Vincent Humili
email: vincent.humiliere@math.polytechnique.fr

On some completions of the space of Hamiltonian maps

, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of R 2n endowed with the standard symplectic form ω 0 = dp ∧ dq. We study the completions of this space for the topology induced by Viterbo's distance and some others derived from it, we study their different inclusions and give some of their properties.

In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous Hamiltonians. We also prove that some dynamical aspects of Hamiltonian systems are preserved in the completions.

Introduction.

Given an open subset U in R 2n , we denote by Ham(U) the set of all 1-periodic time dependent Hamiltonian functions R × R 2n → R whose support for fixed time is compact and contained in U. We will write Ham for Ham(R 2n ).

Given a Hamiltonian function H ∈ Ham, its symplectic gradient (i.e the unique vector field X H satisfying dH = ι X H ω 0 ) generates a Hamiltonian isotopy {φ t H }. The set of Hamiltonian diffeomorphisms generated by an element H in Ham(U) will be denoted by H(U) = {φ H = φ 1 H | H ∈ Ham(U)}, and we will write H for H(R 2n ). Finally, we call L = {φ(0 n ) | φ ∈ H}, the set of Lagrangian submanifolds obtained from the zero section 0 n ⊂ T * R n = R 2n , by a Hamiltonian isotopy with compact support.

As usual, we denote Viterbo's distance on L or H by γ (see [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]). Convergence with respect to γ is called c-convergence.

Our main goals in this paper is to understand the completion H γ of the metric space (H, γ), to give some convergence criterion (section 3) and to compare it with the convergence for Hofer's distance d H (see [START_REF] Hofer | Symplectic invariants and Hamiltonian dynamics[END_REF], chapter 5 section 1). The notion of C 0 symplectic topology has been studied by many authors, starting from the work of Eliashberg and Gromov on the C 0 closure of the group of symplectic diffeomorphisms, to the later results of Viterbo ([15]) and Hofer ([4]).

More recently Oh ([9]) gave a deep study of several versions of C 0 Hamiltonians. However, our definition seems to differ from his, since in all his definitions, he needs the Hamiltonians to be continuous, while our study starts as we drop this assumption.

Let us now state our main results. For convenience, they will be restated throughout the paper. In section 3, we introduce a symplectic invariant ξ ∞ associated to any subset of R 2n , and prove that Theorem 1.1. Let (H k ) be a sequence of Hamiltonians in Ham, whose supports are contained in a fixed compact set. Suppose there exist a Hamiltonian H ∈ Ham and a compact set K ∈ R 2n with ξ ∞ (K) = 0, such that (H k ) converges uniformly to H on every compact set of R × (R 2n -K). Then (φ H k ) converges to φ H for γ.

Examples of sets K with ξ ∞ (K) = 0 are given by compact submanifolds of dimension d n -2.

Viterbo's distance γ is defined on H, but we can define for any H, K ∈ Ham γ u (H, K) = sup{γ(φ t H , φ t K ) | t ∈ [0, 1]}, to get a new distance on Ham (we give several variants of this definition). Then the following proposition allows to extend the notion of Hamiltonian flow.

Proposition 1.2. If we consider the respective completions H

γ and Ham γu of the metric spaces (H, γ) and (Ham, γ u ), then the map (H, t)

→ φ t H , Ham × R → H induces a map Ham γu × R → H γ .
The induced map associates to any element H in Ham γu a path in H γ that we will call the generalized Hamiltonian flow generated by H.

We then show that some aspects of Hamiltonian dynamics can be extended to the completions (section 4): We can define a natural action of a generalized flow on a Lagrangian submanifold. We can also associate to it a support and extend the notion of first integral.

To some of them, it is also possible, as we prove in section 6, to associate a solution to the Hamilton-Jacobi equation:

∂u ∂t + H t, x, ∂u ∂x = 0.
Indeed, a γ 2 -Cauchy sequence of Hamiltonians gives a C 0 -Cauchy sequence of solutions (where γ 2 denotes one variant of the distance γ u we mentioned above).

In section we give examples of elements in both completions H γ and Ham γu that can be described in a much more concrete way than their abstract definition (as equivalence classes of Cauchy sequences). More precisely, we prove Proposition 1.3. There is a one-one map

F ∞ → Ham γu ,
where F ∞ denotes the set of all functions H : R × R 2n → R ∪ {+∞} such that:

(i) H is continuous on R × R 2n , (ii) H vanishes at infinity: ∀ε > 0, ∃r, (|x| > r ⇒ (|∀t, H(t, x)| < ε)),

(iii) there exists a zero capacity set (e.g. an infinitesimally displaceable set), that contains all the points x where H(t, x) is +∞ for time t (iv) H is smooth on R × R 2n -H -1 ({+∞}).

Finally, let us mention that although we developed our theory on R 2n , we can reasonably expect similar results (except those of sections 4.2 and 6) on any compact symplectic manifold satisfying ω| π 2 (M ) = 0 and c 1 | π 2 (M ) = 0. Indeed, on these manifolds, Schwarz defined in [START_REF] Schwarz | On the action spectrum for closed symplectically aspherical manifolds[END_REF] a distance which is entirely analogous to Viterbo's.

Organization of the paper. In Section 2 we give the definitions of the objects used in the paper. For the reader's convenience, we first recall the construction of Viterbo's distance γ (2.2) which is based on the theory of generating functions for Lagrangian submanifolds (2.1). We also remind the reader of the different symplectic capacities constructed from γ (2.3). Finally we introduce our new distances derived from γ (2.4).

Section 3 is fully devoted to the proof of our convergence criterion. Examples of cases where it holds is then given in 3.3.

In Section 4 we define the completions of Ham and H and show that some aspects of Hamiltonian dynamics that can be extended to the completions.

In Section 5 we discuss some interesting examples of elements of the completions.

Our results on the Hamilton-Jacobi equation are given in Section 6.

Finally, we prove in Appendix a "reduction inequality" usefull to prove then all the inequalities between the distances considered in the paper.

Acknowledgments.

I am grateful to my supervisor C. Viterbo for his advices. I also want to thank my friends M. Affre and N. Roy for spending hours correcting my awful English.

Symplectic invariants.

In this section we give the definitions of all the objects we will use in the sequel. We first recall the definition of Viterbo's distance, defined first for Lagrangian submanifolds with the help of generating functions, and then for Hamiltonian diffeomorphisms (see [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]).

Generating functions quadratic at infinity.

Let L be a Lagrangian submanifold of the cotangent bundle T * M of a smooth manifold M. We say that L admits a generating function if there exists an integer q > 0 and a smooth function S : M × R q → R such that L can be written

L = (x, p) ∈ T * M | ∃ξ ∈ R q , ∂S ∂ξ (x, ξ) = 0 and ∂S ∂x (x, ξ) = p .
Such function S is called a generating function quadratic at infinity (or just "g.f.q.i") if there exists a non degenerate quadratic form Q on R q and a compact K ⊂ M × R q such that, ∀(x, ξ) / ∈ K, S(x, ξ) = Q(ξ). For instance, any quadratic form on R q viewed as a function on M × R q is a g.f.q.i of the zero section 0 M ⊂ T * M. J.C. Sikorav proved in [12] that the property of having a g.f.q.i is invariant by Hamiltonian isotopy with compact support. For this reason we will be interested in the set L of Lagrangian submanifolds, images of the zero section by a Hamiltonian isotopy with compact support. Furthermore, C. Viterbo and D. Thret proved that the g.f.q.i's of a given Lagrangian submanifold are essentially unique. Before stating this result, let us introduce the following definitions: For a given function S : M × R q → R, we call a stabilisation of S any function S ′ : M × R q × R q ′ → R of the form S ′ (x, ξ, ξ ′ ) = S(x, ξ) + q(ξ ′ ), where q is a non-degenerate quadratic form on R q ′ . In addition, two functions S, S ′ : M ×R q → R are said equivalent if there exists a diffeomorphism φ of M × R q and a real C such that S ′ = S • φ + C. Theorem 2.1 ( [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF][START_REF] Thret | A complete proof of Viterbo's uniqueness theorem on generating functions[END_REF]). Suppose S, S ′ are two g.f.q.i's of the same Lagrangian submanifold in L. Then, up to stabilisation, S and S ′ are equivalent.

This result allows to associate symplectic invariants to any element of L.

Invariants defined by minimax and a distance on

the group of Hamiltonian diffeomorphisms.

The invariants defined in this section have been introduced by C. Viterbo in [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]. We recall their construction. We first define invariants for Lagrangian submanifolds.

Let L be an element of L and S : M × R q → R be one of its g.f.q.i's. Let us denote S λ = {x ∈ M × R q | S(x) λ}. Since S is quadratic at infinity, the homotopy types of the pairs (S λ , S µ ) and (S µ , S -λ ) do not depend on λ, provided that λ is sufficiently large . Therefore, we will denote S ∞ and S -∞ , instead of S λ and S -λ for λ large enough.

Let us introduce E - ∞ the negative (trivial) bundle of the quadratic form which coincides with S at infinity. We denote B(E - ∞ ), S(E - ∞ ) the ball bundle and the sphere bundle associated to E - ∞ . The Thom isomorphism is given by H

* (M) → H * (B(E - ∞ ), S(E - ∞ )
), and we also have the isomorphism

H * (B(E - ∞ ), S(E - ∞ )) ≃ H * (S ∞ , S -∞
). We will denote by T their composition. For further informations on those isomorphisms, see [START_REF] Husemoller | Fiber Bundles[END_REF] for example. The inclusion j λ : S λ → S ∞ induces a morphism in cohomology j * λ : H * (S ∞ , S -∞ ) → H * (S λ , S -∞ ), for all real number λ. We are now ready for the following. Definition 2.2. Let (u, L) ∈ H * (M) × L, with u = 0. Using a g.f.q.i S of L, we define a real number c(u, L) as follows:

c(u, L) = inf {λ | j * λ • T (u) ∈ H * (S λ , S -∞ ) is non zero}. (1)
Observe that c(u, L) is well defined, and is independent of the choice of S's choice, up to additive constant. Indeed, if we replace S with an equivalent or stabilized generating function, the value of c(u, L) does not change, up to additive constant and we conclude using theorem 2.1. Even if it doesn't depend on the generating function, we sometimes use the notation c(u, S) instead of c(u, L).

