is often called a virtual velocity of the system in the configuration q. Then, the linear form 'rJ D q(t) /d t is called virtual power of inertial forces. The analysis of the dynamics has to take into account external and internal forces. They are usually given as a force distribution on the current real world configuration. This is an E-valued measure which may depend on the current state (q, q) and on timet. We shall denote it by cp (q, q; t) The power of the internal and external forces at timet in the motion q(t) is: le ( rh,d cp (q, q; t) ( m (M,q))) E = le (oq m (M,q) • q, dcp(q, q; t) ( m (M,q))) E , which also defines a linear form f(q, q; t) on TqQ by: (f(q, q;t) , v) q �le (8q m (M,q) • v,dcp(q, q; t)( m ( M ,q))) E ,

for any virtual velocity v E TqQ. This linear form j (q, q, t) E r;Q is called virtual power of external and internal fo rces. The reason for such a modelling of forces by duality is that it ensures the consistency of the forces modelling with the geometrical description of the system. The virtual power mapping f (q, q, t) extracts from the force field ::? only the information which is relevant to the dynamics analysis in the framework of the geometrical assumption of rigidity.

The fundamental principle of classical mechanics asserts that the virtual power of inertial forces should equal at every instant the virtual power of external and internal forces:

V t, P �q(t) = j(q(t), q( t), t).

(1.4) Equation (1.4) is referred to as the equation of motion. It is a second-order differential equation on the configuration manifold. To express it in a particular parametrization of the system, the following is useful.

Proposition 1 (Lagrange) Let 'lj; be a local chart and q ( t) a C 2 motion on Q.

One has: P � q(t) = (:t(j� i K(q(t),q( t)) -8 � i K( q(t) , q ( t))) e i (q(t)) .

Proof. It is straightforward since:

( d ;j rj • k • l ) i 9ij dt •r
+ k l q q e ' .

( d • j + 1 jh ( 8 9h l 8 gh k 8 gk l ) • k • l ) i 9 z j dt q ?, 9 8q k + 8q l -8q h q q e '

( d 8 ( l • j • k ) 8 ( l •J • • k )) i --. -q g • k q --. -q g • k q e . dt 8qZ 2 J 8qZ 2 J D
We are given an initial instant to and an initial state (q0, vo) E TQ. Then, the evolution problem associated with the dynamics of rigid bodies system is the Cauchy problem:

Problem I. Find T >to and q E C 2 ([ t0, T[ ; Q) such that:

• (q(to),q(to)) = (qo,vo) ,

• Vt E [to, T[, P �q(t) = j( q(t),q(t) ,t).

1.3

We ll-posedness of the dynamics

To study the well-posedness (existence and uniqueness of solution) of prob lem I, we have to specify regularity assumptions on Q and f.

Counter-example 1. Consider the evolution equation d 2

1
dt2 q(t) = 6l q(t) l 3 (q E �) with initial condition (q {O ) , q {O )) = {0, 0). It is readily checked that the two motions defined on � + q(t) = 0 and q(t) = t3 provide two distinct solutions.

To get well-posedness, we have to make ::¥her hypotheses. Throughout this paper, we shall distinguish two classes of hypotheses: the constitutive hypothe ses and the regularity hypotheses. A constitutive hypothesis is an hypothesis which conveys physical meaning. A regularity hypothesis conveys no physical meaning and is stated to eliminate mathematical pathologies. The following regularity hypothesis is slightly stronger than necessary.

Regularity hypothesis. The Riemannian configuration manifold is of class C 2 and the mapping

f : TQ x � --+ T* Q is of class C1 .
It should be pointed out that the first part of this hypothesis is actually no hy pothesis at all. The configuration manifold of the three-dimensional rigid solid, of the infinitely thin rigid bar or of the punctual particle, with arbitrary consis tent mass distribution are coo (or, even more, analytic) Riemannian manifolds. The configuration manifold of a rigid bodies system (with no constraint), being a cross-product of such manifolds, can be assumed to have arbitrarily regularity. This is a restriction neither on the geometry nor on the mass distribution of the system, but on the class of admissible parametrizations.

Under this regularity hypothesis, we have the following well-posedness re sult.

Theorem 2 (Cauchy) There exists a unique maximal solution fo r problem I.

More precisely, theorem 2 states that there exists T m > t0 (T m E �U { + oo}) and qm E C 2 {[to, Tm[, Q) being a solution of problem I such that any other solution of problem lis a restriction of qm . Of course, we expect that T m = +oo, in which case the dynamics is said to be eternal. This situation can not be taken for granted, in general.

Counter-example 2. Consider the evolution equation d 2 d t 2 q(t) = (q(t)) 2 10(q E JR) with initial condition (q(O), q(O)) = (0, 1 ) . It is readily checked that the maximal solution is defined on the interval [0, 1 [.

In the usual cases encountered in mechanics, eternal dynamics is ensured by the following general sufficient condition.

Theorem 3 The configuration manifold Q is assumed to be a complete Rie mannian manifold (this is no hypothesis in the case of rigid bodies system with no constraints). The mapping f is supposed to admit the fo llowing estimate:

V(q, v) E TQ, fo ralmost all t E [to, +oo[, ll f(q, v ; t) ll� :S l(t) ( 1 + d(q, qo) + llv llq) , where d(•, •) is the Riemannian distance and l(t), a (necessarily nonnegative) function of Lfo c(I�.; IR) .

Then, the dynamics is eternal: T m = +oo.

The proof of theorem 3 relies on the Gronwall-Bellman lemma which is now recalled. Vt E [to, T], 2 </; 2 (t) :::; 2 a 2 + m (s)<f;(s) ds,

to Vt E [to, T], 1 </J(t) l :Sa + {t m (s) ds.

lto Elementary proofs of lemmas 4 and 5 can be found in BREZIS (1973), p. 156.

Proof of theorem 3. Suppose Tm is finite. From the equation of motion (1.4),

we have, for all t E [to, Tm[, � l l tim(t)J I �m< t )-� J l v ol l �o � { t (J(qm(s), tim(s); s), tim(s))qm (s) ds , lto � r t l (s) (1 +d(qm (s),qo) + l l tim(s)J i qm(s)) ll tim(s)l l qm(s)ds , Vt E [to, Tm [, d(qm(t), qo) � { t JJtim(s)J Jqm(s) ds , lto

Vt E [to, Tm [, d(qm(t), qo) + J J qm(t)J Jqm( t ) � ll v o l lqo + r t l (s) d s + r t d(qm (t),qo) + JJtim(t)JJqm( t) � ( ll v ollqo + 1: l (s) d s ) e ft t o ( l+l (s)) d s , which shows that the function t t--+ ll ti m(t)J i q( t) is bounded over [to , Tm [• By the completeness of Q, we deduce that qr = lim qm(t) t-tT;;;, exists in Q. Then, it is an easy matter to deduce that ( qr ,v r ) = lim (qm(t), qm(t)) exists in TQ, t-tT;;;,

Perfect holonomic bilateral constraints

A constraint describes a type of forces which are not taken into account by the forces mapping f. Indeed, it is possible to specify (partially) some forces by their kinematical effects. These kinematical effects leave in general the associated forces partially undetermined and we have to add phenomenological assumptions on the way the constraint acts, through a constitutive law of the constraint.

2.1

The geometric description

A holonomic bilateral constraint is a restriction on the admissible motions of the system which is expressed by means of a finite number n of smooth real-valued functions t.p i defined on the configuration manifold Q:

ViE { 1, 2,••• ,n} , t.p i(q) =0.

(1.5)

The word constraint in the singular will be used indifferently to speak either of a constraint specifically associated with a single function t.p i or of the constraint associated with all the functions ::Ôi• In this terminology, a finite collection of constraints is still a constraint. We denote by S the set of all admissible configurations:

S ={qE Q ; ViE{1, 2,••• ,n} , t.p i(q)=O } .
The following hypothesis is usual in this framework.

Regularity hypothesis I. The functions l.{) i are functionally independent, that is, for all q E S, the dt.p i ( q) ( i E { 1, 2, • • • , n} ) are linearly independent in T*Q.

A straightforward consequence of this hypothesis is that S is a submanifold of Q of dimension dn. As a result, S inherits a Riemannian structure from Q. We shall say that S is the configuration manifold of the constrained system.

Formulation of the dynamics

The realization of the constraint (1.5) necessarily involves a modification of the equation of motion (1.4). This is done by adding to the vi::Ďual power of forces f(q, q; t) a corrective unknown term R called the virtual power of reaction forces:

V t, � � q(t) = f(q(t), q(t), t) + R(t).

We might expect R to be determined by the geometric constraint (1.5). It does not work in general. We have to add phenomenological assumptions on the MA THEMA TICS: way the constraint acts. This is the constitutive law of the constraint. At this point, we restrict ourselves to the following.

Constitutive hypothesis 11. The holonomic bilateral constraint (1.5) is sup posed to be perfect (one also says synonymous! y ideal), that is, the virtual power of the reaction forces R vanishes in any virtual velocity compatible with the bilateral constraint:

\fv E { v E Tq Q ; ViE { 1, 2, .. • ,n }, ( drÄi (q),v)q = 0} � TS, ( R,v)q = 0.
Hypotheses I and II imply that there exists n real-valued functions Ai , unique, such that: n R(t) = L Ai (t) drÅ i (q) . i=l Now, we formulate the evolution problem associated with the dynamics of rigid bodies systems with perfect bilateral constraints. The initial condition is assumed to be compatible with the realization of the constraint: ( qo , vo) E T S.

Problemii . Find T > to,q E C 2 ( [to,T[;Q) and n functionsr«i E C 0 ( [to,T[;�) such that:

• (q(to),q(to)) = (qo , vo),

• \ft E [t 0 , T[, q(t) E S ,

D n • \ft E [to, T[, IJ d i q(t) = f(q(t), q(t), t) + L A i (t) d rAEi (q(t)).
i =l Here, we used the notation DQ / dt for the covariant derivative to underline the fact the covariant derivative is understood with respect to the Riemannian strùcture of Q (and not to that of S).

Let q be a point of Q, v a vector in TqQ. and Ea subspace of Tq Q. The orthogonal projection of v on E for the scalar product of Tq Q induced by the Riemannian structure of Q is denoted by Proj q [v; E] . Similarly, Proj� [v*; E*] denotes the orthogonal projection of the cotangent vector v* on the subspace E* of T ; Q. If q(t) is a curve on the Riemannian submanifold S of Q and v a vector field on that curve, then we have (CHAVEL (1993), p. 54):

Therefore, any solution of problem II is seen to be a solution of Problem IT'. Find T > t0 and q E C2([t0 , T[; S) such that:

• (q(to), tj (to)) = (qo , vo),

• Vt E [to, T[, � �: tj (t) = Proj � ( t ) [ f(q(t), tj (t); t); r ;( t )s] .

Reciprocally, any solution of problem II' is readily seen to generate a solution of problem II: the two evolution problems are equivalent.

The linear form (cotangent vector) Proj� [f(q, q; t); r; S ] equals the restric tion of the linear form f ( q, q; t) on the space Tq S of virtual velocities compatible with the bilateral constraint. Therefore, it is the virtual power of external and internal forces in any virtual velocity compatible with the constraint.

2.3

Well-posedness of the dynamics

Problem II' has formally the same structure of problem I. Since problems II' and II are equivalent, the results of Section 1(1.3) give the well-posedness of the dynamics of rigid bodies systems with perfect bilateral constraints.

Regularity hypothesis lll. The configuration manifold Q and the functions 'Pi are of class C 2 and the mapping f : TQ x r±-+ T*Q is of class C 1 .

Proposition 6 Problems I/ and I/1 have a unique maximal solution qm. More over, if Q and the functions 'Pi are of class GP (p 2: 2), and f of class CP-l then qm is of class CP. If Q, f and the 'Pi are analytic functions then so is qm.

The second part of proposition 6 follows from standard results on ordinary differential equations (see, for example, CoDDINGTON & LEVINSON (1955)).

The analysis of the eter÷ty of the dynamics is provided by theorem 3.

The regularity hypothesis I could seem very restrictive. However, dropping it would make us run into troubles.

Counter-example 3. Consider a rigid homogeneous bar of length l. The two extremities of the bar are constrained to remain on a fixed circle of diameter l.

The two corresponding bilateral constraints are supposed to be perfect. This is a simple occurøence of bilateral constraint which does not satisfy hypothesis I. At initial instant, the bar is at rest. A constant force is applied at the middle point of the bar. This force is directed in the plane of the circle but not along the bar. The reader will convince himself that the corresponding evolution problem II admits no solution.

Illustrations and comments

The confirÖon manifold Q of the rigid body system with no constraint is often referred to as the primitive configuration manifold, whereas the subman ifold S is called the reduced configuration manifold. In practice, the reduced configuration manifold can be often constructed directly, without introducing first a primitive configuration manifold. In such a case, the forces mapping is directly inrĄoduced with respect to the reduced configuration manifold.

Example 4. Consider a plane system of two homogeneous rigid bars 1 and 2.

The bar 1, of length l 1 and mass m1 is connected to a fixed support by means of a perfect ball-and-socket joint equipped with a spiral spring of stirÐess k1. The bar 2, of length l 2 and mass m 2 is connected to the free extremity of the bar 1 by means of another ball-and-socket joint also equipped with a spiral spring of stiffness k 2 • A force acts on the free extremity of the bar 2. This force remains parallel to the direction of the bar 2 and is of constant magnitude ). > 0 (see The configuration space is JR 2 equipped with its canonical structure of coo manifold (it is not the 2-torus since the spiral springs impose to be able to count the 'number of turns'). This manifold may be represented by a single chart; in other terms, there exists a global parametrization of the system. In the sequel, we shall only use the chart (q 1 , q 2 ) defined by the angular measures associated with each of the joints.

•

The kinetic energy is: K = This kinetic energy defines a Riemannian structure on the configuration space. The expression of the metric tensor in the considered chart is:

911 ( q 1' q 2 ) 912 ( q 1' q 2 ) 922 ( q 1' q 2 ) = (� 1 +m2) l� , -� m 2hl2 cos (q 1 -l) = 921 (q1, q 2 ) , 1 2 = 3m2l 2 .

•

The forces mapping has for expression in the considered chart: f (q , q; t) = [>.h sin { q 1 -q 2 ) -(k 1 + k 2) q1 + k 2 q 2 ] e 1 (q) + [k2q 1 -k 2 q 2 ] e 2 (q).

The equations of motion in the chart under consideration is easily formed by use of proposition 1:

(� + m2) l� i P + �hl2 cos { q 1q 2 ) ;p + �hl2 sin ( q 1 -q 2 ) ( q 2 ) 2

= Ah sin { q1-q 2 )-(k1 + k 2) q 1 + k 2 q 2 , �hl2cos (q1-q 2 ) i/ + !]2l �iP -� hh sin ( q1q 2 ) ( q 1) 2

= k 2 ( q 1 -q 2 ) .

By proposition 6, one can conclude that a unique maximal motion is associated with any initial condition. Moreover, this maximal motion is analytic and is defined for all time. Indeed, it is easily seen that there exists a positive real constant C, depending only on (h, 1 2 , m1, m2) such that:

where 1 • 1 denotes the canonical Euclidean norm on JR 2 . Therefore, the assump tions of theorem 3 are satisfied.

It should be underlined that the framework of pe::Ąect bilateral constraints does not require that there should be no energy dissipation physically associated with a constraint. Indeed, such an energy dissipation can be described, in some cases, in terms of internal forces. For example, suppose that, in the system described above, some viscous damping with coefficients 'f/ 1 and 'f/ 2 is associated with each ball -and-socket joint. Then, it is incorporated in the forces mapping

f which should be changed into f(q, q; t) = [-\h sin (q 1 -q 2 ) -(k 1 + k 2 ) q 1 + k2q 2 -('f! l + 'f/ 2 ) q 1 + 'f/ 2£i 2 ] e 1 (q) + [k2q 1 -k 2 q 2 + 'f/ 2£i 1 -'f/ 2£i 2 ] e 2 (q).
The above remark does not apply to the case of Coulomb type friction.