Since the cohomology of the sets S λ changes when we cross the level c(u, L), it has to be a critical value of S.

Finally, observe that the definition can be extended to classes with compact support u ∈ H * c (M). Then, we can use those invariants associated to Lagrangian submanifold to define other invariants associated to Hamiltonian diffeomorphisms.

Consider a Hamiltonian diffeomorphism ψ ∈ H(R 2n ). Its graph Γ ψ is a Lagrangian submanifold of R 2n × R 2n (= (R 2n × R 2n , -ω 0 ⊕ ω 0 ), where ω 0 is the standard symplectic structure on R 2n ). It coincides with the diagonal ∆ = {(x, x) | x ∈ R 2n }, outside the product B 2n (r) × B 2n (r), for r sufficiently large. When we identify R 2n × R 2n with T * ∆ using the map,

(q, p, Q, P ) → q + Q 2 , p + P 2 , P -p, Q -q ,
we see that the image Γ ψ of Γ ψ is identified with the zero section of T * ∆ outside a compact set. Then, we can associate the previous invariant to Γ ψ (We normalize generating functions by asking their critical value at infinity to equal 0). Let 1 be a generator of H 0 (R 2n ) and µ a generator of H 2n c (R 2n ).

Definition 2.3 (Viterbo,[START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]). We define,

c -(ψ) = -c(µ, Γ ψ ), c + (ψ) = -c(1, Γ ψ ), γ(ψ) = c + (ψ) -c -(ψ), γ(φ, ψ) = γ(ψ -1 φ).
Let us describe now the properties of the numbers γ, c + and c -that we will use in the paper. Proposition 2.4 (Viterbo,[START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]). a)(Sign and Separation) For all ψ in H, we have c -(ψ) 0 c + (ψ).

Moreover, c -(ψ) = c + (ψ) = 0 if and only if ψ = Id. b) (Triangle inequality) If φ is another diffeomorphism in H, then c + (φ • ψ) c + (φ) + c + (ψ), c -(φ • ψ) c -(φ) + c -(ψ), γ(φ • ψ) γ(φ) + γ(ψ). (φ, ψ) → γ(φ, ψ) is a distance on H.
In particular, the separation property and the triangle inequality imply that

(φ, ψ) → γ(φ, ψ) is a distance on H. c)(
(R 2n × [0, 1], R). If H 1 -H 2 ε, then |γ(ψ 1 ) -γ(ψ 2 )| ε.
2.3 Two symplectic capacities on R 2n .

We start this section by reminding the reader of the definition of a symplectic capacity. This is a "symplectic" way of measuring sets that plays an important role in symplectic topology. We will use it in particular for our convergence criterion in section 3.

Definition 2.5 (Ekeland-Hofer).

A symplectic capacity on (R 2n , ω 0 ) is a map associating to each subset U ⊂ R 2n a number c(U) ∈ [0, +∞] satisfying

1. U ⊂ V ⇒ c(U) c(V ) (monotony), 2. c(φ(U)) = c(U) for all Hamiltonian diffeomorphism φ ∈ H (symplectic invariance), 3. c(λU) = λ 2 c(U) for all real λ > 0 (homogeneity), 4. c(B 2n (1)) = c(B 2 × R 2(n-1) ) = π, where B 2n (1) is the unit ball of R 2n (normalisation).
The invariants defined in the previous section allow to define two symplectic capacities as follows ( [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]). Definition 2.6. 1. For any compact subset K ⊂ R 2n , we denote by γ(K) the number defined by

γ(K) = inf{γ(φ) | φ(K) ∩ K = ∅}. If V is not compact, we set γ(V ) = sup{γ(K) | K ⊂ V }.

2.

For any open subset U ⊂ R 2n , we denote by c(U) the number defined by

c(U) = sup{c + (φ H ) | Supp(H) ⊂ U}. If V is not an open set, we set c(V ) = inf{c(U) | V ⊂ U}.
The maps c and γ are symplectic capacities and moreover c γ. We remind the reader of the definition of the displacement energy

d(U) = inf{d H (φ, Id) | φ(U) ∩ U = ∅},
where d H is Hofer's distance defined by

d H (φ, ψ) = inf{ H -K | H generates φ and K generates ψ}, whith H = 1 0 (max x H(t, x) -min x H(t, x))dt.
We are going to define a new symplectic capacity derived from c, but before we need the following lemma.

Lemma 2.7. We consider a subset

V ⊂ R 2n and R 2 × V ⊂ R 2+2n . Then, c(R 2 × V ) c(V ).
That lemma follows from the reduction inequality of Proposition A.1. We postpone its proof to Appendix. The reverse inequality might be true but we are unable to prove it. That leads us to introduce the following object.

Definition 2.8. For any open subset

U ⊂ R 2n , we set c ∞ (U) = lim N →∞ c(U × R 2N ),
and if V is not an open subset, c ∞ (V ) = inf{c ∞ (U) | V ⊂ U}.
We obtain a symplectic capacity that satisfies c ∞ (V ) = c ∞ (V × R 2 ) for all subset V (this property will be useful), and c c ∞ . Moreover, since

d(U) = d(U × R 2k ) and c d, we have c ∞ d.
To summarize the known inequalities between capacities we have, Proposition 2.9. c γ d and c c ∞ d.

Other distances derived from γ.

In this section we introduce several other distances for many reason. One is that we want to consider distances not only on H but also on Ham. Another motivation is our result on the Hamilton-Jacobi equation (section 6) that needs almost all of them. Finally, a stupid but important reason is that we still don't know which is the best one to develop our theory! Let us start with the following distance defined on H already introduced by Cardin and Viterbo in [START_REF] Cardin | Commuting Hamiltonians and Hamilton-Jacobi multi-time equations[END_REF].

Definition 2.10. For all Hamiltonian diffeomorphisms φ, ψ ∈ H, we define

γ(φ, ψ) = sup{γ(ψ -1 φ(L) -L) | L ∈ L}, where γ(L) = c(µ, L) -c(1, L), ∀L ∈ L and L 1 -L 2 = {(q, p 1 -p 2 ) | (q, p 1 ) ∈ L 1 , (q, p 2 ) ∈ L 2 }, for L 1 , L 2 ∈ L.
Then, we define distances not anymore on H, but on Ham. Definition 2.11. For any H, K ∈ Ham, we set

γ u (H, K) = sup{γ(φ t H , φ t K ) | t ∈ [0, 1]} and γu (H, K) = sup{γ(φ t H , φ t K ) | t ∈ [0, 1]}.
Here, the subscript "u" means "uniform". Clearly, γ u and γu are distances on Ham.

For the next two distances, the principle is to add two dimensions by associating to an Hamiltonian H two suspensions defined on R × R 2+2n :

Ĥ(s; t, τ, x) = τ + H(t, x), Ȟ(s; t, τ, x) = tH(st, x).
Here, the new time variable is s, while the former time variable t becomes a space variable (As a consequence Ĥ is an autonomous Hamiltonian). We would like to define our distances by γ(H, K) = γ( Ĥ, K) and γ(H, K) = γ( Ȟ, Ǩ). But since Ĥ and Ȟ are not compactly supported we have to be slightly more subtle. Definition 2.12. Let ρ be a fixed real function defined on [0, +∞), supposed to be non-negative, smooth, decreasing, with support in [0,1], flat at 0 and such that ρ(0) = 1. For every natural integer α and every real number t, we set ρ α (t) = 1 if -α t α, and ρ α (t) = ρ(|t|α) otherwise.

We denote by Ĥα and Ȟα the Hamiltonian functions defined on R×R 2+2n , by Ĥα (s; t, τ, x) = ρ α (τ )τ + ρ α (t)H(t, x), and Ȟα (s; t, τ, x) = ρ α (t)tH(st, x).

Then, for H, K ∈ Ham, we set

γ(H, K) = lim sup α→+∞ γ u ( Ĥα , Kα ),
and γ(H, K) = lim sup α→+∞ γ(φ Ȟα , φ Ǩα ).
Remark that γ(H, K) and γ(H, K) are finite. Indeed, if we denote by B a ball containing both supports of H and K, then Ĥα , Kα , Ȟα and Ǩα have support in R 2 × B, for any integer α. Hence γ(φ Ȟα , φ Ǩα ) 2c(R 2 × B) 2c ∞ (B) (See section 2.3 for notations). It shows that the lim sup in the definition of γ is finite. The same proof shows that γ(H, K) is also finite.

The triangle inequality for γ and γ is a direct consequence of the triangle inequality for γ. The separation property is obtained from the separation property for γ and Proposition 2.13.

For convenience, we will not write the subscript α anymore. In the following, we will denote Ĥ for Ĥα , and Ȟ for Ȟα .

Remarks. By repeating these constructions several times (i.e., by taking suspensions of suspensions), we can construct new distances. For example, we will use in section 6 the distance γ 2 = lim sup α→+∞ γ( Ĥα , Kα ).

Using the invariance of γ, it is easy to verify that the suspended distances γ, γ and γ 2 are invariant under the action of H. Namely, for H, K Hamiltonians and ϕ Hamiltonian diffeomorphism, we have:

γ(H • ϕ, K • ϕ) = γ(H, K), γ(H • ϕ, K • ϕ) = γ(H, K), γ 2 (H • ϕ, K • ϕ) = γ 2 (H, K).
The following proposition gives inequalities between the distances. It will be proved in Appendix. It is based on the reduction inequality (Proposition A.1). Proposition 2.13. γ γ, γu γ u min(γ, γ).

3 The convergence criterion. This is the central section of our paper. We give there the proof of our main result, Theorem 1.1.

A sufficient condition for a Hamiltonian diffeomor-

phism to be γ-close to Id.

We start this section with some formulas concerning Hamiltonian flows. They can be obtained by direct computation (see [START_REF] Hofer | Symplectic invariants and Hamiltonian dynamics[END_REF], page 144).

Lemma 3.1. For all Hamiltonians H and K, with compact support, we have:

φ t H = (φ t H ) -1 , where H(t, x) = -H(t, φ t (x)) φ t H♯K = φ t H • φ t K , where (H♯K)(t, x) = H(t, x) + K(t, (φ t ) -1 (x)) φ t H♯K = (φ t H ) -1 • φ t K . Remark. (H♯K)(t, x) = (K -H)(t, φ t (x)).
The following proposition shows that if a sequence of Hamiltonians (H n ) converges to zero uniformly on every compact set contained in the complement of a set whose capacity is zero, then (φ Hn ) converges to Id for γ. Proof. Let K 1 , K 2 be Hamiltonians with compact support, such that 0 K i 1, i = 1, 2, K 1 equals 1 on the support of H and K 2 equals 1 on the support of K 1 (hence K 1 K 2 ). Denote ψ 1,ε the diffeomorphism generated by H -εK 1 , and ψ 2,ε the diffeomorphism generated by εK 2 . Then we have H εK 2 + (H -εK 1 ). As (ψ 2,ε ) -1 coincides with Id on the support of H -εK 1 , the lemma 3.1 implies that εK 2 + H -εK 1 is the Hamiltonian that generates ψ 2,ε • ψ 1,ε . The monotony, the triangle inequality and the continuity given Proposition 2.4 then give

c + (φ) c + (ψ 2,ε • ψ 1,ε ) c + (ψ 2,ε ) + c + (ψ 1,ε ) ε + c + (ψ 1,ε ).
Denote by ψ 1,ε the diffeomorphism generated by a non-negative Hamiltonian, with support in U, and greater than H -εK 1 . Then by the monotony property, c

+ (ψ 1,ε ) c + ( ψ 1,ε ). Finally, since Supp( ψ 1,ε ) ⊂ U, we get c + ( ψ 1,ε ) c(U) ε.
Using the inequality H -εK 2 + (H + εK 1 ), we obtain the same type of inequality for c -.

For example, if K is a compact submanifold of dimension lower or equal than n -1, then d(K) = 0 (and hence c(K) = 0).

What about non-identity elements that are close for γ?

Unfortunately, the previous result cannot be straightforwardly generalised to obtain a general convergence criterions when the limit is not zero. Indeed, we can find two Hamiltonians that are C 0 -close out of a null-capacity set, but not γ-close.

Example. It is well known that the capacities c and γ of the unit sphere

S = {x ∈ R 2n | x = 1} are π.
It is also true for c ∞ . Then, for all ε > 0, there exists a Hamiltonian H with support in a small neighbourhood U of S, and such that c + (φ H ) > πε. Because of the monotony property (proposition 2.4), H can be chosen non-negative. We set

U + a neighbourhood of {x ∈ S | x 1 0} and U -a neighbourhood of {x ∈ S | x 1 < 0}, such that U = U + ∪U -.
If U, U + and U -are choosen small enough, we have d(U ± ) < ε and by proposition 2.9 c ∞ (U ± ) < ε. Using some partition of unity associated to the decomposition U = U + ∪ U -, we get two functions H ± , with support in U ± and such that H = H + + H -. Now, we see that H + coincides with H outside U -, whose capacity verifies c ∞ (U ± ) < ε, but on the other hand,

γ(φ H ) -γ(φ H + ) γ(φ H , φ H + ) π -ε -γ(φ H -) π -2ε.
It shows that the previous statement is false when the limit is not zero.

Nevertheless, we can introduce a new invariant, in order to extend the result of proposition 3.2. Definition 3.3. For any subset U and any Hamiltonian H ∈ Ham, we define

ξ H (U) = c ∞   t∈[0,1] φ t H (U)   .
We may then set

ξ λ (U) = sup ξ H (U), for 0 < λ ∞,
where the supremum is over all Hamiltonian functions H with γ u (H) λ.

Theorem 3.4. Let H 1 and H 2 be Hamiltonians on R 2n with compact support. Let U be a subset of R 2n , satisfying one of the two following conditions:

1. ξ ∞ (U) ε. 2. ∃λ > 0, ξ λ (U) = 0 If |H 1 (t, x) -H 2 (t, x)| ε for all t ∈ [0, 1] and all x / ∈ U, then we have γ(φ H 1 , φ H 2 ) 4ε. Proof. Consider the Hamiltonian H(t, x) = H 1 (t, φ t 2 (x)) -H 2 (t, φ t 2 (x)). By assumption, |H(t, x)| ε, for all (t, x) with x / ∈ φ -t 2 (U)
and hence for all t and all x /

∈ t∈[0,1] (φ -1 2 ) t (U). Each condition on U implies c   t∈[0,1] (φ -1 2 ) t (U)   c ∞   t∈[0,1] (φ -1 2 ) t (U)   ε.
By proposition 3.2 and lemma 3.1, we get γ(φ

H 1 , φ H 2 ) = γ(φ H ) 4ε.
Important remark. In the proof of theorem 3.4, we see that the important condition is in fact ξ H 2 (U) ε, which is of course implied by both conditions ξ ∞ (U) ε and ξ λ (U) = 0.

Corollary 3.5. The conclusion of theorem 3.4 still holds if we replace γ with γ. For the distances on Ham, we get under the same assumptions

d(H 1 , H 2 ) 4 H 1 -H 2 C 0 ,
where d is either γu , γ u , γ or γ.

Proof. By proposition 2.13 (inequality between distances), we just have to prove it for γ and γ. Then remark that under the hypothesis of theorem 3.4, we have | Ĥ1 (s; t, τ, x)-Ĥ2 (s; t, τ, x)| ε and | Ȟ1 (s; t, τ, x)-Ȟ2 (s; t, τ, x)| ε for all integer α, all s ∈ [0, 1], and all (t, τ, x) / ∈ R 2 × U. Unfortunately, even if U satisfies one of the conditions of proposition 3.4, it is not in general the case for R 2 × U. However, by the above remark, it is sufficient to show that for all real number δ > 0 and all integer α large enough, ξ Ȟ2 (R 2 × U) ε + δ and ξ Ĥ2 (R 2 × U) ε + δ. By letting δ tend to zero and taking limsup with respect to α, we obtain γ(H, K) 4ε and γ(H, K) 4ε as required.

Let us denote F for Ȟ2 or Ĥ2 . The inequalities on ξ F come directly from the expression of φ Ȟ2 and φ Ĥ2 (see computations in Appendix A.2). Indeed, in both cases,

s∈[0,1] (φ -1 F ) s ([-α, α] 2 × U) ⊂ R 2 × s∈[0,1] (ψ -1 ) s (U),
where ψ s is a Hamiltonian isotopy that appears in last coordinate when we compute φ F . Therefore, since ξ F (R 2 × U) = lim α→+∞ ξ F ([-α, α] 2 × U), we get for any δ > 0 and any α large enough:

ξ F (R 2 × U) δ + c ∞   R 2 × s∈[0,1] (ψ -1 ) s (U)   = δ + c ∞   s∈[0,1] (ψ -1 ) s (U)   δ + ε.
That concludes the proof.

Corollary 3.6. Let (H k ) be a sequence of Hamiltonians in Ham, whose supports are contained in a fixed compact set. Suppose there exist a Hamiltonian H ∈ Ham and a compact set K ∈ R 2n with ξ ∞ (K) = 0, such that (H k ) converges uniformly to H on every compact set of R × (R 2n -K). Then (φ H k ) converges to φ H for γ, γ, and (H k ) converges to H for γu , γ u , γ, γ.

Proof. For γ, γ, it is a direct consequence of the remark that follows theorem 3.4. We just have to verify that for all ε > 0, there exists a small neighbourhood U of K such that ξ H (U) ε. This is true because for every neighbourhood V of t∈[0,1] φ H (K), we can choose a neighbourhood U of K such that

t∈[0,1] φ H (U) ⊂ V. Since c ∞ ( t∈[0,1] φ H (K)) = 0 and t∈[0,1] φ H (K) is compact, we can choose V such that c ∞ (V ) ε, and obtain c ∞ ( t∈[0,1] φ H (U)) ε as required.
For γ and γ, we have to verify that for all ε > 0 and all δ > 0, there exists a small neighbourhood U of K such that for all α large enough ξ φ (U) ε+δ, where F is either Ĥ or Ȟ. The proof made above for φ H shows that we can find U such that ξ f (U) ε, where f generates the isotopy ψ s defined as in the proof of corollary 3.5. Therefore we have for all δ and all α large enough,

ξ F (R 2 × U) ξ F ([-α, α] 2 × U) + δ ξ f (U) + δ ε + δ.
By proposition 2.13, corollary 3.6 is also true for γu and γ u .

Remark. Similar proofs give that theorem 3.4 and corollary 3.6 still hold for γ 2 .

3.3 Example of a non trivial ξ-small set. Proof. Let H ∈ Ham. The problem is that t∈[0,1] φ t H (U) is not in general a manifold. To avoid that problem, we are going to add two dimensions and make a suspension in this way. We denote by Φ the Hamiltonian diffeomorphism on R 2+2n = {(t, τ, x)} generated by the Hamiltonian [0, 1] × R 2+2n → R, (s; t, τ, x) → tH(ts, x).

We also set

V = Φ([0, 1] × [-1, 1] × U). The computation of Φ gives Φ(t, τ, x) = (t, τ -H(t, x), φ t (x)).
We see that t∈[0,1] φ t H (U) can be obtained from V by symplectic reduction by the coisotropic manifold {τ = 0}. So we are going to look for a Hamiltonian diffeomorphism φ K that displaces V and preserves {τ = 0} at the same time. If the Hamiltonian does not depend on t, the second condition is verified.

Since V is compact, it is sufficient for K to verify ∀v ∈ V, RX K (v) ∩ T v V = {0}, which is equivalent to ∀v ∈ V, ker dK(v) ⊕ T v V ⊥ = R 2+2n and to ∀v ∈ V, T v V ⊥ ⊂ ker dK(v).
That makes us consider the 1-jet bundle J 1 (R×R 1+2n , R) and its submanifold

W = {(s, q; σ, p; z) | (s, q) ∈ V, z ∈ R, T (s,q) V ⊥ ⊂ ker(σ, p)}.
The dimension of W is exactly 2n + 1. Indeed, the vector space

{(σ, p) ∈ R 2n+2 * | T (s,q) V ⊥ ⊂ ker(σ, p)} has dimension 2n + 2 -dim(T (s,q) V ⊥ ) = n.
By Thom transversality theorem (see [START_REF] Golubitsky | Stable mappings and their singularities[END_REF] for example), there exists a function L whose 1-jet verifies j 1 L ⋔ W . But j 1 L can be seen as a function R × R 1+2n → J 1 (R × R 1+2n , R), and by lemma 4.6 page 53 in [START_REF] Golubitsky | Stable mappings and their singularities[END_REF], we have for a generic choice of s ∈ R, j 1 L(s, •) ⋔ W . We fix s as previously and we denote K :

R 2+2n → R, K(t, •) = L(s, •)
Then, notice that for every s, q, p, z, the set of all σ such that (s, q; σ, p; z) ∈ W is either ∅ or R. It can be shown by direct computation of T V ⊥ , whose first component appears to be always {0}. As a consequence, we get

j 1 K ⋔ W (j 1 K differs from j 1 L(s, •) just by its σ-component which is {0} instead of ∂L ∂s (s, •) for j 1 L(s, •)). Now, since (2n+2)+(2n+1) = dim(j 1 K(R 2+2n ))+dim(W ) < dim(J 1 (R× R 1+2n , R)) = 4n + 5, we get j 1 K(R 2+2n ) ∩ W = ∅.
It follows that K satisfies the two conditions: it preserves {τ = 0} and it satisfies

∀v ∈ V, RX K (v) ∩ T v V = {0}.
As V is compact, for ε small enough, since φ εK = φ ε K , we have φ εK (V ) ∩ V = ∅. In addition εK can be made as C 0 -small as we want.