Remark 2. As problems II and II' are equivalent, we see that the dynamics of the constrained system depends only on the geometry of the submanifold S and not on the particular choice of the fu nctions 'P i used to define it. In other words, consider a constraint, say constraint 1, de:: n ed by n functionally independent functions 'P i and another constraint, say constraint 2, defined by n functionally independent functions 'P i• Suppose, in addition, that:

S = {q E Q ; Vi, 'Pi(q) = 0}= {q E Q ; Vi, 'Pi(q) = 0}.
Then, the dynamics of the system subjected to constraint 1 is identical to the dynamics of the system subjected to constraint 2. Moreover, the reaction forces in the motion are the same in both cases.

Since the modelling of rigid bodies system with no constraint or with per fect holonomic bilateral constraint leads to the construction of mathematical stru ctures of the same type, we state the following definition.

Definition 7 A simple discrete mechanical system is a pair ( Q, f) where:

• Q is a finite-dimensional smooth Riemannian manifold called the config uration manifold.

• f : TQ x lR--+ T*Q is a smooth mapping satisfying: V(q,v) E TQ, Vt E JR, called the fo rces mapping. ITQ( f(q , v; t)) = q, 3.

Perfect unilateral constraints

The consideration of elementary examples shows that the dynamics of rigid bodies systems can lead to some prediction of the motion where some bodies of the system overlap in the real world. Of course, this should not be allowed. Hence, very often, one has to add the statement of non-penetration conditions to a simple discrete mechanical system. This is a simple occurrence of uni lateral constraint. In this section, we shall discuss the consideration of perfect unilateral constraints in simple discrete mechanical systems.

3.1

The geometric description

Consider a simple discrete mechanical system with configuration manifold Q . A unilateral constraint is a restriction on the admissible motions of the system which is expressed by means of a finite number n of smooth real-valued functions <p i defined on the configuration manifold Q:

ViE {1,2, ... ,n} , c.pi (q) � 0.

We denote by A the set of all admissible configurations:

A = {q E Q; ViE {1,2, ... ,n} , c.pi (q) � 0}. (1.6)
The set of all active constraints in the admissible configuration q E A is defined by: J(q) = {i E {1,2, . . . ,n} ; 'Pi (q) = 0}.

The following hypothesis should be brought aside regularity hypothesis I of Section 2.2.1.

Regularity hypothesis I. The functions <p i are functionally independent in the sense that, for all q E A , the dcp i (q) (i E J(q)) are linearly independent in T*Q.

Straightfo::đa::Ă consequences of this hypothesis are:

• A is a closed subset of Q, • oA c U� =1 cpi 1 ( {0}) (oA is the boundary of A), 0 0 • A= J-1 ( {0}) ( A is the interior of A ).
Consider amotion q(t) in A and assume that aright velocity q+ (t) E Tq (t) Q exists at instant t, then we necessarily have:

ViE J(q(t)), ( d c.pi (q(t)), q+(t)) q (t) � 0, or, equivalently, ViE J(q(t)), (V<pi (q(t)),q+(t)) q (t) � 0, where V <p i ( q) is the gradient of <p i at q defined by V <p i ( q) = � ( d<pi ( q)) . Thus, if the system has configuration q and if a right velocity q+ exists, then q+ necessarily belongs to the closed convex cone V(q) of Tq Q defined by: V(q) = {v E Tq Q ; ViE J(q), ( d<p i (q),v)q � 0}.

V(q) is called the cone of admissible right velocities at the configuration q. In particular, 0 q EA (i.e. J(q) = 0) ==> V(q) = Tq Q. Similarly, if a left velocity q -E Tq Q exists, then q -E -V ( q)

Formulation of the dynamics

The formulation of the dynamics follows the lines of MOREAU (1983, ) 1988a).

3.2.1

Equation of motion.

As for bilateral constraints, the rea::Özation of the constraints induces some reaction force R. The following hypotheses are made.

Constitutive hypothesis II. The unilateral constraints are of type contact with out a::esion:

V v E V(q), {R,v)q 2:: 0.

Constitutive hypothesis m. The unilateral constraints are perfect:

Vv E { v E Tq Q ; ViE J(q), ( d<p i (q),v)q = 0 }• {R, v)q = 0.
As an easy consequence of constitutive hypotheses II and m, we get:

Thus, the reaction force R E T* Q must be such that:

-R E N•(q) � {t, .>.; d<p;(q) ; ViE J(q), >.; � 0, Vi� J(q), .>.; � 0 }•

(1.7)

N*(q) is a closed convex cone of T; Q and it is the polar cone of V(q) in the duality (Tq Q, r; Q) . We will also have to consider the polar cone N(q) of V ( q) for the Euclidean structure of Tq Q: N (q) � { t, .1, '1 r; (q) ; Vi E J( q), A; ;:>: 0, Vi <t J(q) , A; � 0 } . 0 Now, consider a motion q(t) starting at qo EA at time to with velocity vo.

0 Assumed to be continuous, q(t) remains in A on a right neighbourhood of t0• 0 By formula (1.7), the reaction forceR vanishes as long as q(t) is in A and the motion is governed by the ordinary differential equation:

(q(to), q(to)) = (qo, vo), � �i = f( q, q; t). Suppose that the solution of this Cauchy problem meets 8A at some instant greater than to. Denote by T the smallest of these instants. The motion admits a left velocity vector v:;. at time T. Of course, there may happen: v:;. fl. V ( q(T)). In this case, no differentiable prolongation of the motion can exist in A for t greater than T. The requiròment of differentiability has to be dropped. An instant such T is called an instant of impact.

However, we are still going to require the existence of a right velocity vec tor q +(t ) E V(q(t )) at every instant t. The right velocity need not to be a continuous function of time and the equation of motion o • + � :t = f( q, q + ; t) + R, should be understood in sense of Schwartz's distribution. Actually, we require R to be a vector valued measure rather than a general distribution.

We denote by MMA{I; Q) (motions with measure acceleration) the set of all absolutely continuous motions q(t) from a real interval I to Q admitting a right velocity q + ( t) at every instant t of I and such that the function q + ( t) has locally bounded variation over I. Naturally, bounded variation is classically defined only for functions taking values in a normed vector space. However, for any absolutely continuous curve q( t) on a Riemannian manifold, parallel translation along q(t) classically provides intrinsic identification of the tangent spaces at different points of the curve and so, the definitions can easily be carried over to this case. The precise mathematical setting is postponed to Appendix A. The reader will notice from Appendix A that any motion q E MMA (I; Q) admits a left and right velocity, qand q +, in the classical sense at any instant.

Moreover, with any motion q E MMA(I; Q) is intrinsically associated the covariant Stieltjes measure Dq+ of its right velocity q+. The equation of motion ta es the form:

I1Dq+ = j(q, q+; t) dt+R,
where d t denotes the Lebesgue measure. We have to give a precise meaning to condition ( 1. 7) with R being a vector valued measure.

Convention. We shall write:

R E -N* (q(t))

to mean: there exist n nonpositive real measures A i such that:

n R = L A i d<p i (q(t)), i = l Vi E { 1, 2, .. • ,n}, Supp ).i c {t ; <p i (q(t)) = 0}. (1.8)
Using this convention, the final form of the equation of motion is: R = I1Dq+-j(q(t), q+(t); t) dt E -N* (q(t))

(1.9)

A straightforward consequence of the equation of motion is that an impact (that is, a discontinuity of the right velocity q+ by proposition 43) can only occur at an instant t such that J(q(t) ::f 0. This fact is a justification for the following definition.

Definition 8 An impact occuring at time t is said simple ::Ä J(q(t) contains exactly one element. If J(q(t)) contains at least two elements, the impact is said multiple.

3.2.2

The impact constitutive equation.

We begin this section by an example. Consider the one degree-of-freedom mechanical system whose configuration space is lR equipped with its canonical Euclidean structure. The forces mapping f vanishes identically and the unilateral constraint is repre sented by the single function <p 1 ( q) = q so that the admissible configuration set A is JR-. At initial time to = 0, we consider an initial state (q0, v0) such that qo < 0 and vo > 0. It is readily seen from the equation of motion (1.9) that an impact necessarily occurs at timet = -qo f v 0 • At this time, the left velocity is v 0• But, the right velocity can take any negative value and whatever it is, it is compatible with the equation of motion.

The reason for this indetermination lies in the phenomenological nature of the interaction of the system with the obstacle. This missing information has to be added.

Constitutive hypothesis IV. The interaction of the system with the obstacle at timet is completely determined by the present configuration q(t) and the present left velocity q -( t) . In other terms, we postulate the existence of a mapping :F : TQ -+ TQ describing the interaction of the system with the obstacle during an impact:

Vt, q+(t) = :F (q(t), q -(t)) .

(1.10)

To ensure compatibility with the equation of motion (1.9), the mapping :F should satisfy:

:F (q, v-) E V(q), :F(q,v-) -v-E -N(q). (1.11) Moreover, we add the assumption that the kinetic energy of the system can not increase during an impact:

VqE A, 'v'v-E -V(q), (1.12)
Let us comment on hypothesis N. ::en two solids hit, their bouncing is actually governed by the propagation of deformation waves in each the two solids. But, from the very beginning, we have adopted the simple framework in which each solid is supposed to be rigid, that is, for sake of simplicity, we have chosen to do not t a ke under consideration any phenomena relying on the deformation of the solids. Thus, we cannot expect the theory to be able to predict the outcome of an impact experiment. The aim of constitutive hyposthesis N is to int::ċduce in the theory the missing information. Of course, in practical situations, we have to identify the unknown mapping :F. This can be done either by means of experiments or by use of a refined theory. For example, the theory of elastodynamics could be used to predict the outcome of an impact in every impact configuration. The result would be an identification of the mapping :F. In any case, there is a very big amount of work to get a precise identification of :F. This is the price we have to pay to describe sophisticated physical phenomena in a very simple framework. Actually, this issue is faced in any mechanical theory (one could think of the theory of elasticity). Naturally, in each mechanical theory, the question arises to know what amount of lacking constitutive information should be introduced. Most of the time, well-posedness of the resulting evolution problem serves as a guideline to state the constitutive hypotheses.

Definition 9 Equation ( 1.10 ), with mapping :F fulfilling both requirements ( 1.11) and (1.12) is called the impact constitutive equation. An impact constitutive equation which ensures the conservation of kinetic energy during an impact: is called elastic. The following proposition is a straightforward and useful consequence of requirements (1.11) and (1.12).

Proposition 10 Let F be a constitutive mapping satisfying requirements ( 1.11) and (1.12). Then, we have:

'V q EA , 'tlv-E V(q) n ( -V(q)) ,
Proof. Define v + = F(q, v-) . By requirement (1.11), we have v--v+ E N(q ) . Since v-E V(q) n (-V(q)), we obtain:

(v--v+ , v-) q = 0, that is, ( v+ , v-)q = ll v-11� •
The use of Cauchy-Schwarz inequality and requirement (1.12) gives the desired result.

0

We conclude this section by a comment on requirement (1.12). At first glance, it could seem to be unnecessary. The following counter-example proves that if it was omitted, then, uniqueness of solution for the resulting evolution problem would surely not hold.

Counter-example 6. Consider the one degree of freedom discrete mechanical system whose configuration space is lR equipped with its canonical structure of Riemannian manifold. The forces mapping is supposed to be constant: f ( q, q; t) = 2. To this simple discrete mechanical system, we add the unilateral constraint described by the single function cp1 ( q) = q . Thus, A = JR -. The impact constitutive equation is given by formula (1.13) where the restitution coefficient is supposed to be the constant 1/2 : e ( q, q-) = 1/2. This mechanical system is a formal description of the physical occurence of a single particle subjected to gravity and bouncing on the floor. Consider the initial instant to = 0 and the initial state ( q 0, v0) = ( -1, 0). It is readily seen that the function q : JR + --+ JRdefined by:

Vt E (0, 1], Vt E (1, 2], Vt E [3 -2 nl_ l , 3-2� ) , Vt E [ 3 , + oo[, q(t) = t 2 -1, q(t) = t 2 -3 t + 2, q(t) = t 2 -(6 -2 � ) t + (3 -2nl_ r) {3-2� ) , q(t ) = 0,
(n E N) belongs to MMA ( JR + ; JR-) and satisfies:

• the initial condition,

• the equation of motion (1.9) (with f (q, q; t) = 2),

• the impact constitutive equation ( 1 .13) (with e(q, q) = 1/2).

This motion is pictured on Figure 1.3. Note, by the way, that it exhibits an infinite number of impacts on a compact time subinterval. It could easily be proved that no motion, defined on [0, oo [, with finite number of impact on every compact interval can exist. Now, we are going to analyse what happens when the flow of time is reversed. Define q by: 1 { [ 0,4] -+ JR- q t r+ q(4-t) Considering the initial state (q0, v0) = (0, 0) at t0 = 0, it is easily seen that q' satisfies:

• that initial condition,

• the equation of motion (1.9) (with f(q, q;t) = 2),

• the impact constitutive equation (1.13) (with e(q, q) = 2). But, q" = 0 is also seen to satisfy the same initial condition, equation of motion and impact constitutive equation. This example demonstrates that we cannot expect uniqueness of solution when adopting the canonical impact constitutive equation (1.13) with restitution coefficient e = 2 (or any real number strictly greater than 1). But the canonical impact constitutive equation with restitution coefficient strictly greater than 1 violates requirement (1.12).

3.2.3

The evolution problem. Now, we formulate the evolution prob lem associated with the dynamics of rigid bodies systems with perfect bilateral and unilateral constraints. The initial condition is assumed to be compatible with the realization of the constraint: v0 E V ( qo). Problem m. Find T > to and q E MMA([ to, T[; Q) such that:

• ( q (to),q + (to)) = (qo,vo), • 'Vt E [ to,T[, q(t) E A, • R � bDq + -f (q(t), q + (t); t) dt E -N*(q(t)) , • 'VtE]t0,T[, q +(t)=F (q(t),q-(t)).
The equation of motion is understood in sense of convention (1.8) and the impact constitutive equation is supposed to fulfill requirements (1.11) and (1.12).

Yet, no regularity assumption has been made on the mapping f . This will be done in the next section where well-posedness of problem Ill is studied. However, we can infer from Section 1.1.3 that f will be assumed to be at least of class G 1 . We can state an elementary property of any solution (if there are any) of problem Ill.

Proposition 11 (Energy inequality) Let (T, q) be an arbitrary solution ofp rob lem Ill. Then, it satisfies:

Vt 1 ,t 2 E [ to, T[, t 1 :S: t 2 , K (q(t 2 ), q+(t 2 )) -K (q(h),q+(h)) = � llti +(t 2 ) 11!(t 2 ) -� llri +(t dll!(h) :::; r t 2 (f (q( s ), q+(s);s) , q+(s)) q( s ) d s ltl Proof. We have the following equality between real measures:

( q+(t) + q-( t) 'D q+ ) = 2 q(t) I q+(t) + q -(t) ' f (q(t), q+(t); t) ) dt + I q+( t) + q-( t) ' R ) 0 \ 2 q(t) \ 2 q(t)
Integrating over ]t 1 , t 2 ] and using proposition 41 of Appendix A, we get:

Consider D = { t E]t 1 , t 2
] ; q+(t) ; q -(t) -:/= q+(t) } .

D is (at most) countable and therefore Lebesgue-negligible. We obtain:

� 11 q+ ( t 2 ) 11 !(t 2 ) -� 11 4 + ( t l ) ll!( tl) = rt 2 (q +(t), J(q( t), q+( t);t) ) q(t) dt + r 14 + ;4-, R) . ltr l]tl h] \ q Therefore, to prove proposition 11, there remains only to prove:

q + q-,R < 0.

1 ( • + 0 ) ] tl h] 2 q - (1.14) But, on one hand, r Jq++q_-,R) = r (q_ + ,R) q = r (q_ -,R) q , 1] t1,t2]\D \ 2 q 1]t1.t2]\D 1]tlh]\D
where the second integral is nonnegative by convention (1.8) whereas the third integral is nonpositive. As a consequence:

q q R =0 1 ( • + + • -)
]tth]\D (1.15)

On the other hand, r (q + (t) + q -(t) R)

1 D 2 ' q( t)
r (q + (t) + q_ -(t ) , n g_ + ) ,

Well-posedness of the dynamics

To study the well-posedness of problem m, we need to state regularity as sumptions on the data. Looking at those of Section 2.2.3, we could expect to be able to prove well-posedness of problem m under the assumption that the func tions <fJi and the mapping f are of class C 2 and C1 respectively. The following counter-example originally due to BRESSAN (1960) and SCHATZMAN (1978) shows that uniqueness does not hold in general even if the data are supposed to be of class coo.

Counter-example 7. Consider a simple discrete mechanical system whose configuration space is lR equipped with its canonical structure of Riemannian manifold. This is the configuration space of a particle with unit mass constrained to move along a line. A fixed obstacle at the origin is taken into consideration. It gives rise to a unilateral constraint described by the single function: cpi(q) = q Therefore, the admissible confi::¬ration set is A = JR-. The impact constitutive equation is supposed to be elastic. Here, the geometry is so poor that this state ment determines completely the impact constitutive equation. It is necessarily the canonical one with restitution coefficient e = 1 . The forces mapping f is supposed not to depend on the state but only on time. It will be denoted by f(t). The initial instant is to = 0 and the initial state is (qo, vo) = (0, 0). The corresponding problem Ill admits here the simple formulation: find T > 0 and q E MMA ( [ O, T[; �) such that:

• (q( O ), q +( o)) = (o, o), • Vt E [0, T[, q(t) � 0, • R � d q + -f(t) d t is a nonpositive real measure such that: Supp R c { t E [0, T[ ; q( t ) = 0}, { q(t) i= 0 => q +( t) = q -( t ) • Vt E]O, T[, q( t) = O => q +( t ) = -q -(t)
Here d q + is merely the classical Stieltjes measure associated with the fu nc tion with locally bounded variation q +. We investigate uniqueness under the assumption that f is of class coo and nonnegative:

VtE�+ , f(t) � O.

Then, it is readily seen that the null fu nction ij = 0 on �+ is a solution of that problem, whatever is the nonnegative coo function f. Now, we are going to construct an explicit example of such a function f in such a way that the associ ated evolution problem Ill admits another solution, distinct from the identically vanishing one.

First, define a Massin function p by: p 0

{ � -+JR. X 1---7 1 Ce z(z-1) if X E] -oo, 0] U [1, +oo[ if x E]O, 1[
where C is a real constant which is chosen to get:

Define: r + oo }_00 p ( x ) d x = 1. T -� (n +5) 2 -� (n + 1)(n + 2)(n + 3)(n + 4) ' (i +5) 2 � (i + 1)(i + 2)(i + 3)(i + 4) ' n + 5 (n + 1)(n + 2)(n + 4) 1 f n = n!' 1 (n + 3)!"
( i.e. c5 n = �:! (an-an + l )) ,

Now, the fu nctions f(t) and v (t), from [0, T[ to JR. are defined by: f(O) = 0 and:

Finally the fu nction q : [0, T[--+ lR is defined by: q(t) = lo t v(s) ds.

The graph of the functions f (t) and q(t) is sketched on Figure 1.4. The reader will easily check that:

• f (t) is a C00 nonnegative function on [0, T[, • (T, q
) is a solution of the considered evolution problem,

• the only instants at which q(t) = 0 are 0 and the an .

Therefore, q and fj provide two solutions of the evolution problem. These two solutions do not coincide on any open subinterval of (0, T[. Therefore, uniqueness of solution for problem m cannot be asserted, even in the case where the data are supposed to be of class 000 • PERCIVALE (1985,1991) was the first to notice that, in the above example, if f ( t) is supposed to be analytic, f(t) f(t) . i Regularity hypothesis V. The Riemannian configuration manifold, the func tions 'P i and the mapping f: TQ x lR--+ T*Q are analytic.

The proof of the following proposition can be found in BALLARD (2000). An earlier proof can also be found in LOTSTEDT (1982). Proposition 12 Let qo E A and vo E V( qo) . Then, there exist Ta > to, an analytic curve qa : [to , T a [--+ Q and n analytic functions Aai : [to, Ta [--+ lR such that:

• ( qa ( 0) , q;_f-( 0)) = ( qo, vo),

• Vt E [ to, T a[, P�qa(t) = f(qa(t), qa(t); t) + t, Aai(t) dcpi(qa(t)),

• � � : ��2� a • [,
• , n , Aai ( t) ::::; 0, 'P i ( Qa ( t)) ::::; 0, Aai( t) 'Pi ( qa( t)) = 0.

Moreover, the solution of this evolution problem is unique in the sense that any other analytic solution ( T, q, .-\ 1 , • • • , .Xn ) is either a restriction or analytic extension of (Ta, qa, Aa l , • • • , Aan) • Corollary 13 There exists an analytic solution ( T a, Qa) fo r problem Ill. Proof. Consider the motion Qa fu rnished by proposition 12. It obviously sat isfies the initial condition, the unilateral constraint and the equation of motion.

The only thing which remains to prove is that it satisfies the impact constitutive equation. Since qa is analytic and satisfies the unilateral constraint, we have:

Vt E]to,Ta[, q;;-(t) = q;_f-( t) E V(qa(t)) n (-V(qa( t))), and therefore,

Vt E]to, T a [, q ;_f-(t) = q ;;-(t ) = :F (qa(t), q ;;-(t)) , by proposition 10.

D

Naturally, the analytic solution furnished by corollary 13 will cease to exist at the first instant of impact. This is the reason why we have considered the wider class MMA which contains motions which are not differentiable in the classical sense. Considering motions in MMA will allow to extend the solution beyond the first instant of impact. But, it must be made sure that admitting the wider class of solutions MMA will not introduce parasitic solutions. This is the aim of the following theorem.

Theorem 14 Let (Ta, qa) be the solution fo r problem Ill furnished by corol lary 13, and ( T, q) be an arbitrary solution fo r problem Ill. Then, there exists a real number To (to < To ::::; min{T a, T}) such that: q f[to ,To[ = q af( t o,To ( • In other terms, there is local uniqueness fo r problem Ill.

The proof of theorem 14 makes extensive use of the following corollary of Gronwall-Bellman lemma (lemma 4).

Lemma 15 Let m be a nonnegative integer, and '1/J : [0, T] --+ ffi. an integrable function. If <P : [0, T] --+ ffi. is any absolutely continuous fu nction such that </J( t) = o( tm+ 1 ) when t tends towards 0 and such that there exists a nonnegative real constant C such that:

fo r almost all t E]O, T[, t : t <jJ (t) :::::; (1 +m+ Ct ) <jJ (t) + t m + 2 '1jJ (t), then,

Proof. This is almost obvious. Dividing each member of the inequality by tm+ 2 , we obtain:

for almost all t E]O, T[, � ( <jJ (t) ) < C <jJ (t) + .! ,(t) dt tm + l -tm + l '!-' •
After integration, Gronwall-Bellman lemma yields:

Vt E]O, T], <jJ (t) :::::; { t '1/J(s)ds + { t C e c (t -s ) r '1/J(o-) do-ds. tm + l Jo Jo l o

Then, an integration by part gives the desired conclusion.

Proof of theorem 14.

Step 1. Pa rametrization of the problem and notations.

D

We denote by do :::::; d the number of elements of J(qo). Consider a local chart '1/J : U C Q --+ ffi. d on Q centered at qo such that the do first components of '1/J(q) are (rpi(q)) iEJ(qo)• Such a chart exists since (drpi(qo)) iEJ(qo) is linearly independent in r; o Q by regularity hypothesis I. Next, choose a > 0, sufficiently small to have, for all t E [t0, t0 + a],

• qa(t) E U, q(t) E U, • ViE J(qo), : t rpi(qa( t)) = (drpi(qa( t)), qa{ t) )q.(t) :S 0, (1.17)

• ViE{1,2,•••,n} \J(qo) , rpi(qa(t)) < O, rpi(q(t)) < 0 .(1.

18)

Such a choice for a is possible because the fu nctions 'Pi ( qa ( t)) are analytic and the functions 'Pi(q(t)) are continuous. We denote by fi the components of f in the natural basis ( e i ) associated with the chart under consideration. Since qa and q are local solutions for problem Ill, we have, for all i E {1, 2, • • • , n }, gij(qa) ( iit + r{ 1 (qa)ti:ti!) = fi(qa , tia; t) + Aai, gi i (q) ( dq+i + r{1(q)q+ k q+ 1 dt) = fi (q, q+; t) dt + J-ti, (1. 20 ) where the Aai and ft i are respectively d nonpositive analytic functions on [to, to+ a] and d nonpositive measures on [to, to + a] . Note incidentally that the Aai and ft i vanish identically for i > d0, by (1.18). We denote by 1-1 the standard euclidean norm on R d . Confusing (abusively) q and 'lj; (q) , we shall write: d l q l 2 = L (q i ) 2 ' and i=l i=l Step 2. There exist some positive real constants C 1 and C 2 such that the fo llowing estimate: { t (l q-qa l 2 ( s ) + l q+qa l 2 (s)) d s ::; By proposition 41 of Appendix A, we have:

d ( � (q+ i -q!) 9ij(q) (q+i -qt)) = (1.21)
( q-i ; q+ i -q!) 9ij(q) ( dq+i -qt dt + r{ l (q)q+ k ( q+ l -q�) dt) ' and, therefore, using equation of motion (1.20), d (� (q+ i -q! ) 9ij(q) (q+i -�) ) =

(q+ i -q!) fi(q , q+; t)dt -(q+ i -q!) Yij(q) ( ;fa + r{ 1 (q)q+ k q�) dt

do ( • -k + • + k ) "" q q •k + L..

-q a ft k • k =l

But, each q :ft k is a nonnegative measure by (1.17), and,

do q -k + q+ k _ ( q -+ q + ) L 2 ft k -2 , R k =l
q is a nonpositive real measure by proposition 11. Therefore, d (� (q+ i -q! ) 9ij(q) (q+i -�) ) ::;

[ (q+ i -q!) f i(q , q+; t) -(q+ i -q! ) 9ij(q) ( qt + r{ 1 (q)q+ k q� ) J dt , in the sense of ordering of real measures. Integrating over ]to, t] (t E [to, to+a]), we get: � ( 4+ i -4 ! ) 9 i j ( q) ( 4+ j -tPa) � 1t [ ( 4+ i -4 !) fi ( q, 4+ ; s) -( 4+ i -4 ! ) Yii (q) ( ifa + r { l (q) 4+ k 4 � )] ds .

to

The term within the integral sign is an analytic function of the three variables q , 4+ and s. Therefore, it is also an analytic fu nction of the three variables q -qa, 4+ -4a and s. It is written under the form :

( • + i •i ) F. (
• + . . ) q -qa i q-qa, q -qa, S • But, each function Fi can be decomposed under the fo rm:

Fi (q-qa, 4+-4a; s) = Fi (O, 0 ; s) + G i (q-qa, 4+-4a; s),

where the G i are analytic and G i (O, 0 ; s) = 0. Hence, there exists a positive constant M such that, for all t E [t0, t0 + a],

/ G i (q(s) -qa(s), 4+ (s) -4a(s); s)/ � M V r-l q -( s -) ---qa -( s -) -1 2-+ -l q -. + -( s -) ---4 -a( -s ) -1 2

Hence, we have proved:

� ( 4+ i -4 !) 9 i j ( q) ( 4+ j -tPa) � 1 t { ( 4+ i -4 ! ) [fi (qa, 4ai s) -9 i j (qa) (;fa + r { l (qa)4 ! 4 � )] to +Md / 4+-4a / V lq-qal 2 + 14 + -4al 2 } ds.

Moreover, by a compactness argument, there exists a positive constant m such that for all t E [to, to +a],

� (4+ i -4 !) 9 i j (q) ( 4+ j -tPa ) 2: m /4+-4a/ 2 •

We obtain :

/ 4+ -lia / 2 (t) < �d 1: (l q-qal 2 (s) + /q+-lia / 2 (s)) ds l l t do . .

--L Aai (s) ( 4+� -4! ) ds, m to i=l where equation of motion (1.19) has been used. Note that, actually:

Vi E { 1, 2, • • • , do }, Aaiq! = 0,
and, so, by the analyticity of functions q! and Aai :

Vi E { 1, 2, • •• , do }, Aaitl! = 0.
By use of Cauchy-Schwarz inequality, we get:

Defining:

1

Md cl = -, c 2 = -+ a, m m
multiplying each term of the above inequality by e-02 t and integrating, we obtain estimate (1.21).

Step 3. Estimate (1.21) implies that the function t I-+ Ef� 1 .X�(t) q+ i (t) van ishes identically on a right neighbourhood of to Indeed, by estimate (1.21):

V t E (to, to + a], { t e -02 8 t t Aai (a) q+ i (a) dads � 0, fto fto i==l which is, after integration by parts:

But, since,

Vi E { 1 , 2, ••• , do} , V s E [to, to + a],
Aai (s) � 0 and q i (s) � 0, the two members of inequality (1.22) are nonnegative and, therefore, the in equality is preserved when taking the absolute value of each member. We get:

£ e-c ,, t. >..; ( s) q'(s) d s <; 1,: e-c,, f t. Q i (s ) -e-C2(s + t o ) q i (s + to), L i (s ) = -Aa i (s + to).

With these notations, we obtain:

t d o t sdo VtE [ O,a], 1 "f_ L i (s) Q i (s)ds$, 11 "f_ l £ i (s ) IQ i (s)dads, (1.23)
where the L i are nonnegative real-analytic functions and the Q i are nonnegative continuous functions which all vanish at t = 0 and are right-differentiable at t = 0. We are going to prove that inequality (1.23) implies that:

:3{3E]O,a], VtE [ 0,{3], Vi E { 1, 2,••• ,do} , L i (t) Q i (t ) =O.
The fu nctions L i being analytic nonnegative, there exist nonnegative integers

n1 < n2 < ••• < nm, apartitionh,h,••• Jm of { 1,2,•••
,do}, and analytic nonnegative functions G i such that:

Vk E { 1, 2,••• ,m} , Vi E Ik ,
with either G i ( O) > 0 or G i = 0. Inequality (1.23) may be rewritten as:

But, by the analyticity of the functions G i : :3{3 > 0, :JN > 0, Vi E J(q0), V a E [ 0,{3],

Therefore, for all t E [0, ,8],

Integrating by parts the left member of the inequality, we obtain:

t 1 t f L unk-l G i ( u ) Q i ( u ) d u ::; a k =l i El k 1 t 1 8 f L ( nk + 1) unk -1 G i ( u ) Q i ( u ) d u ds a a k=l i El k + Nt 1 t 1 8 f L unk-I G i ( u ) Q i ( u ) d u ds o (1. 2 4)
a a k =l i Eh Since each fu nction G i ( u) Q i ( u ) j u is bounded over [0, .8] , there exists a nonneg ative real constant H such that, for all k E {1, 2 , 0 0 0 , m} and for all t E [0, ,8],

Since it can be assumed that ,8 < 1, inequality (1.24) gives, for all t E [0, ,8], t 1t L unt -l G i ( u ) Q i ( u ) d u ::;

a i Elt (1 + n l + N t ) r r L O"n1 -l G i ( u ) Q i ( u ) d u ds+ Hlt n 2 + 2 ' l a l a i Eit
where H1 is a non negative real constant. Applying lemma 15, we get:

and the assertion of step 3 is proved.

Step 4.

Conclusion of the proof of local uniqueness.

Bringing together the results of steps 2 and 3, we get:

Vt E [to, to + ,8], 1 t (Jq -qa J 2 (s) + lti + -ti a l 2 (s)) d s � 0, to which yields the desired conclusion:

Vt E [to, to + ,8], q(t) = qa(t).

Corollary 16 There exists a unique maximal solution fo r problem Ill.