We are now ready for the reduction by {τ = 0}. Since it preserves {τ = 0}, εK induces a Hamiltonian on the reduction R 2n . This Hamiltonian is C 0small and generates a diffeomorphism ψ whose Hofer's distance to identity d H (ψ, id) is small, and that satisfies

ψ   t∈[0,1] φ t H (U)   ∩ t∈[0,1] φ t H (U) = ∅.
This Hamiltonian is not compactly supported, but any Hamiltonian with compact support which coincides with it on a sufficiently large ball, would have the same properties. That proves d t∈[0,1] φ t H (U) = 0, and since c ∞ d, we get ξ H (U) = 0.

Completions and extension of Hamiltonian dynamics

In this section, we introduce the completions and give the first properties of their elements: the existence of a flow that acts on Lagrangian submanifolds, the notion of first integral and the existence of a support. The full section 6 will be devoted to another property related to the Hamilton-Jacobi equation.

Notations, inclusions and definitions

Action on Lagrangian submanifolds

Recall that the set L of Lagrangian submanifolds isotopic to the zero section by compactly supported Hamiltonian isotopy, can be endowed with Viterbo's distance, also denoted γ (set γ(L 1 , L 2 ) = γ(L 1 -L 2 ), see definition 2.10). Let us denote L the completion of L with respect to this distance.

Proposition 4.3. The groups H γ and H γ naturally act on the set L. This action extends the natural action of H on L.

Proof. It is a simple consequence of the inequality γ γ (Proposition 2.13 proved in Appendix).

Let L ∈ L represented by a sequence (L k ) and φ in H γ (the proof is the same for H γ ), represented by a sequence (φ k ). We are going to show that (φ k (L k )) defines an element of L that we will denote φ(L).

This follows easily from the fact that for φ, ψ ∈ H and L, M ∈ L,

γ(φ(L) -ψ(M)) γ(φ(L) -ψ(L)) + γ(ψ(L) -ψ(K)) γ(ψ -1 φ(L) -L) + γ(L -K) γ(φ, ψ) + γ(L -K).
Remark

Notion of first integral

This property has been first mentioned in [START_REF] Cardin | Commuting Hamiltonians and Hamilton-Jacobi multi-time equations[END_REF], in the definition (3.3) [START_REF] Cardin | Commuting Hamiltonians and Hamilton-Jacobi multi-time equations[END_REF] that this definition extends the usual definition of commuting Hamiltonian functions.

Existence of a support

In this section, we state a lemma which makes it possible to define a support for the elements of the different completions. Proof.

a.

Thanks to Proposition 2.13, we just have to prove the assertion in the case of γ. Suppose supp(φ) ⊂ U. Then there exists an x in R 2n -U such that φ(x) = x. Let ψ be a Hamiltonian diffeomorphism whose support is included in R 2n -U and which does not contain φ(x). Suppose in addition that ψ(x) = x. Then, since the supports of φ n and ψ are disjoint, we have ψ • φ -1 n • ψ -1 • φ n = Id, for all integer n. Taking limit, we get on one hand ψ • φ -1 • ψ -1 • φ = Id. But on the other hand, we have by construction,

ψ • φ -1 • ψ -1 • φ(x) = ψ(x) = x, which is contradictory. b.
We use the first part of the lemma to conclude that for all time t, supp(φ t ) ⊂ U. This implies that supp(H) ⊂ U .

Remark. A similar argument shows that the property of letting globally invariant any sphere centered at 0, is invariant by taking γ or γ limits. Similarly, a γ u , γu , γ or γ limit of radial Hamiltonians is radial. Then, it is easy to see that V U = V U = supp(η).

Description of some elements of the completions

The elements of the different completions are by definition equivalence classes of Cauchy sequences. So they are defined in a very abstract way. In this section, we show that many elements of the completions can be seen in a more concrete way.

Examples in the completion of Ham

The inequalities between Hofer's distance and our four distances γ u , γu , γ and γ on Ham imply inclusions of the completions. In particular any continuous time-dependent Hamiltonian can be seen as an element of Ham γu , Ham γu , Ham γ and Ham γ . In view of Theorem 3.4 and Corollary 3.6, we can conjecture that if

K ∈ R 2n satisfies ξ ∞ (K) = 0 a sequence (H k ) converges uniformly on compact sets of R × (R 2n -K) to a function H continuous on R × (R 2n -K), then (H k )
is Cauchy for either γ u , γu , γ or γ (compare with Corollary 3.6). We are still unable to prove it, but if we restrict to a family of Hamiltonians which converge to +∞ at their discontinuity points, this result can be established. Definition 5.1. We denote by F the set of all functions H : R × R 2n → R ∪ {+∞} such that: andA, A ∞ the subsets of time-independent elements of F and F ∞ .

(i) There exist K ∈ R 2n with c ∞ (K) = 0 such that H(t, x) = +∞ ⇒ x ∈ K, (ii) H vanishes at infinity: ∀ε > 0, ∃r, (|x| > r ⇒ (∀t, |H(t, x)| < ε)), (iii) H is continuous on R × R 2n . We also set F ∞ = {H ∈ F | H is smooth on R × R 2n -H -1 ({+∞})},
For the elements of F ∞ , the set of discontinuity is somehow "stable" under the Hamiltonian flow. This property allows to consider functions with a larger discontinuity set than what could be expected in the general case (c ∞ (K) = 0 instead of ξ ∞ (K) = 0). Lemma 5.2. Suppose H is an element of A and K = H -1 ({+∞}). Then there exists a sequence of smooth autonomous Hamiltonians (H k ) ∈ Ham with the following properties: a. (H k ) converges to H uniformly on every compact subset of R 2n -K.

b. (H k ) is Cauchy for γ u , γu , γ and γ.

Moreover, if H ∈ A ∞ , then any sequence (H k ) that converges to H uniformly on the compact subsets of R 2n -K, does not converge in Ham, for none of the distances γ u , γu , γ and γ.

Proof. Fix k > 0. Properties (ii) and (iii) in Definition 5.1 imply that K is compact. Since c ∞ (K) = 0, there exists an open neighborhood U of K such that c ∞ (U) 1 k . Then, if we denote H >A = {x | H(x) > A}, we have for A large enough, K ⊂ H >A ⊂ U.
Indeed, if it was not true, then for all integer for all integer l > 0, there would exists a point a l in H >l , but not in U. Then, the sequence (a l ) would take values in H 1 ∩ (R 2n -U) which is compact, and hence it would have a subsequence that would converge to an element of K ∩ (R 2n -U), which contradicts our assumption. Let us fix a real number A k such that

H >A k ⊂ U. Now, let H k be a smooth function with compact support such that |H k - H| < 1 k on R 2n -H >A k + 2 k , and such that |H k -A k -2 k | < 1 k on H >A k + 2 k .
The sequence (H k ) clearly converges to H uniformly on every compact subset of R 2n -K. Let us see why it is Cauchy.

By Proposition 2.13, we just have to prove it for γ and γ. We write F k for either Ȟk or Ĥk . We also denote, as in the proof of Corollary 3.5, ψ k for the third coordinate of φ F k . Since H k is an autonomous Hamiltonian, its flow φ t H k preserves its level sets. Hence, the isotopy ψ s k preserves the level sets of F k (see the computations in Appendix A.2). Therefore, since by construction

H >A k + 2 k ⊂ H >A k + 1 k k , we have t∈[0,1] ψ t k (H >A k + 2 k ) ⊂ H >A k + 1 k k .
Let δ > 0 and suppose α is sufficiently large. Then, as in the proof of Corollary 3.5,

ξ F k (R 2 × H >A k + 2 k ) δ + c ∞   R 2 × s∈[0,1] (ψ -1 k ) s (H >A k + 2 k )   δ + c ∞   s∈[0,1] (ψ -1 k ) s (H >A k + 2 k )   δ + c ∞ (H >A k + 1 k k ).
Since

H >A k + 1 k k ⊂ H >A k ⊂ U and c ∞ (U) 1 k , we obtain ξ F k (H >A k + 2 k ) 1 k +δ. Now, pick an integer l k.
If l and k are large enough, then we have

| Ĥk -Ĥl | 1 k and | Ȟk -Ȟl | 1 k on R 2+2n -(R 2 ×H >A k + 2 k
). Therefore, by the remark that follows Theorem 3.4, we get γ(H l , H k ) Suppose now that H is smooth on R 2n -K. Then we can choose H k such that it coincides with H on B k -

H >A k + 2 k
, where B k is the ball of radius k, centered at 0. Suppose that (H k ) converges to a Hamiltonian L ∈ Ham for γu , γ u , γ, and γ. Then for any integer k, H k ♯H l converges to H k ♯L while l tends to infinity for γu (see Lemma 3.1 for notations). According to Lemma 4.5, since H K ♯H l has support in the complementary of B k -

H >A k + 2 k , H k ♯L has support in its closure and hence L coincides with H on B k -H >A k + 2
k . Since it is true for any k, L has to coincide with H on R 2n -K. Therefore L cannot belong to Ham, which contradicts our assumptions.

Finally, if (L k ) is another sequence of Hamiltonians that converges to H uniformly on the compact subsets of R 2n -K, then, similarly as in the above proof that (H k ) is Cauchy, we obtain that γ(L k , H k ) and γ(L k , H k ) converge to 0, where H k is the particular sequence defined in the previous paragraph. Since (H k ) does not converge, (L k ) does not converge either.

Remark. As usual, the results of Lemma 5.2 still hold for γ 2 . Proof. Let us first consider the autonomous case (elements of A ∞ ). Since Ham γ ⊂ Ham γu ⊂ Ham γu and Ham γ ⊂ Ham γu ⊂ Ham γu , it is enough to prove it for γ and γ. We will make the proof for γ and the proof for γ will be exactly the same. Let J be the function that associates to any H ∈ A ∞ the element of Ham γ represented by any sequence (H k ) that converges uniformly to H on the compact sets of R 2n -H -1 ({+∞}). As we noticed at the end of the proof of Lemma 5.2, two such sequences are equivalent and hence J is well-defined.

Let us now prove that J is one-one. Let H, G ∈ A ∞ and let (H k ), (G k ) be two sequences respectively associated to them, precisely constructed as in the last but one paragraph of the previous proof. Suppose that G = H, we are going to show that γ(H k , G k ) does not converge to zero, that will imply that γ(H k , G k ) does not converge to zero.

We can define almost everywhere the flows φ t G , φ t H and

ψ t = φ -t G • φ t H . Let ψ k = φ -1 G k • φ H k . Since G = H, there exists a point x such that ψ(x) = x.
Hence, there exists a small ball B around x such that ψ(B) ∩ B = ∅. Let K be a compact neighborhood of t ψ t (B). For k large enough, H k and G k coincide respectively with H and G on K, and thus

ψ k (B)∩B = ∅ too. Since γ(H k , G k ) = γ(ψ k ) γ(B) > 0, γ(H k , G k ) cannot converge to zero.
To achieve the proof, we just have to notice that the map H → Ĥ is a oneone map F ∞ 2n → A ∞ 2n+2 (the subscript denotes the dimension of the ambient symplectic space). Thus, according to the autonomous case, if

H is in F ∞ 2n then Ĥ is in Ham γu (R 2n+2
φ H (r, θ) = (θ -r 2 f ′ (r 2 ), r),
for r > 0 where f (s) = i>0 h(2 i s) (for any s, all the terms in this sum are 0 except maybe one). We see that φ H is a homeomorphism. Let us prove that it cannot be smooth at zero.

If we denote by (q, p) the coordinates in R 2n , and by φ 1 the projection of φ H on R n × {0}, we have for q ∈ 2 -i [1/2, 1], ∂φ 1 ∂q (q, 0) = cos(q 2 2 i h ′ (2 i q 2 ))-2(q 4 2 2i h ′′ (2 i q 2 )+q 3 2 i h ′ (2 i q 2 )) sin(q 2 2 i h ′ (2 i q 2 )).

(2) Suppose that h is chosen so that there exists q 1 and q ′ 1 in [1/2, 1] such that ∂φ 1 ∂q (q 1 , 0) = ∂φ 1 ∂q (q ′ 1 , 0) (we denote by A their difference), and define q i = √ 2 -i q 1 and q ′ i = √ 2 -i q ′ 1 . Then, (q i ) and (q ′ i ) vanish, but from [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF] we see that ∂φ 1 ∂q (q i , 0) -∂φ 1 ∂q (q ′ i , 0) converges to A. Therefore φ H cannot be smooth at 0. Example of a discontinuous element in H γ .

In the previous example, the sequence of diffeomorphisms (φ H k ) was converging almost everywhere to a homeomorphism (which was not a diffeomorphism). Therefore, one could think that the class of (φ H k ) in the completion H γ can be represented by a homeomorphism. However, with the help of Proposition 5.2, we can show that it is not true in general, at least in dimension 2n 4. Indeed, consider H : R 2 × R 2n → R,

(x 1 , x 2 ) → 1 | x 1 2 -1| + x 2 2 χ( (x 1 , x 2 ) ),
where χ is smooth with compact support and equals 1 on the ball of radius 2 centered at zero. Clearly, H ∈ F ∞ (because K = H -1 ({+∞}) = S 1 × {0} satisfies c ∞ (K) = 0 as required). Consider the sequence (H k ) constructed in the proof of Lemma 5.2. Since (H k ) is Cauchy for γ u , (φ H k ) is Cauchy for γ. Suppose it converges to an element φ. Then, Lemma 4.5 implies that for any neighbourhood U of K and for k large enough, φ coincides with φ H k on R 2+2n -U. Therefore, we can compute the explicit form of φ on R 2+2n -K.

In polar coordinates (s 1 , θ 1 , s 2 , θ 2 ) with s 1 = x 1 2 and s 2 = x 2 2 , we get for s 1 < 1: 6 Application to the Hamilton-Jacobi equation.

φ(s 1 , 0, 0, 0) = s 1 , s 1 (1 -s 1 )
Let H be a smooth Hamiltonian function on R × R 2n . We consider the evolution Hamilton-Jacobi equation (HJ):

∂u ∂t + H t, x, ∂u ∂x = 0,
where u : R × R n → R, (t, x) → u(t, x) satisfies an initial condition u(0, x) = u 0 (x). First, we remind the reader of the construction of a variational solution of (HJ) (see for example [START_REF] Viterbo | Solutions d'quations de Hamilton-Jacobi[END_REF] or [START_REF] Ottolenghi | Solutions généralisées pour l'équation d'Hamilton-Jacobi dans le cas d'évolution[END_REF]).

Recall on variational solutions of (HJ).

Let us denote by Λ 0 the graph of du 0 and call it the initial submanifold. In fact, the following construction can be made for any Lagrangian submanifold Λ 0 ⊂ R 2n . We consider Σ = Ĥ-1 ({0}) ⊂ R 2+2n . A geometric solution of (HJ) is a Lagrangian submanifold L that satisfies Λ 0 L Σ. For example, the graph of the differential of a smooth function u is a geometric solution if and only if u itself is solution of (HJ).

With the help of the flow φ t Ĥ , we can construct a geometrical solution L H = t∈I φ t Ĥ (Λ 0 ), where I is an open interval containing [0, 1] and such that ρ α = 1 on I. The Lagrangian submanifold L H obtained is an element of L(R 2+2n ).

For any element L ∈ L(R 2k ), we can associate a function u L on R 2k by the following method.

Let S : R k × R q → R be a g.f.q.i of L. Denote by 1 z the fundamental class in H 0 (z). Then, we define u L by u L (z) = c(1 z , S| z×R q ), with notations of section 2. The function u L is everywhere C 0 , and it is proved in [START_REF] Ottolenghi | Solutions généralisées pour l'équation d'Hamilton-Jacobi dans le cas d'évolution[END_REF], that u L is C k on a dense open set, for k 1. Moreover, when it is defined, we have (x, du L (x)) ∈ L. Therefore, the function u L H is a solution of (HJ) on any open set on which it is smooth.

We are now going to prove an interesting property of the elements of Ham γ 2 , which is the fact that we can extend to them the construction of a variational solution of (HJ). In particular, any Hamiltonian function in F ∞ (see definition 5.1) admits a unique generalized variational solution.

Extension to the completion

Proof.

Let (H k ) ∈ Ham be a Cauchy sequence for γ 2 representing an element H ∈ Ham γ 2 . Then, proposition 6.1 implies that (u L H k ) is a Cauchy sequence in C 0 and hence converges to a continuous function u. Moreover, if (H k ) and (F k ) are two equivalent Cauchy sequences for γ 2 , then proposition 6.1 also implies that (u L H k ) and (u L F k ) are equivalent, and hence converge to the same limit. It gives the existence and the unicity. The continuity of the map Ham γ 2 → C 0 is also an immediate consequence of Proposition 6.1.

To prove proposition 6.1, we first prove the following lemma: Lemma 6.4. For any L ∈ L, we have u L C 0 γ(L).

i λ : H * (S λ , S -∞ ) → H * (S λ X , S -∞ X ). The naturality of Thom isomorphism and the fact that all different inclusions commute make the following diagram commutative. Lemma A.3. Let W be a coisotropic linear subspace of R 2n . Denote by N the zero section of R 2n = T * R n . Then there exists a decomposition in linear isotropic subspaces

H * (B) T ---→ H * (S ∞ X , S -∞ X ) j * X,λ ---→ H * (S λ X , S -∞ X ) i *     i∞   i λ
R 2n = N 1 ⊕ V 1 ⊕ N 2 ⊕ V 2 ⊕ N 3 ⊕ V 3 , where N = N 1 ⊕ N 2 ⊕ N 3 and each N i ⊕ V i , i = 1, 2, 3 is a symplectic subspace, such that W = N 1 ⊕ V 1 ⊕ N 2 ⊕ V 3 .
Proof. Let us first recall that if W is coisotropic with symplectic orthogonal W ω ⊂ W , any subspace F such that F ⊕ W ω = W is symplectic. Indeed, since

F ⊂ W , F ∩ F ω = F ∩ F ω ∩ W = F ∩ (F ⊕ W ω ) ω = F ∩ W ω = {0}.
where I H (s, t, x) = ρ α (t)sH(st, x(s))ρ ′ α (t)t s 0 H(σt, x(σ))dσ and x(s) is solution of ẋ(s) = ρ α (t)tX H (st, x(s)). For t ∈ [-1, 1] and α 1, it gives x(s) = φ ts (x).

Similarly as above, we set J(s, t, x) = I H (s, t, x) + I K (-s, t + s, (φ H ) ts (x)), the set U = [-1, 1] × R × R 2 × R 2n × R 2n and the identification Ψ : R 4+4n → T * ∆ 2+2n . The graph can be written this way: Γ φ -s Ǩ φ s Ȟ ∩ Ψ(U) = {(t, J(s, t, x), τ + 1 2 J(s, t, x), 0, z(x))

| (t, τ, x) ∈ [0, 1] × R × R 2n , z(x) ∈ Γ φ -st K φ st H }.
Now, we see that Γ φ -t

K φ t H
is the reduction of Γ φ Ǩ -1 φ Ȟ by the coisotropic manifold W = {t} × R × {0} × R × R 4n . Using lemma A.2 twice, we conclude that for all t ∈ [0, 1], γ(φ -t K φ t H ) γ(φ -1 Ǩ φ Ȟ ).

A.3 Proof of lemma 2.7.