D

It was noticed above that the analytical solution for problem m furnished by corollary 13 stops to exist at the first instant of impact. To overcome this fact, we have proved that local uniqueness still holds in the wider class of motion MMA which allows impacts. But, this does not guarantee that the maximal solution for problem m is not going to stop to exist at finite time for unphysical reasons. In other terms, we still do not know if the class MMA is wide enough.

Actually, it is wide enough as shown by the following theorem which should be brought aside theorem 3.

Theorem 17

The configuration manifold Q is assumed to be a complete Rie mannian manifold and the mapping f is supposed to admit the fo llowing esti mate:

V (q, v) E TQ, fo ralmost all t E [to, +oo[, l lf(q , v; t )JJ; � l(t) ( 1 + d(q , qo) + l l vl lq) , where d(• , • ) is the Riemannian distance and l(t) , a (necessarily nonnegative) function of Lfo c (l�; ffi! ).

Then, the dynamics is eternal, that is, the maximal solution fo r problem Ill is defined on [to, +oo[. Proof. We proceed as for the proof of theorem 3. We assume that the maximal solution q is defined on [to, T[, with T finite and try to obtain a contradiction.

Step 1. The function t t-t IJq+ (t)IJq ( t ) is bounded over [to, T[: 3 V > O, VtE[to, T [, llti +(t) llq( t) �V .

( 1 .25)

By proposition 11, we have:

� llti +(t) ll�( t ) � � J J vo J I� o + 1 t llf (q( s ), q+(s); s ) ll;(s) llti +( s)ll q( s ) d s . to Vt E [ to, T[, d(q(t), qo) � { t li ti + (s)ii q(s ) ds, l to d(q(t), qo) + ll ti + (t) ll q( t) � ll vo llq o + lo t l(s) ds + l: (1 + l (s)) ( d(q(s), qo) + ll ti + (s)ll q(s ) ) ds.

By Gronwall-Bellman lemma (lemma 4), we have, for all t E [ to, T[: d(q(t),qo) + ll ti + (t) ll q( t) � ( ll vo llq o + lo t l(s) ds ) ef t t o (l+l( s)) ds , which yields (1.25).

Step 2. The right velocity q+ has bounded variation over [ to, T[ :

Var (q+ , [ to, T[ ) < oo.
By step 1 , we have:

Since Q is assumed to be complete, we deduce that:

qr = lim q(t) t-+T- (1.26)
exists in TQ. We denote by dr the number of elements of J ( qr). Let ( U, '1/J) be a local chart on Qa t qr such that the dr first components of '1/J(q ) in � d are ('Pi ( q) )iEJ( Q T ) . Consider a compact neighbourhood K of qr in Q such that:

• KcU ,

•

V q E K, J(q) C J(q r ).

or, equivalently, We deduce: for all i E { 1 , 2, • • • , d} and all si, s2 E [ t�, T[ with si < s2. There results that the Ai are d bounded measures on ]t�, T[ . Thanks to equation (1.28), it is readily seen that the measures dq + i are also bounded measures on Jt�, T[. Therefore, the d functions q + i : ]t�, T[--7 lR have bounded variation over the interval ]t�, T[. Then, corollary 36 of Appendix A yields the desired result.

Step 3. Conclusion of the proof of theorem 17.

By Steps 1 and 2 and by proposition 38 of Appendix A,

(qr,v Y, ) = lim (q(t), q +(t ))

t---+ Texists in TQ. Define: vr = F (qr, vY,) Take it as a new initial condition at t = T. Then, theorem 13 furnishes T' > T and an extension of q on [ T, T'[ such that q E M M A ([ to, T'[; Q) is a solution of problem Ill. But, this contradicts the definition of T. D

3.4

Illustrative examples and comments

It is readily seen that the fu nction q displayed in counter-example 6 is the unique maximal solution of problem Ill corresponding to the situation under consideration. This solution exhibits an accumulation of impacts on the left side of instant t = 3. However, as predicted by corollary 13, for each instant t E JR+, there exists aright neighbourhood [ t, t+ 17[ oft, such that the restriction of q to [ t, t + 17[ is analytic. A straightforward and general consequence of this is the following.

Proposition 18 Let q be the maximal solution of problem Ill fu rnished by corol lary 16. Although infinitely many impacts can accumulate at the left of a given instant, such an accumulation of impacts can never occur at the right of any instant. Moreover, in the particular case where the impact constitutive equation is elastic, the instants of impact are isolated and therefo re in finite number in any compact interval of time.

Proo f. Since for each instant t E [ t0, T[, there exists a right neighbourhood [ t, t + 17[ of t, such that the restriction of q to [ t, t + 17[ is analytic, we get the first part of the proposition. For the second part, let T be an arbitrary instant in ]t0, T[ and consider the problem m associated with the initial condition ( q( T ) , -q-( T )), the elastic constitutive impact equation and the force mapping g(q, v; t) defined by: g(q, v; t ) = f(q, -v; T-t ) which is analytic. By theorem 14, there exists an analytic function qa : [ 0, Ta [ ---+ Q which is a solution of this problem m. Any other solution of problem m coincides with qa on a right neighbourhood of t = 0. Actually, as seen in the proof of local uniqueness (theorem 14), a little bit more is proved: any function q' E M M A ([ O, T[; Q) satisfying the initial condition, the unilateral constraint, the equation of motion (1.9) and the energy inequality (proposition 11) has to coincide with qa on a right neighbourhood of t = 0. But, it is readily seen that the function defined by: q 1 (t) = q(T -t), t E [ 0, Tto[ fulfill these requirements. Thus, q' can not have right accumulation of impacts at t = T and, therefore, q can not have left accumulation of impacts at t = T and the instants of impact are isolated. Of course, if q is the maximal solution defined on [to , T[, impacts can still accumulate at the left of T, as seen on simple examples.

D

The fact that infinitely many impacts can accumulate at the left of a given instant but not at the right is a specific feature of the analytical setting that is lost in the C 00 setting as seen in counterexample 7. Actually, this counterexam ple shows that pathologies of nonuniqueness in the coo setting are intimately connected to the possibility of right accumulations of impacts. The fact that the analytical setting prevents from such right accumulations is the thorough reason why we could prove uniqueness in this case.

We conclude this section by a come back to the double pendulum of exam ple 4. The aim of the following example is to illustrate the generality of the above theory.

Example 8. Consider the double pendulum described in example 4 and add a rigid obstacle on the vertical coordinate axis as represented on Figure 1.5. This obstacle may be represented by two analytic functions whose expressions in the global chart of Q described in example 4 are: 'P l ( q l ' q 2 ) 'P 2 (q l , q 2 ) -h sin q 1 � 0, -h sinq 1 -1 2 sin q 2 � 0. It is readily seen that, except in the particular case where h = 1 2 , these constraints are functionally independent, that is, they satisfy regularity hypoth esis I. An arbitrary initial state ( qo , vo) such that vo E V ( qo) is given at time t0 = 0. To fix ideas, we adopt the canonical constitutive equation with arbi trary restitution coefficient e ( q, rì-). Then, writing the evolution problem in the chart under consideration is straightforward. By corollary 16, we get a unique maximal solution for this evolution problem. By theorem 17, we can state that this maximal solution is defined all over JR+ , that is, the dynamics is eternal.

Perfect non-holonomic bilateral constraints

In this section, we come back to simple discrete mechanical systems. Perfect holonomic bilateral constraints were defined to be constrr¿nts on the configu rations of type:

'Pi (q) = 0.

Considering an arbitrary motion satisfying the constraint and differentiating wirĂ respect to time, gives: Thus, the constraint may be viewed as acting on the velocity. There are practical situations where the constraint is given in this way. A typical occurence is the "rolling without slipping". Thus, we are led to consider general constraints of type:

(ai(q),q) q = 0, where the ai(q) are cotangent vector fields (we say also 1-form) on the config uration manifolds.

The reason that makes here desirable the study of non-holonomic constraints is that the "rolling without slipping" can be seen as a fri ctional bilateral con straint with a friction of infinite magnitude. Therefore, this section prepares the full discussion of frictional constraints in the sequel.

4.1

The geometric description

A non-holonomic bilateral constraint is a restriction on the admissible mo tions of the system which is expressed by means of a finite number n of smooth 1-form ai defined on the configuration manifold:

ViE{1,2,•••,n}, (ai(q), q) q =O.

(1.29)

As in the case of holonomic constraints, the constraints are required to be independant in the following sense:

Regularity hypothesis I. For all q in Q, the ai(q

) (i E {1, 2, • • • , n}) are linearly independent in T*Q.
A straightforward consequence of this hypothesis is that the set E of all admissible velocities: E = { (q, v ) E TQ ; ViE {1,2, • • • ,n}, (ai(q), v ) q = 0 } , (1.30) is a tangent subbundle of Q (that is, a vector bundle over Q which is also a submanifold of TQ) of dimension 2d -n.

Of course, the terminology is a little bit confusing (but it is classical) since a non-holonomic constraint may turn out to be holonomic ('holonomic' is greek for 'integrable'). A trivial example is provided in the case n = 1 when the 1-form a1 is exact (that is, there exists <p1 such that d<p1 = a1). In this case, the non-holonomic constraint is equivalent to the holonomic one: <p1 ( q) = constant. The constant is determined by the initial configuration qo. The non holonomic constraint definrÍ by a1 may turn out to be holonomic even in the case where a1 is not exact. Indeed, even if a1 is not exact, there may exist some real valued function h(q) such that h(q)a1(q) is exact. We shall say that the non-holonomic constraint defined by the ai is holonomic if there exist (locally) n real-valued functions <pi such that (1.29) is equivalent to: ViE {1,2,••• ,n} , (d<pi(q), q) q =0. takes values in E as well.

Hence, the study of non-holonomic bilateral constraints is more general than the study of holonomic ones, since the former contains formally the latter. However, the handling of holonomic constraints is simpler since it allows im mediately the elimination of the redundant parameters in any parametrization. So, each time a non-holonomic constraint turns out to be holonomic, it should be integrated.

Formulation of the dynamics

Here also, the realization of the consrăaints necessarily involves some reac tion forces R which should be specified through a constitutive assumption.

Constitutive hypothesis II. The non-holonomic bilateral constraint (1.29) is supposed to be perfect, that is, the virtual power of the reaction forces R vanishes in any virtual velocity compatible with the bilateral constraint:

'v' (q, v) E E, (R, v)q = 0.

Hypotheses I and II imply that there exists n real-valued functions Ai, unique, such that:

n R(t) = L Ai(t) O:i(q).
i=l Now, we formulate the evolution problem associated with the dynamics of rigid bodies systems with perfect bilateral constraints, either non-holonomic or holonomic (the holonomic constraint is included in the definition of the configuration manifold Q). The initial condition is assumed to be compatible with the realization of the constraint: ( q0, v0) E E. Problem lV. Find T > to, q E C 2 ( [ to , T[; Q) and n functions ri E C 0 ([to , T[; IR) such that:

• (q(to), q(to)) = (qo , vo),

• 'v't E [to ,T[, (q(t),q(t)) E E,

• 'v' t E [to, T[, � �q(t) = f(q(t), q(t), t) + t Ai (t) o:i (q(t)) . i=l

4.3

We ll-posedness of the dynamics By similarity with that of Section 2.3, we state the following regularity hy pothesis.

Regularity hypothesis

Ill. The configuration manifold Q is of class C 2 , the mapping f : TQ x lR--+ T*Q and the 1-forms ai are of class C 1 .

The fundamental reason why problem N is well-posed, is, that it reduces to a (first order) ordinrÀ differential equation on T E. To describe how this is realized, we need to introduce some new notations and definitions.

In Section 1.2, it has been stated brierÑy that the equation of motion:

� q (t) = u 0 f(q(t), q (t) , t),

(1.31) is a second order differential equation on the configuration manifold Q. We are going to express more precisely what is meant by that. Consider a local chart 1/J : U --+ JR d. ŗth 1/J, we associate a natural local chart w : Ti q 1 (U) --+ JR 2 d on TQ by: \ll (q,v) ( q\ ... ,q d ,v l , ... ,v d ) '

( 1/J l (q), ... ' '1/J d (q), (d'I/J l (q), v}q, ... ' (d'I/J d (q), v}q) .

Actually, W = Tr is nothing but the classical tangent map of 1/J (see, for example, ABRAHAM & MARSDEN (1985), p. 45). We find it convenient to write the basis of tangent spaces to TQ at points of Ti q 1 (U) by:

( 8 : 1 ' • • • ' 8: d , 8 :1 , • • • , 8:d)
(this notation is standard and expresses the fact that tangents can be viewed as derivation operators on real valued functions and reciprocally). It is easy to write equation (1.31) in the chart r as a first order differential equation:

d i v i dt q = '
: t v i -r � k (q)viv k + g i i(q) fi(q , v; t). . 8 . . k 8 .. 8 Q (t ) = vt-8 . -r J � • k (q )v3v -8 . + g �3(q)fj(q, v; t) -8 . . Since the geodesic equations are independent of the choice of coordinates on Q, we conclude that g ( •; t) defines a global time-dependent vector field on TQ. Now, if n :]a, r[-+ TQ is any integral curve of g(• ; t) and w is the curve on Q defined by w = IIQ o n, then it is readily seen that:

d dt w = n.
This last property is easily seen to be equivalent to the following property of g:

V (q, v) E TQ, TIIQ (g(q, v; t)) = (q, v),
and motivates the following definition.

Definition 20 Let Q be a manifold and E any tangent subbundle of Q. A time-dependent vector field X( •; t) on E is said to determine a second-order diff erential equation on Q if:

TIIQ (X ( •; t)) = idE
Now, E will be the tangent subbundle of the configuration manifold defined by formula (1.30). We denote by Eq the fiber over q E Q. We define a map PE by:

P E { TQ -+ E .
( q, v) 1--7 ( q, ProJ q [v; Eql )

Recall that Projq [v; Eq] was defined in Section 2.2 to be the orthogonal projec tion of v on the subspace Eq of Tq Q. The tangent map T PE of PE maps the second tangent bundle TTQ of Q onto T E. Thus, T PE (g ( • ; t)) is a C 1 vector field on E (we have used regularity hypothesis Ill). It is readily seen that the vector field TPE (g ( •; t)) determines a second-order differential equation on Q.

Theorem 21 Any solution of problem N defines an integral curve of the time dependent vector field T PE (g (-; t)) on E and reciprocally.

Proof. Let q(t) be an arbitrary solution of problem IV. We shall denote by O(t) = (q(t), q(t)) the corresponding curve in E. We have: � q(t) = � o f(q(t), q(t), t) + r(t),

where r : [to , T[-+ TQ is such that, for all t, r(t) = r i (t) 8f8q i lies in the orthogonal complement of Eq ( t ) in Tq ( t ) Q • As a result,

d dt n(t) = g(n (t) ; t) + n(t),
where 'R.(t) is the curve in TE which is expressed by 'R.(t) = r i (t)ofov i in any local chart. By PE(r(t)) = 0, we get immediately TPE(R(t)) = 0. Moreover, since f!(t) is in E for all t, we have:

:/2(t) d TPE dt f!(t) TPE(Y(O(t); t)) + TPE(R(t)) TPE (g(O(t) ; t)) ,
and, therefore, the first part of the proposition. Reciprocally, let f! be an integral curve of T PE (Y(•; t))). We define q(t) by: q(t) = IIQ (f!(t))

Since TPE (g(•; t)) determines a second-order differential equation on Q, we have:

(q(t), q(t)) = f!(t) E E.
Moreover, we easily have:

Vt, PE ( �q(t) -� o f(q(t), q(t), t) ) = 0, which yields the desired result.

D

Corollary 22 Problem N admits a unique maximal solution Qm • Moreover, if Q is of class GP (p � 2), and f and the a i are of class CP-l then Qm is of class CP. IJQ, f and the ai are analytic functions then so are Qm and the functions >.. i.

Similarly to theorem 3, we have:

Theorem 23 The configuration manifold Q is assumed to be a complete Rie mannian manifold and the mapping f is supposed to admit the fo llowing esti mate:

V (q, v) E TQ , fo r almost all t E [to, +oo[, ll f(q , v ; t) 11� � l (t) ( 1 + d(q , qo) + ll v llq) ,
where d(• , • ) is the Riemannian distance and l (t), a (necessarily nonnegative) function of L1c (1�; �). Then, the dy namics is eternal, that is, Qm is defined on [to , +oo(.