It is sufficient to show that c(V ) c(R 2 × V ) for all open subset V ∈ R 2n . Let H be a Hamiltonian function with support in V . We just have to find a Hamiltonian function K with support in V × R 2 satisfying the inequality c + (H) c + (K). If we set K = Ȟα for α large enough, K has support in R 2 × V , and we saw in particular in the previous proof that Γ φ 1 H is the reduction of Γ φ Ȟ . Therefore, by the remark that ends section A.1, we have c + (H) c + (K) as required.

Proposition 3 . 2 .

 32 Let H be a Hamiltonian on R 2n with compact support. If U is an open subset of R 2n , such that c(U) ε and |H(t, x)| ε for all t ∈ [0, 1] and all x ∈ R 2n -U, then γ(φ H ) 4ε.

Proposition 3 . 7 .

 37 Let U be a closed submanifold of R 2n whose dimension d verifies d n -2. Then ξ ∞ (U) = 0.

Lemma 4 .

 4 [START_REF] Hofer | Symplectic invariants and Hamiltonian dynamics[END_REF]. a. Let (φ n ) be a sequence in H converging to a Hamiltonian diffeomorphism φ, with respect to γ or γ. Assume that there exists a set U ∈ R 2n such that supp(φ n ) ⊂ U. Then supp(φ) ⊂ U .b. Let (H n ) be a sequence in Ham converging to a smooth Hamiltonian function H, with respect to γ u , γu , γ, γ, etc. Assume that there exists a set U ∈ R 2n such that supp(H n ) ⊂ U. Then supp(H) ⊂ U .

Definition 4 .

 4 6. a. Let ψ be an element of H γ or H γ . Then we define support(ψ) as {U | U open set, such that there exists (ψ n ) representing ψ such that ∀n, supp(ψ n ) ⊂ U}, where "supp" denotes the usual notions of support for Hamiltonian diffeomorphisms. b. Let K be an element of Ham γu , Ham γu , Ham γ , Ham γ or Ham γ 2 . Then we define support(K) as {U | U open set, such that there exists (K n ) representing K such that ∀n, supp(K n ) ⊂ U}, where "supp" denotes the usual notions of support for smooth Hamiltonians. These new notions of support coincide with the usual notions for smooth Hamiltonians and Hamiltonian diffeomorphisms. Indeed, let η be either a Hamiltonian diffeomorphism viewed as an element of H γ or H γ , or a smooth Hamiltonian seen as an element of Ham γu , Ham γu , Ham γ , Ham γ or Ham γ 2 . Let (η n ) be a sequence representing η, and U an open set with supp(η n ) ⊂ U for all n. Then lemma 4.5 gives supp(η) ⊂ U. Hence supp(η) ⊂ support(η). Conversely, for any neighbourhood U of supp(η) the constant sequence (η) converges to η and has support in U. Therefore support(η) ⊂ V U , where the intersection is over the set V of all open neighbourhoods of supp(η).

4 k

 4 and γ(H l , H k ) 4 k , after taking limsup with respect to α. It proves that (H k ) is a Cauchy sequence for γu , γ u , γ, and γ.

Proposition 5 . 3 .

 53 The set F ∞ can be embedded into each completion Ham γu ,Hamγu , Ham γ and Ham γ .

H

  * (M) T ---→ H * (S ∞ , S -∞ ) j * λ ---→ H * (S λ , S -∞ ) Suppose now that j * X,λ • T (1) = 0. Then i λ • j * λ • T (1) = j * X,λ • T • i * (1) = j * X,λ • T (1) = 0 hence j * λ • T (1) = 0. That proves c(1, L X ) c(1, L).In the case of L Y , we also have an explicit generating function, constructed as follows. Since R k is contractible we can suppose that the fibers of π do not depend on the second coordinate of M. Denote by i : B ≃ B × {0} → E the inclusion and by τ : B × R k → B the trivial bundle of rank k over B. Consider the vector bundle over B, ρ = τ ⊕ i * π whose total space isF = π -1 (B × {0}) × R n . Then, the function S Y , defined for all v ∈ B and (x, ξ) ∈ ρ -1 (v) by S Y (v; x, ξ) = S(v, x; ξ) is a g.f.q.i for L Y . The map f : E → F, (v, x; ξ) → (v; x, ξ) is a diffeomorphism and satisfies S Y •f = S. Therefore, we have S λ Y = f (S λ ), an isomorphism H * (S λ , S -∞ ) ≃ H * (S λ Y , S -∞ Y ) and a commutative diagram H * (B) T ---→ H * (S ∞ Y , S -∞ Y ) → H * (S ∞ , S -∞ ) j * λ ---→ H * (S λ , S -∞ )The previous argument gives c(1, L Y ) c(1, L). The reverse inequality is obtained from the same diagram with the inclusion i replaced by the projection p : M → B (which reverses vertical arrows). Finally, c(µ B , L N ) c(µ M , L) = c(µ B , L Y ) is obtained from c(1, L X ) c(1, L) = c(1, L Y ) by Poincaré duality, by noticing that L X = L X and L Y = L Y .

  Monotony) Let ψ 1 and ψ 2 be two Hamiltonians generated by H 1 and H 2 . Suppose that for all (t, x) ∈ R × R 2n , we haveH 1 (t, x) H 2 (t, x). Then, c + (ψ 1 ) c + (ψ 2 ) and c -(ψ 1 ) c -(ψ 2 ).As a consequence, if H is a non-negative Hamiltonian, then c -(φ H ) = 0. If H is in addition non zero, we deduce c + (φ H ) > 0. d) (Continuity) Let H 1 and H 2 be two compactly supported hamiltonians, generating ψ 1 and ψ 2 . Let • be the usual norm on C 0

  Indeed, by construction of the distances, H → φ 1 H is Lipschitz both as a map (Ham, γ u ) → (H, γ) and as a map (Ham, γu ) → (H, γ).

	The inequalities between the different distances, proved in Proposition
	2.13, induce inclusions between the completions which may be summarized
	by the following diagram. Here, H distance d H (which satisfies d H	d H denotes the completion of H for Hofer's γ) and C c the set of continuous (not
	necessarily smooth) Hamiltonians with compact support.
	C c _ Ham γ 2	/ / Ham γ r J J J J J J J J J J $ $ / / Ham	γu	/ / Ham	γu
		H	d H	/ / H
	The sets H metric induced by the natural structures on (H, γ) and (H, γ). Moreover we γ and H γ have a natural structure of group with bi-invariant
	have the following fact:
	Proposition 4.1. The map H → φ 1 H induces Lipschitz maps Ham γu → H

Let us denote respectively H γ , Ham γu , H γ , Ham γu , Ham γ , Ham γ and Ham γ 2 the completions of (H, γ), (Ham, γ u ), (H, γ), (Ham, γu ), (Ham, γ), (Ham, γ) and (Ham, γ 2 ). γ and Ham γu → H γ . Proof. γ / / Ham γ / / H γ As in Proposition 4.1, the map (H, t) → φ t H , Ham × R → H induces maps Ham γu ×R → H γ and Ham γu ×R → H γ . Therefore, any element H in one of the completions Ham γu , Ham γu , Ham γ , Ham γ or Ham γ 2 can be associated a path in either H γ , or H γ . This path of course has the semi-group property. That leads us to the following definition. Definition 4.2. Such a path will be called the generalized Hamiltonian flow generated by H.

  . A consequence of Proposition 4.3 is that we can define what is a Lagrangian submanifold invariant under a generalized flow. That leads us to another question which is: Can we define what is an invariant hypersurface of a generalized flow? A (partial) answer to this question is that we can define what is a first integral of a generalized Hamiltonian flow.

  of the so-called c-commuting Hamiltonians. Let us restate it with our notations.An element in one of the completions Ham In other words, K is a first integral of H if there exists two Cauchy sequences (H n ) and (K n ) representing H and K, such that for all s and t,

	φ s Kn φ t Hn φ -s Kn φ -t Hn c-converges to Id.
	It is proved in

γu , Ham γu , Ham γ , Ham γ and Ham γ 2 will be said autonomous if it can be represented by a Cauchy sequence of time-independent Hamiltonian functions. Definition 4.4. Let H, K be two elements in one of the above completions, generating two respective generalized flows φ t H and φ t K . Then we will say that H and K commute, or that K is a first integral of H if φ s K φ t H φ -s K φ -t H = Id.

  ). Moreover, according to Lemma 5.2, we may construct a Cauchy sequence H k of the form Fk for some Hamiltonians F k . That means that H is actually an element of Ham γ (R 2n ). Inclusions between completions give that it is an element of Ham γ (R 2n ) and Ham γ (R 2n ) too. finally, a similar reasoning using γ 2 instead of γ allows to see H as an element of Ham γ (R 2n ). Now, if we denote by C 0 the set of continuous Hamiltonians that vanish at infinity, we can improve the diagram of section 4.1: Examples in the completions of H In the completions of Ham easy examples was given by continuous Hamiltonian functions. In the completions of H there are no similar result. Indeed, there are no known relation between C 0 -distance and γ. However, we can give concrete examples of elements of the completion of H by Corollary 5.3. Indeed, it implies that the (generalized) flows generated by elements of F ∞ are in both H γ and H γ . Let us give some examples (in their construction, γ can be replaced by γ without any problem). R 2n and H(x) = ∞ i=1 h(2 i |x| 2 ) (the "sky-scrapper" Hamiltonian). Let us see why φ H can be seen as a non-smooth homeomorphism. Lemma 4.5 implies that φ H coincides with φ H k out of B 2 -k . So, we can compute the explicit form of φ. In polar coordinates, we obtain:

	x ∈
	F
	F ∞ ∪ C 0 k K y y s s s s s s s s s s s γ 2 Ham / / Ham γ	( ( / / Ham γ _ / / Ham γu	/ / Ham	γu
	H	d H	/ / H
	γ . We consider a decreasing function h : [0, +∞) → [0, +∞), with support in Example of a non smooth homeomorphism in H [0, 1], and equal to 1 on [0, 3/4]. Then we define H k (x) = k i=1 h(2 i |x| 2 ), for

γ / / H γ 5.2

  If we let s 1 converge to 1, we see that φ is not continuous.