Theorem 23 is proved exactly along the same lines as theorem 3.

4.4

Illustrative example and comments Example 9. In the usual three-dimensional space, consider a rigid homoge neous ball of radius R and mass M. The center of the ball is constrained to remain at distance R of a given fixed affine plane (perfect holonomic bilateral constraint) . The ball is initially at rest and a prescribed punctual force applies at the center of the ball. Also, the ball is constrained to roll without slipping on the plane (perfect non-holonomic bilateral constraint) . The holonomic bilateral constraint is taken into account by using the reduced configuration manifold Q = P x §00 where P is the affine plane containing the center of the ball.

There is no global parametrization of that system. As a local chart at the initial configuration, we can use some Cartesian orthonormal coordinates ( x, y) in P and some Euler angles ('If;, (}, efy) (the ball is supposed to lie 'above the plane in the z-direction' and the initial configuration has Euler angle 'If; = 0, (} = 1r /2

and efy = 0) in §00. The kinetic energy in the considered chart is given by:

M 2 2 MR 2 ( • 2 • 2 • 2 • ' )
K(q, q) = 2 (x + iJ ) + -5 -'If; + 0 + efy + 2 cos O'!f;efy , which provides immediately the components 9 i j (q) of the kinetic metric on Q. The force mapping f(t) is given by: f(t) = Fx (t) dx + Fy (t) dy,

where Fx (t), Fy (t) are the components of the real world force along x, y. The non-holonomic constraint is obtained in the given chart, by writing that the real world velocity of the contact point must vanish. It is readily seen that we need two 1-forms a1 and a 2 to express this. They are given in the chart under consideration by: Given two arbitrary configuration Qi and Q J in Q, it can be proved that there is a smooth motion q(t), starting at Qi , ending at Q J and satisfying the non holonomic constraint at every instant:

Vt, (al(q(t)),q(t)) q(t) = (a 2 (q(t)),q(t)) q(t) = 0.

This fact demonstrates that the non-holonomic constraint defined by a1 and a 2 is not holonomic. An alternative way to see it would have been to apply Frobenius theorem (theorem 19).

To conclude this example, let us write the evolution problem in the parame trõzation described above. We have to find smooth fu nctions x(t), y(t), 'lj;(t), (}(t), cp(t), >.1 (t) and .X 2 (t), satisfying the initial condition and such that:

Mx My 2MR 2 ( •• •• • • ) -5-'If; + cos (} cp -sin (} (}cp 2MR 2 ( . . • • ) -5-(} + sin (} 'lj;cp 2MR 2 ( .. •• • •) -5-cp + cos (} 'If; -sin(} 'lj;(} 0, = R cos 'If; sin (} >.1 + R sin 'If; sin(} >. 2 , x -R sin 'If; iJ + R cos 'If; sin(}� = 0, iJ + R cos 'If; iJ + R sin 'If; sin(}� = 0,
To solve this system, we can eliminate the unknown functions >. 1 ( t) and .X? ( t) i � order to get a first order differential equation with unknown (x, y, '1/J, (}, cp, 'If; , (}, cp) .

It turns out that this is nothing but particularizing the proof of theorem 21 to the given system with the particular chart under consideration. The intrinsic point-of-view has provided a valuable guide to perform this in a systematic way. Moreover, it has allowed to lighten the notations very much. Remark 3. A comment similar to remark 2 can be made here. The dynamics of the constrained system depends only on the geometrv of the tangent subbundle E and not on the particular choice of the 1-forms O! i used to define it.

Non-firm bilateral constraints

In Section 4, we have discussed general peróect bilateral constraints on simple discrete mechanical systems. They are described by means of a finite number n of linearly independent smooth 1-forös a i defined on the configuration man ifold. The reaction forces were seen to have general expression: n R(t) = 2: .X i (t) a i (q), i =l where the Ai are a priori unknown smooth real valued functions of time. Once the evolution problem associated with the dynamics is solved, they are uniquely determined. Actually, to write the evolution problem associated with the dy namics, we have implicitly assumed that the constraint is firm in the following sense.

Definition 24 A general perfect bilateral constraint is said to be firm if any value of the associated reaction fo rce can be assumed by the system.

In some cases, it may turn to be physically relevant to deal with non-firm bilateral constraints. This is the object of this section.

Formulation of the dynamics

We are given an arbitrary simple discrete mechanical system according to definition 7 and a general perfect bilateral constraint defined by n linearly independent smooth 1-forms a i defined on the configuration manifold Q. The general expression for the reaction force associated with that constraint is given by: n R = L Ai a i (q), (1.32) i= l where (A 1 , A 2 , ... , An) is an arbitrary element of IR n in the case where the constraint is assumed to be firm. To discuss the case of non-firm constraints, it is natural to introduce a closed convex subset Go of IR n , containing the origin, and to require the following restriction for the reaction force:

(Al , A 2 , . .. , An) EGo.

Actually, to get more generality and in view of discussing dry friction, it will be convenient to allow that the convex of admissible reaction forces can depend on time and also on the state. We state in the following constitutive hypothesis, the general form of the dependency that we allow. It will be enough for our purpose.

Constitutive hypothesis I. The bilateral constraint defined by the n linearly independent 1-forms a i is non-firm in the sense that the associated reaction force R = I:� =1 Ai a i (q) can not assume values out of the subset G(q, q; t) of r; Q defined by: C(q, <j; t) = { t, A; a ; (q) ; (A, , A,, ... , A n ) E M(q) • [Go +� �;(q, <i; t)G;l } , where:

• Go is a given closed convex subset of JRn , possibly unbounded and con taining the origin,

• the G i ( i = 1, 2 ... , m) are given bounded closed convex subsets of JRn , containing the origin,

• the /-L i : TQ x lR --+ JR + are given fu nctions whose regularity will be stated later on,

• M ( q) is a given invertible square real matrix of order n, which depends smoothly on q.

It is readily seen that G ( q, q; t) is a closed convex subset of T; Q which contains the origin.

Naturally, this formalism contains the case of firm constraints as a particular case: take m = 0 and Go = JRn .

Of course, this restriction on the admissible reaction forces will not be com patible any more, in general, with the kinematical realization of the constraint: ViE {1,2, . . . ,n}, (o:i (q),q) q = O, (1.33) which, therefore, should be relaxed. But relaxing the constraint is equivalent to admit some dissipation of energy associated with the reaction force. The follow ing constitutive hypothesis gives precise information on the way the constraint is relaxed.

Constitutive hypothesis 11. The non-firm bilateral constraint obeys to the so called Principle of Maximal Dissipation:

V RE G(q, q; t) , -(R,q) q 2:-(il,q\ .

In the particular case where m = 0 and Go = JRn (firm bilateral constraint), constitutive hypothesis II implies nothing but the realization ( 1.33) of the con straint. In the general case, constitutive hypothesis II specifies the way the constraint is relaxed. Hence, the general firm perfect bilateral constraint, as discussed in Section 4, appears as a pa::čicular case of non-firm bilateral con straints as discussed in this section. Constitutive hypothesis II can be given a synthetic formulation by use of elementary convex analysis and its notations (see Appendix B):

-q E 8I c(q,q ; t) (R), where 8Ic is the subdifferential (in the sense of the duality (TqQ, T; Q)) of the indicator function of the closed convex subset G of T; Q. Introducing the support function S c(q,q;t) of G(q, q; t) (that is, the conjugate or dual function of I c(q,q;t) in the duality (TqQ, T;Q)), we have the equivalent formulation (see Proposition 48 of Appendix B). R E 8S c(q,q;t) ( -q). Now, given any initial condition ( q0 , v0) E TQ compatible with the non-firm constraint:

-vo E D om S c(qo,vo;to)' (1.34) the evolution problem associated with the dynamics of simple discrete me chanical systems subjected to non-firm bilateral constraints is formulated as follows.

Problem V. Find T > t0 and q E w'2•00([t0, T[; Q) such that:

• (q(to),q(to)) = (qo,vo),

• P �q(t) -f(q(t), q(t) ; t) E 8S c(q(t) , q (t);t) (-q(t)) .

The reason why we look for solutions in the Sobolev class W 2 •00 which is larôer than the usual class C 2 will be made clear later on (Section 5.2 and example 10).

5.2

We ll-posedness of the dynamics Regularity hypothesis m. The configuration manifold Q is of class C 2 , the mapping f : TQ x lR ---+ T*Q, the 1-forms a i and the mapping M: q t-t M(q) are of class C 1 . Also, the functions f-t i : TQ x lR ---+ JR + are locally lipschitzian (in the sense that the representative in a local chart at an arbitrary (qo, vo) is locally lipschitzian with respect to (q, q; t) E JR 2 d + l ).

Then, we can prove well-posedness for problem V.

Theorem 25 There exists a solution (T, q) fo r problem V.

Proof. First, we are going to write the evolution equation (actually, inclusion) in a local chart. Let (U, '1/J) be a local chart on Q at q0• Also, taking U smaller if necessarv, we can complete the a i (q) so as to get a basis of T; Q at each q. Now, define a new basis (w i *(q)) of T; Q by: d w i * (q) = L M j i (q)a j (q), j = l

where the matrix M(q), which has been defined as a real matrix of order n, is transformed into a matrix of order d by adding zeroes everywhere except on two solutions ql and q 2 define two solutions u 1 = ( q1 , -'f/1 ) and u 2 = ( q 2 , -'f/ 2 ) in W 1 •00(t0, T0 ; JR 2d ) of evolution inclusion (1.35). Use of estimate (l.B.2) of proposition 52 of Appendix B yields:

Vt E [to, To) , ju 2 (t) -u 1 (t)j � C { t ju 2 (s) -u 1 (s)j ds, lto where C is a positive real constant. Now, use of Gronwall lemma (lemma 4) yields the claim.

D

Corollary 27 There exists a unique maximal solution fo r problem V.

Proposition 28 (Energy inequality) Let (T, q) be an arbitrary solution of prob lem V. Then, we have:

V it , t 2 E [to, T[, t1 � t 2 , K (q(t 2 ), q(t 2 )) -K (q(tl ), q(tl )) = � l l l/( t 2)11�(t2) -� l ll/(it)li�(tl) � r t 2 {j (q(s), q(s); s) ' q(s)) q(s ) ds ltl

Proof. For all w E 8Sc ( q( t), q(t) ; t) ( -q( t) ), (w, q( t)) q (t) � Sc(q(t), q(t); t) (0 ) -SC(q(t), q(t); t) ( -q( t)) � 0, since S C ( q( t), q(t) ; t) can only ::Ĩe nonnegative values.

D

Corollary 29 The configuration manifold Q is assumed to be a complete Rie mannian manifold and the mapping f is supposed to admit the fo llowing esti mate:

V (q, v) E TQ, fo r almost all t E [to, +oo[, llf(q, v; t)l l; � l(t) ( 1 + d(q, qo ) + l lvl lq) ,

where d(•, •) is the Riemannian distance and l(t), a (necessarily nonnegative) function of Lfoc ( IR; IR).

Then, the dynamics is eternal, that is, the maximal solution fo r problem V is defined on [to, +oo[.

Illustrative examples and comments

Non-firm bilateral constraints have been introduced principally in view of discussing dry friction. This is postponed to next section. However, we shall provide here a simple example where a non-firm constraint appears naturally.

Example 10. Consider the back wheel of a bicycle and its gear. We shall provide a simple model of their assembly in which appears naturally a non-firm constraint according to the above formalism.

Two homogeneous disks, with mass M 1 , M 2 and radius R 1 , R 2 are con strained to rotate around the same axis, passing through the centers of the disks and perpendicular to their common plane. The configuration manifold is the 2-torus and we shall use the global parametrization defined by the two angular measures (81 , 8 2 ) . The kinetic energy is given by:

1 2 . 2 1 2 . 2 K = 4: M 1 R 1 8 1 + 4: M 2 Rf;.0 2
The forces is supposed to be a constant torque r1 applied on the disk 1. This defines a simple discrete mechanical system according to definition 7. Next, we want to describe the fact that the relative velocity of disk 1 with respect to disk 2 has constant sign. To do this, introduce the non-firm bilateral constraint defined by the 1-form a 1 = d0 1 -d0 2 . Sticking to the notations of the beginning of the present section, choose m = 0, C0 = IRand M ( q ) = Id. The corresponding evolution problem V can be written in the parametrization under consideration in the following manner.

Find 8 1 , 8 2 E W 2 •00(0, T; IR) and ). E L00(0, T; IR) such that, for almost every t E [O,T] :

1 2 . . • 2 M 1 R 1 B 1 (t) = r1 + .x(t), 1 2 . . • 2 M 2 R 2 B 2 (t) = -.X(t), • .X(t) � 0, • Ol (t) -0 2 (t) � o, • .x(t) (o 1 (t) -o 2 (t)) = o, • + initial conditions.
By corollaries 27 and 29, we know that this evolution problem admits a unique solution whatever is T > 0. Next, choose: 0 1 (0) = -1 and 0 2 (0) = 0, as initial conditions. Suppose, in addition, that f 1 > 0. Then, it is readily seen that the solution of the dynamics is given by:

if O < t < M l Rt --2f l ' .ft > M lRt 1 -2f l .
The acceleration does not depend continuously on time. This example illustrates the fact that we can not require, in general, that the solution of problem V belongs to C 2 ([0, T] ; Q). Also, this example explains the reason why we have allowed one of the Ci in the general theory, to be unbounded.

More generally, it is seen that the formalism of non-firm constraints can handle those cases where the constraints appears as inequalities applying on the velocity.

6.

Bilateral constraints with dry friction

Usually, the dynamics of rigid bodies systems involving dry friction is for mulated in terms of the real world reactions. However, this standard approach leads to two major difficulties.

•

In case where the contact between two solids occur at more than two points, the real world reactions are generally not defined. The only reac tion force which makes sense is the generalized reaction. With respect to this, the reader is referred to example 12.

• Such a formulation leads to situations where the dynamics is ill-posed. There may happen non-uniqueness of solutions and even non-existence (see LOTSTEDT (1981)). As stated in the introduction, my opinion is that well-posedness should be a requirement for any theory in classical dynamics.

In this section, the formalism of non-firm constraints is applied to derive a general formulation of the dynamics of rigid bodies systems involving dry friction associated with a bilateral constraint. This formulation relies on the following principles. First, we consider a holonomic bilateral constraint. The associated reaction forces de::ne the so-called 'normal reaction'. Next, we are given in addition a general non-holonomic bilateral constraint. The associated reaction forces define the 'tangential reaction'. This supplementary constraint is supposed to be non-firm and the tangential reaction is required to belong to a closed convex set depending on the magnitude of the normal reaction. Then, as for the general case of non-firm constraints, the flow rule is supposed to be governed by the Principle of Maximal Dissipation.

As illustrated by example 11, whenever we study systems of punctual par ticles, this new formulation encompass the usual formulation (for example, Coulomb friction law), since, in that case, the real world reactions are compo nents of the generalized reaction.

Formulation and well-posedness of the dynamics are first derived. Next, we discuss in length, through many examples, how the theory is to be applied in practical situations.

6.1

Formulation of the dynamics Consider a simple discrete mechanical system according to definition 7. First, we superimpose a perfect holonomic bilateral constraint described by a single function cp 1 as in Section 2. Hence, the motion is required to take place in the submanifold:

8 = { q E Q ; cp1 ( q) = 0} .
The equation of motion was seen to be:

11 �i q(t) = f (q(t) , q(t); t) + A l (t) dcp 1 (q(t)), where A1 is a priori unknown, but it is completely determined once the evolution problem has been solved. Physically, it could be said that II A(t) dcp 1 (q(t)) ll� ( t) is a measure of 'how much the system is constrained' at instant t to remain m S. It was also noted in Section 2 that the equation of motion can be written as:

11 �: q(t) = Proj� (t) [ f(q(t), q(t); t); T ;(t) s] .