	Questions. The previous examples lead us to natural questions: Are all the γ (R 2 ) homeomorphisms? Conversely, can we see any symplectic elements of H homeomorphism (element of the C 0 closure of symplectic diffeomorphisms in
	the homeomorphisms in general dimension, area-preserving homeomorphisms γ ? in dimension 2) as an element of H This last question is related to Oh's still open question whether his group
	of "Hamiltonian homeomorphisms", called Hameo, equals or not the group
	of symplectic homeomorphisms [9].

2

, 0, 0 .

  Proposition 6.1. Let H and K be two Hamiltonian functions, and u L H , u L K the solution obtained by the above method with the same initial submanifoldΛ 0 . Then, u L Hu L K C 0 γ 2 (H, K).

	That leads us to the following definition.

Definition 6.2. Let H ∈ Ham γ 2 . A continuous function u will be called generalized variational solution of (HJ) for H, if there exists a Cauchy sequence (H k ) in Ham representing H and such that the sequence of solutions (u L H k ) C 0 -converges to u. Therefore, proposition 6.1 implies the following statement: Theorem 6.3. For each initial condition u 0 , any element H in the completion Ham γ 2 admits a unique generalized variational solution u H . Moreover, the so constructed map Ham γ 2 → C 0 is continuous.

Proof. Since L coincides with the zero section out of a compact set, u L has a compact support. It follows that u L C 0 max(u L )min(u L ). We will prove that min(u L ) c(1, L). It will also imply that max(u L ) c(µ, L) by Poincar duality. Indeed, using the fact that c(µ, L) = -c(1, L) and that for all z, µ z = 1 z , we have u L = -u L .

Let z ∈ R k , and S : R k ×R q → R be a g.f.q.i of L ⊂ R 2k . Then, S| {z}×R q is a g.f.q.i. of the reduction of L by the coisotropic submanifold {z}×R k ⊂ R 2k . Therefore, by lemma A.2, we get c(1 z , S| {z}×R q ) c(1, S), for all z and hence min(u L ) c(1, L) as required.

Proof of proposition 6.1. The proposition comes from a sequence of inequalities:

The third inequality comes from the first inequality in proposition 2.13. The second one is proved in [START_REF] Cardin | Commuting Hamiltonians and Hamilton-Jacobi multi-time equations[END_REF]. Finally, the first one comes from the lemma 6.4 above and proposition 3.3 in [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF], which states that for all u, v ∈ H

By exchanging H and K and taking the supremum over (t, x), we obtain

Remark and Question. Joukovskaia proved in [START_REF] Joukovskaa | Singularits de minimax et solutions faibles d'quations aux drives partielles[END_REF] that for Hamiltonian functions that are convex in p, variational solutions of (HJ) coincide with viscosity solutions (These are a notion of weak solution introduced by Crandall and Lions in [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF] that has shown its efficiency in a lot of domains of applications including optimal control and differential games, front propagation problems, finance, image theory.... ). We are tempted to use it together with some convergence result on viscosity solutions, to prove that our generalized variational solution is a viscosity solution. This would give another interpretation of our notion of solution, and since our solution is continuous, it would also give a continuity result on viscosity solutions.

However, since we developed our theory in the context of compactly supported Hamiltonians, we cannot reason on Hamiltonian functions convex in p. That leads us to our question : Can one define a completion with similar properties for a class of Hamiltonian functions convex in p?

A Appendix: Proof of inequalities

In this appendix we prove proposition 2.13 and lemma 2.7. All those inequalities are based on the reduction inequality stated in proposition A.1.

A.1 Inequality between γ and γ.

We first prove the inequality γ γ.

Let ϕ be a Hamiltonian diffeomorphism, and L ∈ L. We wish to show that γ(ϕ(L)-L) γ(ϕ). If we denote by N the zero section of R 2n = T * R n , there exists a Hamiltonian isotopy ψ t such that L = ψ 1 (N). Therefore, we just need to prove γ(ϕ(N)) γ(ϕ). Indeed, if we assume this inequality, then

Let us prove now that γ(ϕ(N)) γ(ϕ). We denote by ∆ p the diagonal in R p × R p , and by Φ the symplectic identification

, and the proof will be achieved if we prove the following proposition.

Proposition A.1 (Reduction Inequality). For every Lagrangian submanifold L in R 2n and every linear coisotropic subspace W of R 2n , we have γ(L) γ(L W ), where L W denotes the image of L by reduction by W .

We first prove the following lemma.

Lemma A.2. Let L be a Lagrangian submanifold in a cotangent bundle of the form

Proof.

We start the proof by showing that c(1, L X ) c(1, L). Let us fix λ ∈ R and consider the inclusion i : B ≃ {0} × B → M. Let S be a g.f.q.i. of L defined on a bundle π : E → M. Then the function S X = S| π -1 (B×{x 0 }) is a generating function for L X . Since S X is a restriction of S, we have an inclusion of the sublevels S λ X ⊂ S λ , which induces a morphism

If there exists a decomposition as in the lemma, then W ω = N 2 ⊕ V 3 . Therefore we set N 2 = W ω ∩ N. Then, we define N 1 as one complementary of N 2 in W ∩ N, and F 1 as one complementary of W ω in W , containing N 1 . By the above remark, F 1 is symplectic, and we can choose V 1 as one Lagrangian complementary of N 1 in F 1 .

Then, we define V 3 as a complementary of N 2 in W ω . Since W ∩ N = N 1 ⊕ N 2 , V 3 ∩ N = 0, and we can define N 3 as a complementary of

Finally, we define F 2 as a complementary of

Then, F 2 is symplectic for a similar reason as F 3 , and we can define V 2 as a Lagrangian complementary of

Proof of proposition A.1. Since the linear symplectic group acts transitively on the set of all pairs of complementary Lagrangian subspaces (see proposition 7.4 in Chapter 1 of [START_REF] Libermann | Geométrie symplectique, Bases théorique de la mécanique, Tome I[END_REF]), and since the space of Lagrangian subspaces which are complementary to the zero section N is path connected, there exists a symplectic isotopy Ψ t of R 2n such that Ψ 0 = Id and that Ψ 1 lets all the elements of N invariant and maps

The reduction of L by W is identified with the reduction of Ψ 1 (L) by Ψ 1 (W ). Therefore, applying twice the lemma A.3, we get γ(L W ) γ(Ψ 1 (L)). But, by proposition 2.6 in [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF], we have γ(L) = γ(Ψ 1 (L)). That concludes the proof of proposition A.1.

Remark. Note that in the end of the previous proof, lemma A.3 also implies c(1, L W ) c(1, L). That will be useful in the proof of lemma 2.7.

A.2 Inequalities involving the "suspended distances".

We now prove the inequality γ u (H, K) γ(H, K), for any H,K Hamiltonian functions. It is sufficient to prove that for all Hamiltonian functions H,K, all s in [0, 1], and all α large enough, γ(φ -s K φ s H ) γ(φ -s K φ s Ĥ ). We will prove that the graph of φ -s K φ s H can be obtained by reduction of the graph of φ -s K φ s Ĥ , and then use proposition A.1.

We denote by Φs the flow at time s of the Hamiltonian Ĥ : (s; t, τ, x) → ρ α (τ )τ + ρ α (t)H(t; x). By direct computation, we get Φs (t, τ, x) = (t(s), τ (s), x(s)), with

] and α is large enough, then ρ α (τ (s)) = 1 and t(s) = t + s. Hence x(s) = (φ H ) t+s t (x). We set

), and J(s, t, x) = I H (s, t, x) + I K (-s, t + s, (φ H ) t+s t (x)). Then, we can write the expression of the composition:

, and its image by the natural identification Ψ : R 4+4n → T * ∆ 2+2n . We get

Consider the coisotropic submanifold W = {0} × R × {0} × R × R 4n . Since τ + 1 2 J(s, t, x) = 0 implies τ ∈ [-M 2 -2M, M 2 + 2M], and since W ⊂ Ψ(U), we see that Γ φ -s K φ s H is obtained from Γ φ -s K φ s Ĥ by reduction by W . By proposition A.1, we get γ( Γ φ -s

and hence γ(φ -s K φ s H ) γ(φ -s K φ s Ĥ ). We are now going to prove γ u (H, K) γ(H, K). The idea of the proof is the same as the previous one: we show that for any s ∈ [0, 1], Γ φ -s K φ s H is obtained by reduction of Γ φ Ǩ -1 φ Ȟ , for α large enough.

Recall that by definition, Ȟ(s; t, τ, x) = ρ α (t)tH(st; x). As above, we compute the flow : φ s Ȟ (t, τ, x) = (t(s), τ (s), x(s)), and we obtain t(s) = t τ (s) = τ + I H (s, t, x)


Nguyen Quoc

On Friction-Induced Instabilities and Vibrations

Some problems of friction-induced instabilities, vibrations and noise emissions are considered in this paper. The presentation is focussed on the possibility of flutter instability of the steady sliding equilibrium of an elastic solid in unilateral contact with a moving obstacle. The system of governing equations is given to obtain the steady sliding equilibrium and to discuss its stability. It is shown that the steady sliding equilibrium is generically unstable by flutter. This instability leads to a dynamic response which can be periodic or chaotic. Principal results of the literature are reported here. In the spirit of Hopf bifurcation, the existence problem of periodic solutions has been discussed in my research team for an academic problem of coaxial cylinders. It has been shown by a semi-analytical approach that an interesting family of these periodic solutions are stick-slip-separation waves propagating on the contact surface in a similar way as Schallamach waves in statics. The transition from a given position near equilibrium to the propagation of waves has been illustrated by a numerical approach using finite element simulations. Some phenomena of nonlinear vibration and noise emission in the daily life, such as brake squeal for example, can be discussed in this spirit.

Elastic Solids in Frictional Contact

Coulomb's Law of Dry Friction

At a contact point of a solid with an obstacle, the relative velocity v = v sv o is by definition the difference between the velocities of the material points of the solid and of the obstacle in contact. The relative velocity v and reaction R can be decomposed into normal and tangential components

v = v T + v N n, R = T + N n, ( 1
)
where n denotes the external normal vector to the obstacle at a contact point, v T is the sliding velocity vector, T the tangential reaction vector and N the normal reaction, N and v N are scalars. The unilateral condition of contact implies that the normal reaction must be non-negative N ≥ 0. When there is contact, Coulomb's law of dry friction states that the friction criterion must be satisfied and that the friction force must have the opposite direction to the sliding velocity,

φ = T -f N ≤ 0, and T = f N, T = -av T , a ≥ 0 if v T = 0, (2)
where f denotes the friction coefficient. In particular, the dissipation by friction is -T v T = f N v T . Coulomb's law has often been interpreted in the literature as a non-associated law since the velocity (v T , v N ) is not a normal to the domain of admissible forces. A second interpretation consists of saying that Coulomb's friction is a standard dissipative law with a state-dependent dissipation potential, cf. [START_REF] Moreau | On unilateral Constraints, Friction and Plasticity[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF]. Indeed, normality law is satisfied by the flux v T and the force T since T and v T are related through a state-dependent dissipation potential D(v T , N)

T = -D, vT with D = f N v T , (3)
where D, vT is understood in the sense of sub-gradient. The set of admissible forces, which is a sphere of radius f N , depends on the present state through the present value of N .