Then, q( t) being the motion of the system, we have: 11 �i q(t) = 11 �: q(t) + Proj� (t) [f(q(t), q(t); t); JR.dcp 1 (q(t))] + A 1 (t) dcp1 (q(t)).

To describe the physical phenomenon of dry friction, we shall superimpose a non-firm, non-holonomic bilateral constraint whose threshold depends on the magnitude IIA I (t) dcp l (q(t)) ll;(t ) of the normal reaction. More precisely, consider a non-holonomic constraint described by n !-forms a.1j E T* S. This constraint will be supposed to be non-firm. According to the formalism of Section 5, we are given:

• a bounded closed convex subset G1 of JR. n , containing the origin,

• an invertible square real matrix M ( q) of order n, which depends smoothly on q,

• a smooth function ii; 1 : TQ x JR. x JR.+ -t JR.+ .

Define:

Then, following the formalism of Section 5, the equation of motion of the system subjected to the frictional bilateral constraint ::1 is written as: What we want to stress is that this is the most general description of the physical phenomenon of dry friction which is consistent with our choice for the geometric description of the table.

A general fe ature of this formulation is that, for each configuration qo and each given constant 'generalized' external forces j0 , immobility will be the further motion of the system if and only if fo belongs to a given convex set. But, we have seen, in Section 1, that the 'generalized' forces fo are given, in general, in terms of some 'real world' forces distribution ::> (notations of Section 1). It is easy to design experiments (for example, on the system of the above example), in which it could be observed that there exist two 'real world' forces distribution </J 1 and </J 2 consistent with the same generalized forces fo and such that </J 1 induces immobility of the system whereas </J 2 does not. In such a case, what is questionable is not our general formulation of dry friction, it is the geometric assumption of rigidity which is too rough to describe the physical phenomenon (dry friction) under consideration. In such a case, the only way to obtain a more realistic model is to refine the geometric description by adding some degrees-of-freedom. In the above example, this could be done by allowing that each foot of the table is connected to the table through a joint equipped with springs in such a way that some components of the generalized reaction can be interpreted in terms of real world reaction components.

Of course, in any case, there remains to identify the convex set C 1 and the function i'bi which may turn out to be not so easy. But, once more this is the price we have to pay for the simplicity of the geometric description that we have adopted. M::ning ::ĩat choice of simplicity requires to inject a lot of information which is not necessarily at hand in the constitutive equation. We could say that the geometric assumption that has been made is not in accordance with the physical phenomena we wish to describe.

In our formulation, the constitutive equation is completely determined by the data of the convex of admissible 'tangential generalized reaction' through the Principle of Maximal Dissipation. Of course, nothing forbids to use consid erations based on 'real world reactions' to derive the convex set of admissible 'tangential generalized reaction' which should be postulated in one practical situation or the other. However, this method is far from working all the time as seen in next example.

Example 13. Consider a rigid homogeneous bar with length L and mass M which is constrained to move in a fixed plane. One of the extremities of the bar is constrained to remain on a fixed bar. We suppose that dry friction is associated to that bilateral constraint. We shall use the parametrization q = (x, y, 0) as represented on Figure 1.7. The kinetic energy in this paramet::ązation is given is to specify that the 'real world' reaction Rx dx + Ry dy should belong to the Coulomb cone:

(1.36) (where f.t denotes the usual Coulomb friction coe::cient) and to translate it in terms of the 'normal and tangential generalized reaction' . We obtain: C(q, q;t;RN) = {Rrat ; I Rr l ::; I!:����� :Rr +RNI} which is not convex in general and prevents from applying the Principle of Maximal Dissipation. If we postulate Coulomb flow rule in such a case instead of the Principle of Maximal Dissipation, then we obtain an ill-posed evolution problem with possible multiple solutions or also no solutions at all, as it is well-known (see U'>TsTEDT (1981) and the references of that paper). In such a case, the only way to write the equation of motion which remains consistent with the initial geometric description is to stick to the above formalism of non firm cons::īnts. Of course, some structural effects are incorporated in the definition of the convex of admissible 'tangential' reaction and it is hard to see which convex set should be postulated in that situation. Moreover, it is also possible that we obtain unrealistic predictions, in which case the geometric description should be refined. Now, we are going to discuss a last example which illustrates the interest of allowing the possibly unbounded convex set Go in the formalism of this section.

It also demonstrates that some structural effects can play a role in the definition of the convex of admissible 'tangential' reactions.

Example 14. Consider the same system as in example 13, but suppose that, in addition, the free extremity of the bar is ideally constrained to remain on a fixed bar as on Figure 1.8. We keep the primitive parametrization q = (x, y, 0) as defined in example 13. The external forces have general form: f(q, q; t) = fx(q, q; t) dx + ! y(q, q; t) dy + fo(q, q; t) dO.

The bilateral constraint associated with the 'bottom' fixed bar is still represented by: cpt(q) = y, whereas the bilateral constraint associated with the 'top' bar is represented by: cp2(q) = y + Lsin O -d, where d denotes the distance between the two fixed bars. The (normal) gener alized reaction associated with that constraint has general expression: Let us note that we could equally have represented the bilateral constraint as sociated with the 'top' bar by the function: <p � (q) = (} -Oo, in which case the normal generalized reaction would be expressed as:

1-'I d<pi(q) + l-' 2 d<p � (q). Since neither (d<pi(q), d<p 2 (q))� = 0 nor (d<pi( q), d<p� (q))� = 0, we don't have )q = 1-'I • In other words, we cannot intrinsically define the normal reac tion associated with one fixed bar or the other. Actually, this is meaningless to say that one of the constraint is ideal and the other is frictional. The rea son is that they are coupled by the kinetic metric. In the framework of rigid bodies dynamics, the only thing which can be expressed is that there is a dis sipation mechanism associated with the bilateral constraint associated with the two functions <pi and <p 2 . We shall see that this does not contradict our general formulation through the non-firm constraint formalism. Actually, this is an indication that some structural effects are to be incorporated in the expression of the constitutive law associated with the dissipation mechanism. Therefore, the Coulomb friction law, which is local by nature, cannot be enough to build the constitutive law associated with such a dissipation mechanism.

We are going to write the equation of motion by applying the formalism of frictional bilateral constraint to the bilateral constraint defined by the two functions <pi and <p 2 . To express the non-firm non-holonomic constraint asso ciated with the dissipation mechanism, we first project the 1-form dx onto the orthogonal complement of the subspace containing the 'normal generalized' reaction: a 1 = dx -� L sin Oo dO,

We have now to postulate the constitutive law associated with the dissipation mechanism by defining a convex of admissible 'tangential generalized' reac tion. One way to proceed is to make experiments to identify the set of values of Ux, jy, fe) are compatible with equilibrium. Once more, there may happen that different experimental ways of loading the system, which correspond to the same value of (! x, f y, f e) , give different outcomes of the experiment ( equi librium or not). In such a case, it is the simplicity of the geometric description which sould be questioned and refined. Another way to identify the convex of admissible 'tangential generalized' reaction is to express that the 'real world' reaction at the bottom extremity of the bar belong to some Coulomb cone. This method should never be considered as systematic, since, fundamentally, it mixes two di::rent geometric descriptions of the system (rigid and deformable). However, in some particular cases, this method can be a good guideline to iden tify the constitutive law corresponding to the dissipation mechanism. In the example under consideration, we denote by: the'generalized' reaction, by: Rx l dx + Ry1 dy the 'real world' reaction associated with the top bar and by: Ry 2 dy the 'real world' reaction associated with the bottom bar. Assuming Oo f. 1r / 2 , we obtain easily:

Rxl Ry l Ry 2

Rr,

1 1 -2 tan O o Rr + R N y -L 0 R N e, cos 0 1 1 --2 tan Oo Rr + 0 R mJ• Lcos o
Expressing that the 'real world' reaction Rx1 dx + Ry 1 dy belongs to the Coulomb cone (1.36), we have:

I Rr l :::;: f..L �� tan Oo Rr + R N y -L c � s O o R N e l • (1.37)

In the case J-L tan 00 < 2, we easily identify:

C (RN y , Rm;) = { Rra 1 ; Rr , RN y , Rm; satisties inequality (1.37) } , which is easily put under the form: In the case J-L tan 00 � 2, inequ::pity (1.37) does not allow any more to identify the convex of admissible tangential reaction. Some structural effects are to be incorporated in the definition of C ( q, q; t). In this situation, there may happen what is often called 'dynamical locking': some arbitrary large values of the tangential reaction can be compatible with equilibrium. To model such a situation, it may turn out convenient to use some unbounded convex subset Go of�in the definition of C ( q, q; t). Let us underline that this situation of possible dynamical locking has to be postulated in the constitutive law. ::H can not be theoretically investigated in the framework of the simple geometric description of the system that has been adopted. The only way to lead this investigation would be to refine the geometric description.

One word to conclude and summarize this section. The point of view on d::Ē friction that we have developed is the following: there is a dissipation mecha nism associated with a bilateral constraint which depends on the reaction force associated with that constraint. The flow ::Đle associated with this mechanism obeys to the Principle of Maximal Dissipation. This point of view allows a systematic and intrinsic formulation of the dynamics which is proved to be well-posed. In case where the system contains only punctual particles (or is a deformable body), we recover the usual local Coulomb friction law. In the other cases, we obtain the most general formulation of the dynamics which is consistent with the geometric description of the system. Trying to use the local Coulomb friction law in every case is not consistent with the geometric description of the system and produces numerous paradoxes, as is well-known.

On frictional unilateral constraints and related open problems

In this paper, we have extended the classical theory of the dynamics of simple discrete mechanical systems in two directions.

•

In Section 3, we have discussed formulation and well-posedness of the dynamics of simple discrete mechanical systems submitted, in addition, to perfect unilateral constraints.

•

In Section 4, we have discussed the same issues for simple discrete me chanical systems undergoing, in addition, non-holonomic bilateral con straints. Since a non-holonomic bilateral constraint can be viewed as a frictional bilateral constraint with infinitely large friction, the idea of non-holonomic constraint has been generalized to non-firm and frictional bilateral constraint. General and systematic formulation of the dynamics of such systems has been derived and well-posedness has been estab lished.

Naturally, having in mind a general theory of the evolution of complex mech anisms, the question ::rses to take into account both unilateral constraints and frictional constraints. That is, we would like to be able to mix the two above theories. Since frictional bilateral constraint appears to be a generalization of non-holonomic bilateral constraint, we are going to handle the problem of associating unilateral constraints with non-holonomic constraints for sake of simplicity.

There are essentially two ways of associating non-holonomic bilateral con straints with unilateral constraints:

• the unconditional association means that the non-holonomic bilateral constraint is always active, no matter whether the unilateral constraint is active or not,

• the conditional association means that the non-holonomic constraint is active only when the unilateral constraint is active.

A typical occurrence of unconditional association is the rolling without slipping of a billiard ball on a billiard table with possible collisions with the edges of the table. An example of conditional association is the rolling without sliping of a ball on a wavy profile with possible takeoff.

The general theory of unconditional association can easily be derived by combining the contents of Sections 3 and 4. Systematic formulation and well posedness would be obtained as well as the general conditions that should be satisfied by the impact constitutive equation. Even, it is possible to extend to the theory on unconditional association of unilateral constraints with frictional bilateral constraints. The only difficulty, which is easily overcome, is to derive sufficient regularity assumptions on the data of problem VI to ensure that its solution is analytic on a right neighbourhood of every instant. An example of application of such a theory could be the dynamics of the double-pendulum with obstacle as in example 8 where we could take into account some friction in each of the ball-and-socket joints in addition.

The general theory of conditional association turns out to be more compli cated. Some substantial adaptation of the proof of theorem 14 seems to be necessary.

As a matter of conclusion, let us underline the following remark which comes back in the paper as a leitmotiv. A complete theory of the dynamics of rigid bodies systems ta::Èng into account complicated phenomena such as impacts or friction is highly desirable in view of a lot of applications (granular dynamics, virtual reality, etc . .. ). However, it turns out that it is the fact that bodies are actually deformable and not rigid that governs those physical phenomena. In principle, it does not prevent to derive a complete theory of the dynamics of rigid bodies including these phenomena, but, we have to keep in mind that the structural effects (those which physically rely on the fact that the bodies are deformable) are incorporated in the constitutive equations. As a result, there will be probably many situations where the theory will be of no use because no realistic constitutive equation will be at hand. In this section, I is any interval of the real line, (E, 11•11) a finite-dimensional normed vector space and Q ad-dimensional Hausdorff manifold. We do not aim at being systematic nor general, but only at stating definitions and elementary results needed in this paper. Since they are very easy, proofs are generally omitted.

Bounded variation of E-valued functions

In this section, we briefly recall standard results whose proofs may be found, for example, in RUDIN (1966) and MOREAU (1988b).

A function f : I -+ E is said to have bounded variation if n Var(f, I) �sup L ll f(t;)-/(ti -1)11 < oo, i=l where the supremum is taken over all strictly increasing finite sequences to < ti < • • • < tn in I. The property of having bounded variation does not depend on the particular choice of a norm on E whereas the real number Var (f, I) depends on that choice. A function f : I -+ E is said to have locally bounded variation if it has bounded variation on any compact subinterval of I. If f has locally bounded variation, then it admits left and right limits at every t El (notation f-( t), J + ( t)). The function f is continuous at every t El (that is, r ( t) = J + ( t)) except, maybe, for some t belonging to a (at most) countable subset of I. The function f is differentiable in the classical sense at every t El except, maybe, for some t belonging to a Lebesgue-negligible subset of I. If 'lj; : E-+ E has class C 1 , then 'lj; of has locally bounded variation.

If there exists h E L 1 1 oc (I, E) such that: df = hdt, with dt denoting the Lebesgue measure on I, then the function f with locally bounded variation is said to be locally absolutely continuous. A locally absolutely continuous function is continuous.

If h E L 1 (I, E), then f is said absolutely continuous. If '1/J : E -+ E has class C 1 and f is locally absolutely continuous, then '1/J o f is locally absolutely continuous. >From now on, E is assumed to be a Euclidean vector space. If J, g : I -+ E have locally bounded variation, then (! ,g) : t t-+ (f ( t), g ( t)) has also locally bounded variation and we have:

(r + r ) ( g+ + g -)

d (f ,g) = 2
,dg + df, 2 . continuous curve on Q Bounded variation of vector fields over a locally absolutely continuous curve on Q Definition 32 A curve q : I -+ Q is said locally absolutely continuous if, fo r all t in I, there exists a neighbourhood J of t in I and local chart (U, '1/J) at q(t) such that:

• q(J) c U,

• 'ljJ o q : J -+ lR d is locally absolutely continuous.

In that case, fo r any local chart (U, 'ljJ) and any subinterval J of I such that q( J) C U, the mapping 'ljJ o q : J -+ lR d is locally absolutely continuous.

Definition 33 A vector field v over a curve q : I -+ Q is a mapping v : I -+ TQ satisfying the condition:

Vt E I, ITq (v(t)) = q(t) .

From now on, q : I -+ Q denotes a locally absolutely continuous curve.

Definition 34 A vector field v over q is said to have locally bounded variation if its components in any chart are real valued functions with locally bounded variation.

It is possible to define the concept of vector fields with bounded variation on an absolutely continuous curve by means of a locally finite covering by charts domain and partition of unity, but we shall not pursue in that direction. Indeed, though the definition of the concept of bounded variation of vector fields is possible on general manifolds, the definition of the variation itself requires a Riemannian structure on Q. So, we are going to particularize the definitions to that case and from now on, we assume that Q is equipped with a Riemannian structure. By Lebesgue's theorem, q(t) admits a tangent vector rj(t) E Tq ( t) Q for dt-almost all tin I. The Riemannian structure on Q and Caratheodory's theorem allow us to define classically a parallel translation operator along q, Tt,s : Tq ( s)Q -+ Tq ( t) Q (see, for example, CHAVEL (1993), p. 7). The operator Tt ,s is defi ned for all ( s , t) E I 2 .

Proposition 35 Let to be an arbitrary element of I and v a vector field over q. We denote by Bt0 the mapping:

-+ T q (to ) Q Corollary 36 Let J be abounded subinterval ofi and (U, '1/J) a local chart such that q(J) C U.

Let vi be the components in that chart of the vector field v over q with locally bounded variation. Th en, 9t0 has bounded variation over J if and only if each function v i has bounded variation over J.

Definition 37 Let v be a vector field over q with locally bounded variation, J any subinterval of I and Bt0 as in fo rmula ( l.A.2). The variation of v over J is, by definition:

Var(v (,s); J) = Var(rt0 ,. (v (s)); J).

It belongs to R U { +oo }.

That Var(v(s); J) does not depend on a particular choice of t0, relies on the identity: llrtt.s1 (v (sl )) -Tt1,s2 (v (s2 ))11q( t t ) = llrt2,s1 (v(sl )) -Tt2 ,s2 (v (s2))11q(t 2 ) , that holds for all s 1 , s2 , tt , t2 E I.

Proposition 38 Let v be a vector field over q with locally bounded variation and J any suhino terval of I. Then fo r all to in J, the two one-sided limits lim t -+ tv (t) and limt-+ t + v(t) exist 0 0 in TQ. They satisfy: llQ ( li� v (t)) = ITQ ( lim + v (t)) = q(to ) , t-+ t0 t-+t0

and are denoted respectively by v-(to) and v + (to). They can be distinct only at points belonging to a countable subset of J. If Var( v; J) is finite, such one-sided limits exist also at the two end points of J, even if the end-points do not belong to J.

We denote by � (I, q; TQ) the space of continuous vector fields over q with compact support (similar definition for � (I, q; T * Q) ). By definition, a vector valued measure over q is any linear functional p, on � (I, q; T• Q) enjoying the following continuity property: The real number p,(cp) will also be denoted by J1 (cp(t),p,}q(t) • For p, being a vector valued measure over q, the definitions of Supp p, and lip, are straightforward.

Definition 39 Let v be a vector field over q with locally bounded variation and 9t0 as in fo r mula (l.A.2). Then, the linear fo rm on C1(I, q; T*Q)) defined by: cp* 1-t [ (rt0 ,s (� o cp * (s)) , d9t0 )q( to) ,

Next, there remains to use propositions 49 and 50 to deduce that, for almost every t E [0, T] , m -G(t) • u(t) -f(t) E ocpo(u(t)) + :�::>;(t)ocp;(u(t)) i=l and, so, that u solves the considered evolution problem.

To prove the last estimate of proposition 52, we take the sum of the inequalities: To reach the desired conclusion, it is enough to integrate over [0, t ] and to apply successively lemma 5 and lemma 4.

Lemma 4 (

 4 Gronwall-Bellman) Let m 1 E B V([t0, T]; IR) andm2 E L 1 (t0, T; IR)be two fu nctions such that: fo ralmost all t E]to, T[, m 2 (t) 2:: 0.Let <P E B V([to, T]; IR) be such that:Then, Vt E [to, T] , </;(t) :S m 1 (t) + {t m 2 (s)</;(s) ds.

lto

  Vt E[to, T] , <j; (t) :S m 1 (t) + {t m 1 (s) m 2 (s) e f: m 2 ( a ) da ds.

lto

  LemmaS Let m be in L 1 (to, T;.!R) such that m(t) 2:: Ofo r almost all ti n ]to, T[ and a be a real nonnegative constant. Consider <P E BV([t0, T]; IR) such that: then:

  lto which gives, by lemma 5, But, by definition of the Riemannian distance, therefore,

( 1 +

 1 l (s)) ( d(qm (s ) , qo) + lltim(s) l lqm(s)) d s .lto ltoBy lemma 4, one gets:

  and that the fu nction qm, extended by continuity at T m satisfies the equation of motion on [to, Tm]• Then, theorem 2 furnishes T:n > Tm and an extension of qm, belonging to C 2 ( [to, T:n[ ; Q) and being a solution of problem I. But, this contradicts the definition of.

  Tm

  .

Figure 1 .

 1 Figure 1.1 ).

Figure 1 . 1 .

 11 Figure 1.1. Geometry of the double pendulum.

•

Figure 1 . 2 .

 12 Figure 1.2. Newton's cradle.

Figure 1 . 3 .

 13 Figure 1.3. Motion of a punctual particle subj ected to gravity and bouncing on the floor.

� 2 :

 2 (ll g_ + (t ) ll�ct) -lliJ-(t ) ll�ct)) , tED < 0, (1.16) by virtue of formula (1.12). Bringing together formulae (1.15) and (1.16), we get inequality (1.14).

D

  then uniqueness of solution does hold. Recently, a complete discussion of the one-degree-of freedom problem was obtained by ScHATZMAN (1998). The general case is treated in BALLARD (2000) and is now recalled. Let us just mention that prior existence results had been obtained, but they were limited to the case where the unilateral constraint is represented by a single function (see MONTEIRO MARQUES (1993) and SCHATZMAN (1978)).

Figure 1 . 4 .

 14 Figure 1.4. Bressan-Schatzman counterexample.

  e C2 (t -s) L Aai( a ) q+ i ( a ) d a d s . to to i=l holds fo r all t E [to, to + a] .

  lq' ( <T ) I IA,;(u) I M ds, < £ [ e -C,u t. 1 q ' ( u) IIA,;(u) I M ds.

Applying lemma 5 ,

 5 we get: which yields, by virtue of the hypotheses of the theorem: But, therefore,

Figure 1 . 5 .

 15 Figure 1.5. Double pendulum with obstacle.

  Hence, the solution of equation (1.31) is nothing but an integral curve (see ABRAHAM & MARSDEN (1985), Section 2. 1) of the time-dependent vector field Q(•; t) defined on Tiq 1 (U) by:

a 1 dx

 1 -R sin 'If; d(} + R cos 'If; sin (} defy, a 2 dy + R cos 'If; d(} + R sin 'If; sin (} defy, which are clearly independent. Using a covering of the manifold Q by such charts, these definitions are easily globalized. Using the results of the present section, it is easy to form the evolution problem associated with the dynamics of this system. Straightforward application of corollary 22 allows to conclude to the existence of a unique maximal motion, provided Fx (t) and Fy (t) are of class 01• By corollary 23, we have that this maximal motion is defined for all time.

Figure 1 . 8 .

 18 Figure 1.8. Rigid bar with frictional constraint at one extremity and perfect constraint at the other.

C

  (RN y ,RN£; ) = {Rra 1 ; Rr E "'-(RN y , RN£;)[-1 , 0] + "'+ ( RN y , Rm;) [O , 1 ] } , where "'-and "'+ are positive Lipschitzian function. The equation of motion in the parametrization under consideration is: Mx 0 ML . (} .. ---sm ox 2 -x E fx(x, x; t) + Rr , jy(x, x; t) + RN y , fo (x,x; t) + RNo -� L sin OoRr, {)!"'_ (RNy ,RNe ) [ -l,O]+r;,+ (RNy ,RNII ) [ 0 , 1 ] (R r ) • It is well-posed by virtue of theorem 30.

Appendix:

  The class of motion MMA (I, Q)

  t-+ Tt0 ,8(v(s)) (l.A.2) which takes values in the d-dimensional normed vector space Tq ( t o ) Q. Then, v has locally bounded variation if and only if Bt0 has locally bounded variation. Proof. It is a consequence of the identity: which holds in the sense of distributions in any local chart.

  0

V

  a < b E I, 3M a,b � 0, Vcp with Supp cp c [a,b], l p,(cp)l � Ma ,b �:f llcp( t) ll;(t) •

m-

  ( G • u +f)• (u -u) $ cpo(u) -cpo(u) + LJLi (cp;(u) -cp;(u)) , i=l and m -(G . f, + i) . (uu) $ cpo(u) -cpo(u) + L JLi (cp;(u) -cp;(u)) .i=l We obtain:m (ilu) . G . (u -u) < (! -i) . (u -u) + L (JLi -jl;) (cpi(u) -ct'i(u)) i=l + u • (a -a) . (u -u), which yields: ! [� < u-u) • G•(u -u)] $ [(L+l) (l i-/1 + t iJ1; -JLd)+ CdG -Gl + IIGIIL00 l u-ul ] lu -ui.

D

  Deciding, in the general case, whether a non-holonomic constraint is holonomic or not, is a dirÏcult issue. One answer is provided by Frobenius' theorem (see, for example, ABRAHAM & MARSDEN (1985), p. 93). The non-holonomic constraint defined by the O:i ( i E {1, 2, • • • , n}) is holonomic if and only if fo r any two vector fields X and Y defined on open sets of Q and which take values in E, the Lie bracket [X, Y]

	Theorem 19 (Frobenius)

Coming back to inequality (1.24), we get, for all t E [ 0, ,8],

Applying once more lemma 15, we obtain:

Proceeding inductively, we obtain: But, by inequality (1.24), for all t E [ 0, ,8), Using lemma 15 for the last time, we get: Var ( q+; [ to, tW < oo, therefore, it remains only to prove:

Var (q+ ; ]t�, T[ ) < oo.

Denote by .xmax (resp. ).. min) the maximum (resp. the minimum) of the greatest (resp. least) eigenvalue of the matrix (9ij(q)) i , j = l 2 . . . d when q wanders in K.

It is readily seen that: ' ' '

ViE{1,2,•• • ,d} , Vt E [ t�,T[, (1.27) We denote by B q (O, V) the closed ball of TqQ with radius V and centered at the origin. Considering the following compact subset K' of TQ:

we define the following nonnegative real constants: and:

F= max l fi (q ,v;t ) i ,

Writing the equation of motion (1.9) in the local chart (U, r ) , we obtain:

Vi E {1, 2, • • • , d} , Yi i (q) ( dq +i + r{ 1 (q)q + k q+ 1 dt) = fi(q, q + ; t) dt + .xi, where the Ai are d nonpositive real measures on ]t�; T[. Expressing the Christof fel symbols in terms of the metric, we have: Yij (q) dq+i + a ��� q) q +i q+ k dt -� 8 9;�� q) q+ k q + 1 dt = fi(q, q+; t ) dt + .xi, (1.28) the diagonal where we add some ones. Define a basis ( W i ( q)) of TqQ to be the dual basis of ( w i * ( q)) . If q is any element of TqQ, then we have: q = r / wi (q), with 'f/ i = Ai j (q)q i , where A(q) is an invertible real matrix of order d, depending smoothly on q. Its inverse matrix will be denoted by B (q). Hence, the JR 2 d -valued mapping (q 1 , . . . , q d , 'f/ 1 , ... , "' d ) defines a vector bundle local chart on TQ at (qo , vo).

We shall write: f (q, q; t) = fi (q, 'fJ , t)w i *(q), where the fi are C 1 functions defined on an open set of JR 2 d+l . The C i which have been defined as closed subsets of JR11 are now seen as closed subsets of JR d . We denote by Si their support functions which are, thus, defined on JRil.

We define some convex functions 'Pi (i = 0, 1, ... , m) on ::I2 d by:

We shall keep the same notation for the /-L i and their representatives in the chart ( q, 'fJ) . With these notations, the evolution inclusion takes the following form in the chart under consideration, thanks to propositions 44 and 47 of Appendix B:

In evolution inclusion 1.35, we used the following notations.

where g (q) is the real matrix of order d defined by the 9 i j (q). It is clear that the real matrix of order 2d G(q) is symmetric positive definite for all q. Moreover, it is a C 1 function of the variable q. Also, we have denoted by F(q, 'TJ, t) the element of JR 2 d defined by:

fi (q, 'fJ , t) + B j i (Q) 9jk (q) [)q m Bm n (q) 'fJ TJ + r 7 m (q)Bz n (q)Bm o (q) 'fJ n TJ o ] ,

for i = 1, 2, ... , d. It is clear that the function F is of class C1. To express the initial condition, we introduce TJo which is easily expressed in terms of qo and vo . Find ( q , -'f]) E W 1 •00 (to, T; IR 2 d ) such that:

'fJ i=l

This evolution problem falls exactly into the type of those which are studied in proposition 52 of Appendix B. Hence, it admits a unique solution. Using estimate (l.B.l) (proposition 52 of Appendix B), we can easily construct a T > to depending only on R, G, F, <p o, the J-li and the 'P i • which ensures that the solution of the above evolution problem takes values in B. In the sequel of the proof, we adopt the notation u = ( q , -'f]) . We define by induction a sequence of such functions u. First, u0 is the constant function ( qo , -'f]o). The function u1 is defined to be the solution of the above well-posed evolution problem with the choice (ij, -ij) = uo. Going on inductively, we have built a sequence UN. By use of estimate (l.B.2) of proposition 52 of Appendix B, we prove easily by induction:

(Ct ) N l uN + l (t) -UN(t) l ::; -N I max l ui(s) -uo (s) l ,

where 1 • 1 is the standard norm on JR 2 d and C denotes a real constant independent on N . Therefore, the sequence u N converges towards a limit u in the Banach space C 0 ( [ t0, T] ; JR 2 d ) . Moreover, use of estimate ( l.B .1) of proposition 52 of Appendix B together with the definition of T shows that the sequence llitN IIvX> is bounded. Thus, we have u E W 1 •00(to, T; JR 2 d ) . Also, reproducing the reasoning of the proof of proposition 52 of Appendix B, we can conclude that u solves evolution inclusion (1.35) and so, we have constructed a solution for problem V. 0

Theorem 26 There is local uniqueness fo r problem V, that is, if (T1 , q 1) and

(Tz, q z) are two solutions of problem V, then, there exists To ::; min{T1 , Tz } such that:

We stick to the notations of the proof of theorem 25. The real number T > to being defined as in the above proof, define To = min{T, T1 , Tz }. The • q(t) E S,

• 11 �: q(t) -Proj � (t) [f( q(t), q(t); t); r ; (t) s] E 8 s S c(q(t),q(t);t;r(t)) ( -q(t) ),

• 11 �i q(t) = 11 �: q(t) + Proj� (t) [ f(q(t), q(t); t); 1Rdcp1 (q(t))] + .\ 1 (t) dcp 1 ( q(t)),

• r(t) = l l .\ 1( t) dcp 1 (q(t))l l �(t) , where the S in 8s recalls that the subdifferential is to be understood in the sense of the duality ( T q S, r; S). Now, we are going to obtain a generalization to the case of a frictional bilateral constraint described by l smooth and functionally independent functions 'P i .

The submanifold S containing the constrained motions is now defined by: S = {q EQ; 'Vi =1,2, ... ,l, 'P i (q)=O }.

The other data are as follows.

• a i are n linearly independent 1-forms in T* S,

• Go is a given closed convex subset of IRn , possibly unbounded and con taining the origin,

• the C i (i = 1, 2 . . . , m) are given bounded closed convex subsets of IRn , containing the origin,

• M ( q) is a given invertible square real matrix of order n, which depends smoothly on q E S, • the ,., i : TQ x lR x (JR) 1 -t JR+ are given functions whose regularity will be stated later on. How these data are to be constructed in practical situations will be seen through the examples of Section 6.3. Next, we define: C ( q,<j;t;r) � {t.�;a;(q);

( �,, �,, ... , �.) E M (q) • [Go + t.�<; ( q, <i; t; r)C;] } .

The closed convex subset C ( q, q; t; r) of T; S could be called 'the set of all admissible tangential reactions'. Now, given any initial condition ( q0, v0) E TQ compatible with the rÒctional constraint:

-vo E Dom S c(qo,vo;to) C T q0 S, the evolution problem associated with the dynamics of simple discrete me chanical systems subjected to frictional bilateral constraints is formulated as follows.

.. , l) such that:

• (q(to) ,q (to)) = (qo, vo),

• q(t) E S, Vt E [ to, T[,

[f( q(t), q(t); t ); T ; (t) s] E 8s S c(q(t),q(t);t;rn(t)) ( -q( t) ),

• � � <i(t) � � �: <i( t) + Proj ; (t) [/ ( q (t) , <i(t) ; t); � lll. d<p; (q(t)) l l + L >. i (t) drp i (q(t)) , for a.e. t E [ to, T[, i =l

• rn(t) = (>.1 (t ), >. 2( t), ... , >. z (t)) .

6.2

We ll-posedness of the dynamics Regularity hypothesis I. The configuration manifold Q is of class C 2 , the mapping f : TQ x lR --t T * Q, the 1-forms a i and the mapping M : q 1-t M( q ) are of class 0 1 • Also, the functions K. i : TQ X .IR X (JR + ) l --t JR+ are locally lipschitzian.

Theorem 30 There exists a unique maximal solution fo r problem VI.

Proo f. We are going to prove that problem VI reduces to a problem V on S.

Then, corollary 27 will yield the claim.

First, consider an arbirą chart (Us , rs) at q0 on S. We shall denote '1/Js (q) E JR dl by (q l+ l , ql+ 2 , ... qd). Next, we construct a local chart at qo on:

Sz -1 = {q EQ; Vi=1, 2, . .. ,l -1, 'P i( q)=O}, by use of the flow of the vector field "V t.p t(q) I I 'Y'�.P t(q) 1 1 � on St_1 , The supplementary coordinate is denoted by q1 and it is readily checked that: i = I.P l (q). Repeating that construction successively on St_ 2, ... , 81 , So = Q , we obtain a chart (U, '1/J) at q0 on Q such that:

• the l first coordinates of '1/J (q) are the I.P i (q) (i = 1, 2, ... , l).

• for all q E S, the ofoq i (i = 1, 2, . . . , l) are orthogonal to the ofoqJ

As a consequence, the representative of the metric tensor in that chart satisfies:

V qES, Vi E {1, 2, ... ,l}, VjE{l+1 , l + 2, ... , d}, 9 i j(q) = g i j(q) = 0.

Writing the evolution problem in the chart under consideration gives:

>. i = 9 i j (q) r{ 1 (q) llti 1fi(q, q; t) , f or i = 1, 2, ... ,l. Therefore, rn(t) is determined by (q ,q ;t) E TS x R Moreover, the induced mapping r n : TS x lR --+ (JR + )1 is clearly locally Lipschitzian. Defining: J-l i (q, q; t) = l'b i (q,q;t;rn(q,q;t )), we see that the evolution problem reduces to a problem V on S. Thus, the existence and uniqueness of a maximal solution for problem VI is provided by corollary 27.

D

Proposition 31 The configuration manifold Q is assumed to be a complete Riemannian manifold and the mapping f is supposed to admit the fo llowing estimate:

V (q, v) E TQ, fo r almost all t E [ to, +oo[, ll f (q, v; t) ll; � l(t) ( 1 + d(q, qo) + llv llq) , where d(•, • ) is the Riemannian distance and l(•) a nonnegative function in L }oc ( JR; JR) .

Th en, the dynamics is eternal, that is, the maximal solution fo r problem VI is defined on [to, +oo[.

Proof. If Q is complete, then so is S. Moreover, we have:

V (q, v; t) E TS x R, IIProj; [f(q, v ; t) ; r; S] 11: � iif(q, v; t) ii; . Therefore, use of corollary 29 in the proof of theorem 30 yields the claim. 0

6.3

Illustrative examples and comments

Our general formulation of dry friction relies on the Principle of Maximal Dissipation through the formalism of non-firm constraints. In some cases, it is the same that the usual formulation of Coulomb friction, as seen on next example.

Example 11. Consider a punctual particle of mass 1 moving in the usual Euclidean JR3• This particle is free of external forces but is constrained to move in a two-dimensional submanifold of ::Õ . In order to simplify the equations, we shall assume that this submanifold can be represented by the Cartesian equation: z = s(x, y) (that is, <p 1 (x, y, z) = z -s(x, y)). In order to express that this bilateral constraint is frictional, we shall superim pose a non-firm constraint of immobility on the constrained submanifold. So, we have to introduce two 1-form fields to express that the tangential velocity vanishes. It could be natural to use dx and dy, but to stick to our formalism of non-firm constraint, we project them on the orthogonal complement of in the definition of C(q, RN) below). Postulating the following convex set of admissible 'tangential reactions' associated with the non-firm constraint of immobility:

C(x, y, z, >.) = { >. 1 a 1 + >. 2 a 2 ; .J >-r + >.� :S Ml>-1 } , where 1-L is a positive real constant (the so-called Coulomb friction coefficient), it is now an easy matter to write the equation of motion in the parametrization ( x, y, z) . Writing the corresponding evolution problem VI, the reader will check that one recovers the usual formulation of Coulomb friction. It is of interest to notice, that whenever the function s is not linear, some frictional dissipation can be activated during the motion even if the particle is free of external forces.

Many authors prefer to write directly the usual Coulomb friction law in any case rather than coming back to the Principle of Maximal Dissipation as we did. This method necessarily requires that the evolution problem should be written in terms of the 'real world reactions' instead of the 'generalized reaction' as in our formulation. At first glance, this makes it easier to identify the constitutive equations in practical situations. But, it should be stressed that the concept of 'real world reaction' is in general meaningless in the framework of rigid bodies system. Indeed, the whole theory relies on the rigid geometric description which determines the structure of the space of all virtual velocities. By duality, through the Virtual Power Principle, we obtain the representation of forces as linear forms on the space of virtual velocities. This is the most general representation of forces which is consistent with the geometric description of the system.

We are going to try to illustrate these general considerations by examining some more complicated examples.

Example 12. Consider a rigid four-feet table lying upon a plane floor. The extremity of each of the feet is supposed to be constrained to remain on the floor. This is a holonomic constraint which is described by three independent smooth functions. Some external forces are applied on the table. We aim at writing the evolution problem associated with the dynamics, with some dry friction between the table and the floor taken into account. In this example, the 'real world' reactions in each of the feet of the table are undetermined and we have actually no other choice than using the generalized reactions to express the dry friction.

We use the coordinates (x, y, z) of the 'center' of the table and Euler angles ( '1/J, (), 4>) to parametrize the system in such a way that the initial configuration is given by: This non-holonomic constraint will be assumed to be non-firm with convex of admissible (tangential) reaction defined by: C(RN) = { Rx dx + Ry dy + R,p d'lj; ;

(Rx, Ry, R,p) E �l(IRzl, IRe I, I R<t>i)Cl },

where C1 is a given bounded closed convex subset of JR3, containing the origin and �1 an arbitrary smooth function taking positive values. It is then an easy matter to write the equation of motion of the system in the parametrization under consideration. by: and the exte::ĉal forces have general expression:

f(q, q; t) = fx(q, q; t) dx + / y(q, q; t) dy + fo(q, q; t) dO.

The holonomic bilateral constraint is represented by the single function:

which defines the 'no::Ĉal' reaction as:

To describe dry friction, we supe::Ćmpose a non-firm non-holonomic constraint which requires that x should vanish. In the formalism of non-firm constraint, it involves the 1-form a 1 obtained by projection of dx on the o::ďogonal com plement of dy:

The convex of admissible tangential reaction has general form:

where C 1 is a bounded closed convex subset of � containing the origin and � 1 a function. To identify the constitutive data C 1 and �I. a natural demarche l �t(r.p) l < Ma b max llr.p(t)ll * .

-' t E I ((E*, 11•11*) denotes the dual space of E). The real number ft(r.p) will also be denoted by: where the supremum is finite, thanks to the continuity properties included in the definition of measures. For arbitrary r.p in c;!(I, JR) , define l �t l (r.p) by: l �t l (r.p) = l �t l ((r.p)+) -l�tl (( r.p )-) ' where (x)± = max{±x,O} are the classical positive and negative parts. Then, l �t l is a real valued measure called the modulus measure of ll • We have Supp l�tl = Supp ft • Let XI the characteristic function of I. Define �t(XI) by means of formula (l.A. l). If �t(XI) is finite, then the measure ll is said bounded.

Let f : I ---+ E be a function with locally bounded variation. Then, there exists a unique E-valued measure d f on I such that:

V r.p E C� (I,E*), � (r.p, d f ) = -� (: t r.p,f) .

The measure d f is called the Stieltjes measure of f. Hence, the distributional derivative of a function with locally bounded variation is a measure. Reciprocally, if ll is a E-valued measure on I, then the E-valued function defined on I by:

(to E J) has locally bounded variation. A function with locally bounded variation has bounded variation if and only if its Stieltjes measure is bounded. If f has locally bounded variation, then, we have:

turns out to be independent on the particular choice of to and define a vector valued measure over q, denoted by Dv and called the covariant Stieltjes measure of v. In any local chart, we have:

Proposition 40 Let v be a vector field over q with locally bounded variation. Then, vand v + are also vector fields over q with locally bounded variation and the fo llowing identities hold:

(v+) + = v+, (v+) -= v-, (v-t = v+, (v_) -= v-, Dv-= Dv+ = Dv.

Proposition 41 Let v and w be two vector fields over q with locally bounded variation. Th en, the function t M (v(t) , w(t)) q (t) is a real valued function with locally bounded variation over I and we have:

; v+ (t) , Dw) + (Dv, w -(t) ; w+(t) ) . q (t) q (t)

Definition of the class MMA

In thls section, Q is a d-dimensional Riemannian manifold.

Definition 42 We denote by M M A(I; Q) (motions with measure acceleration) the set of all locally absolutely continuous motions q : I--+ Q such that the right velocity q+(t) exists (in the classical sense) fo r all t in I and defines a vector field over q with locally bounded variation.

The following proposition ensures the consistency of our notations.

Proposition 43 Let q be in M M A( I; Q). Then, q + :I--+ TQ is right continuous:

Moreover, q(t) admits a left velocity vector at each instant and:

Proof. Use the Mean Value Inequality in a local chart.

0 Appendix: Some convex analysis

In this appendix, we do not aim at providing a systematic list of the main theorems in convex analysis. We just want to recall those that are needed in the paper and also to prove some technical results which, if they had been proved in the paper when needed, could have masked the logical train of ideas.

Basic convex analysis

We denote by (E, 11• 11) a finite-dimensional normed vector space. The dual will be denoted by (E*, 11•11*). The bidual of E is systematically identified with E. We briefly recall some standard definitions and results whose proofs may be found, for example, in RocKAFELLAR (1970).

A function l/J : E -t lR U { +oo} is said proper if its domain:

def Dom l/J = {v E E ; l/J(v) :;l: +eo} , is non-void. A function l/J : E -t lR U { +oo} is said lower-semi-continuous if l/J-1 (]-oo, �] ) = {v E E; l/J(v) ::; �} is closed in E for all � E JR. If l/J is convex, its domain is convex. If l/J is lower-semi-continuous, its domain does not need to be closed. If C is any convex subset of E, then C is the domain of its indicator function le :

If l/J : E -t lR U { +oo} is a proper convex function, its conjugate (or dual) function l/J* : E * -t R U {+eo} is defined by: It is a closed convex subset of E * .

Let l/J and t/J be two proper, lower-semi-continuous, convex functions. We obviously have:

but the equality does not hold in general. However, a sufficient condition to get the equality is that one of the two functions has domain E.

Proposition 47 Let cf> and 'if; be two proper, lower-semi-continuous, convex functions. If Dom tf; =

E, then

Actually, in the finite-dimensional case that we consider, a convex function with domain E is necessarily lower-semi-continuous and even more continuous.

Proposition 48 Let cf> : E -+ lR U {+eo} be a proper, convex, lower-semi-continuous function.

We have:

v* E acf>(v) {:::::::} v E {)cj>* (v*) {:::::::} cf>(v) + cf>*(v*) = (v* , v) .

Most of the above definitions and results can be extended to the infinite-dimensional case. We shall need few results of this sort. In the following proposition, a proper convex lower-semi continuous function is built on the space £2 (0, T; E). The proof can be found in BREZIS (1973), p. 47).

Proposition 49 Let cf> be a proper, lower-semi-continuous, convex function on E and p, be a non-negative integrable function on [0, T). Fo r u E £ 2 (0, T; E), we define:

Then <I> is a proper, lower-semi-continuous, convex function on L 2 ( 0, T; E). Moreover, we have:

Evolution problems associated with subdifferentials

The systematic reference for this section is BREZIS (1973). In the sequel, for H being a Hilbert space, we shall systematically identify the dual of H with H.

First, we recall a well-known result whose proof is to be found, for example, in BREZIS (1973), p. 54.

Theorem 51 Let H be a Hilbert space, cp : H -+ R U { +oo} be a convex, proper, lower semi-continuous function, and uo be any element of Dom cp. Then, there exists a unique u E W 1 •""(0, T; H) such that:

Moreover,

• the solution u admits a right-derivative u+(t), at all t E [0, T[ and:

• the function:

is non-increasing. Now, we are going to derive a modified version of theorem 51 which is adapted to our needs. In the sequel, we denote by ( u, v) = tu • v the canonical scalar product of Rn and by I • I the associated norm. The n 2 -dimensional space of real square matrices of order n is denoted by M n (JR) .

We are given some data as follows.

• <po : Rn --+ R+ U { +oo} is convex, proper, lower-semi-continuous and positively homogeneous.

• <p; : Rn --+ R (i = 1, 2, ... , m) are convex and positively homogeneous.

• p,; E W1 '00 (0, T; R+ ) (i = 1 , 2, . . . , m).

• f E W 1 '00 (0, T; R n ) .

•

By proposition 45, we have that the functions <p; (i = 1 , 2, ... , m) are all L-Lipschitzian for some L. We denote by .\� in > 0 (respectively .\� ax) the minimum (respectively the maximum) of all the eigenvalues of G(t) for t wandering in [0, T] .

Proposition 52 There exists a unique u E W 1 '00 (0, T; Rn ) such that:

• -G (t) • u(t) -j (t) E o<p o (u(t)) + :E;: 1 p,; (t)o<p; (u(t)), for a. e. t E [0, T) . Moreover, �eii611Loo T f>. ';J " { min

Finally, if u is the solution associated with the data ( G, f, p,; ) and u the one associated with the data ( G, j, ji; ), then, fo r all t E [0, T), the fo llowing estimate holds: l ii(t) -u(t) l :::; eii611L00T/>."/;" { c,:;. A;} 1

t [1 / (s) -f (s) l + t IJ};(s) -p,;(s)l] ds } .

Proof.

The proof of proposition 52 is derived from theorem 51 by means of very classical arguments.

To prove uniqueness of solution, consider two solutions u and u and define 8 ( t) = u( t)u( t).

As a consequence of the monotonicity of subdifferentials, we have easily: 8(t) • G(t) • 8(t) � � � ���"" t 8(s) • G(s) • J(s) ds, a J o for all t E [0, T]. Applying the Gronwall lemma (lemma 4), we obtain that 8(t) vanishes identically. Therefore, the functions u and u coincide identically.

To prove existence, we define, for all N E N an approximant UN of the solution in the following way. To see that UN is well-defined, it is enough to apply proposition 47 and theorem 51 with H being Rn equipped with the scalar product induced by the matrix G((k -1 )T /2 N ) and the function cp being defined by: cp(v) = t f ck�� ) T ) • V+ cpo (v) + tjti ( (k�� ) T ) cp; (v). i=l It is obvious that, for all N E N, UN E W 1 '00 (0, T; Rn ) . Also, by use of the second part of theorem 51, we obtain, after a tedious but easy calculation: (l.B.3) where C 1 is the real constant defined in the statement of proposition 52. Now, we are given two arbitrary integers M � N. A standard but tedious calculation yields: where C 2 is a real constant which does not depend on M and N. Actually, we may take:

Applying lemma 5, we obtain that the sequence UN converges in the Banach space 0° ( [0, T]; Rn ) towards a limit u. Coming back to uniform estimate (l.B.3), we can conclude that u E W 1 •""(0, T; Rn) and also that a subsequence of (uN) converges towards u in £"" weak-*. This yields estimate (l.B.l), but there remains to prove that u is a solution of the considered evo lution problem. Using the lower-semi-continuity of cpo, the Fatou lemma and the convergence properties of the sequence (uN ) , we easily establish that, for all v E £1 (0, T; Rn ), -1 T (G • u + f)• (v -u) :S 1 T { cpo (v)-cpo(u) + t J.t;(t) (cp; (v)-cp; (u))} .