Governing Equations

The simple case of an elastic solid occupying a volume V in the undeformed position is considered. The solid is submitted to given forces and displacements

r d = r d (λ(t)), u d = u d (λ(t)
) respectively on the portions S r , S u of the boundary S, and λ(t) denotes a control parameter defining the loading history. On the complementary part S R , the solid may enter into contact with a moving obstacle h(m, t) < 0 and the non-penetration condition is The mechanical response of the solid are governed by unilateral contact conditions, Coulomb's law and classical equations of elastodynamics. In Lagrange description, the governing equations are

h(x + u(x, t), t) ≥ 0 ∀x ∈ S R . ( 4
)
Div b = ρü, b = W, ∇u ∀x ∈ V, bn s = r d ∀x ∈ S r , bn s = N n + T ∀x ∈ S R , u = u d ∀x ∈ S u , ( 5
)
where W (∇u) and b denote respectively the elastic energy per unit volume and the unsymmetric Piola-Lagrange's stress. In particular, the unknowns (u, N ) must satisfy the local equations

u = u d ∀x ∈ S u , h ≥ 0, N ≥ 0, Nh = 0 ∀x ∈ S R (6)
and the variational inequality

V (∇u * -∇ u) : W, ∇u (∇u)dV - Sr r d • (u * -u)da + V ρü • (u * -u)dV - SR (v * N -v N )N da + SR f N ( v * T -v T )dS ≥ 0 ∀u * ∈ U ca , ( 7
)
where U ca denotes the set of kinematically admissible rates

U ca = {u * |u * = ud on S u } (8)
and v * and v are the relative rates

v N = n u-v oN , v T = u-(n u)n-v oT , v * N = n u * -v oN , v * T = u * -(n u * )n-v oT ( 9
) with n = h, m / h, m and v oN = -h, t / h, m . The system of solid in frictional contact under loads is associated with an energy potential and a dissipation potential: [START_REF] Crolla | Brake noise and vibration: state of art[END_REF] where μ denotes the Lagrange multipler associated with the constraints (4) and N = μ h, m . The variational inequality [START_REF] Chateau | Buckling of elastic structures in unilateral contact with or without friction[END_REF] can be condensed as

E(u, λ, μ) = V W (∇u)dV - Sr r d (λ) • uda - SR μh(x + u)da, D(N, v T ) = SR f N v T da,
(J + E, u )(u * -u) + D(N, v * T ) -D(N, v T ) ≥ 0 ∀u * ∈ U ca , ( 11
)
where J denotes the inertial terms. Some regularizations of the frictional contact problem have been proposed in the literature: -Non-local Coulomb's law, proposed by Duvaut [START_REF] Duvaut | Les inéquations en Mécanique et en Physique[END_REF], in which the local normal reaction N is replaced by its mean value N on an elementary representative surface S re

N = 1 S re Sre N (x) ds.
-Normal compliance law, discussed by Oden and Martins [START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF], Kikuchi and Oden [START_REF] Kikuchi | Contact Problem in Elasticity: a Study of Variational Inequalities and Finite Element Methods[END_REF], Andersson [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF], Klarbring et al. [START_REF] Klarbring | A global existence result for the quasistatic frictional contact problem with normal compliance[END_REF], .... It consists of replacing Signorini's relations of unilateral contact by a relationship giving the normal reaction as a function of the gap h(x + u(x)). For example, nonlinear springs of energy ϕ(h) per unit surface may be added to the system where ϕ(h) is a regular function permitting an approximation of Signorini's conditions. In particular, with ϕ(h) = k 2 h 2 -, then the normal compliance law consists of writing that

N = k h, m h -.
General discussions on the existence of a solution of the quasi-static problem in small deformation have been given in the literature, when there is regularization by normal compliance cf. [START_REF] Andersson | A quasistatic frictional problem with normal compliance[END_REF][START_REF] Klarbring | A global existence result for the quasistatic frictional contact problem with normal compliance[END_REF] and by nonlocal Coulomb's law cf. [START_REF] Cocu | Analysis of an Incremental Formulation for Frictional Contact Problems[END_REF]. In these cases, it has been proved that the existence of at least one solution is ensured when the friction coefficient is small enough.

Divergence and Flutter Instabilities of an Equilibrium

There is flutter (or divergence) instability if, under disturbances, the system will leave the equilibrium position with (or without) growing oscillations. As usual, the possibility of divergence instability can be discussed in a purely static approach for the case of fixed obstacles in the same spirit as in plasticity [START_REF] Chateau | Buckling of elastic structures in unilateral contact with or without friction[END_REF][START_REF] Hill | A general theory of uniqueness and stability in elastic/plastic solids[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF][START_REF] Petryk | A consistent approach to defining stability of plastic deformed processes[END_REF]. The static approach consists of defining the stability as the absence of additional displacement when the load does not vary λ = 0. A stability criterion has been discussed in the literature [START_REF] Chateau | Buckling of elastic structures in unilateral contact with or without friction[END_REF][START_REF] Klarbring | Stability and critical points in large displacement frictionless contact problems[END_REF][START_REF] Martins | Some notes on friction and instabilities[END_REF][START_REF] Mroz | Contact Friction Models and Stability Problems[END_REF][START_REF] Nguyen | Bifurcation and stability in dissipative media (plasticity, friction, fracture)[END_REF]. The condition of positivity

I(u * ) = V ∇u * : L : ∇u * dV + SRc N u * T • C • u * T da + SRc f Ṅ (u * ) u * T da > 0 for all u * = 0 ∈ V o ad , ( 12
)
where Ṅ = Ṅ (u * ) is obtained from the expression N = n s bn s , is a criterion of stability since it ensures the static stability of the considered equilibrium. However, for moving obstacles, the possibility of flutter instabilities are the principal action induced by friction. In particular, the instability by flutter of the steady sliding equilibrium has been recently discussed. For the sake of clarity, the instability of the steady sliding equilibrium of an elastic solid in contact with a moving rigid half-space, in translation motion at a constant velocity w parallel to the free surface, is considered here in small deformation.

Steady Sliding Equilibrium

The steady sliding equilibrium u of the solid must satisfy

V ∇δu : L : ∇udV - Sr r d δu da - SR (δu N N + f N τδu T )da = 0.
This equation leads formally to a system of reduced equations of the form

N = k NN [u N ] + k NT [u T ] + N d , T = f N τ = k T N [u N ] + k T T [u T ] + T d .
The principal unknown u N must satisfy

u N = A[N ] + B, N ≥ 0, u N ≥ 0, Nu N = 0 (13) with A = (k NN -k NT k -1 T T k T N ) -1 (I -f k NT h T N ), h T N [N ] = k -1 T T [N τ], B = (k NN -k NT k -1 T T k T N ) -1 N d -k NT k -1 T T [T d ] . ( 14
)
It is clear that the linear operator A is not symmetric if f = 0:

(N * , A[N ]) = SR N * (x)A[N ](x) dS = (N, A[N * ]). ( 15
)
Thus, a linear complementary problem (LCP) must be considered. In particular, the existence and uniqueness of a steady sliding solution are ensured if A is positive-definite or P-positive [START_REF] Cottle | The linear Complementarity Problem[END_REF][START_REF] Klarbring | A mathematical programming approach to three-dimensional contact problems with friction[END_REF][START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF].

Flutter Instability of the Steady Sliding Equilibrium

The stability of the steady sliding position can be obtained from the study of small perturbed motions. However, the equations of motion cannot be linearized without the assumption of effective contact. Indeed, in the presence of a loose contact, a small perturbed motion is not necessarily governed by linear equations because of the possibility of separation. Under the assumption of an effective contact, if the sliding speed is never zero, the dynamic equations can be written as

V δu • ρüdV + V ∇δu : L : ∇udV + SR N δu N dS + SR f N uT -w uT -w • δu T dS = 0 ∀ δu, δN. ( 16
)
The linearization is then possible for sliding motions. The nature of this particular problem can be better understood in the discretized form. After discretization, the linearized equations for sliding motions are

U * N = 0, M Y Y Ÿ * + K Y Y Y * = f Φ N * + f Φ Ẏ Ẏ * , M NY Ÿ * + K NY Y * = N * ( 17
)
where u = (U N , Y ) and Φ( Ẏ ) is a matrix dependent on the direction of sliding. The general expression u * = e st U with U = (U N = 0, X) then leads to a generalized eigenvalue problem

s 2 (M Y Y -f ΦM NY )X -sf Φ Ẏ X + (K Y Y -f ΦK NY )X = 0. ( 18
)
Thus, the considered equilibrium is asymptotically stable (with respect to sliding motions) if (s) < 0 for all s and unstable if there exists at least one value s such that (s) > 0. This generalized eigenvalue problem can be written as (s 2 M + s C + K) X = 0 with non-symmetric matrices M , K and complex eigenvalues and eigenvectors. This analysis leads to the definition of a critical value f d ≥ 0 such that the considered equilibrium is unstable when f > f d .

Example on the Sliding Contact of Two Elastic Layers

The simple example of the frictional contact of two elastic infinite layers is considered here as an illustrating example. This problem was discussed analytically by Adams [START_REF] Adams | Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction[END_REF], by Martins et al [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF] and by Renardy [START_REF] Renardy | Ill-posedness at the boundary for elastic solids sliding under Coulomb friction[END_REF]. The contact in plane strain with friction of two infinite elastic layers, of thickness h and h * respectively as shown in Figure 2, is considered. The lower face of the bottom layer is maintained fixed in the axes Oxyz. The upper face of the top layer, assumed to be in translation of velocity w in the direction Ox, is compressed to the bottom layer by an implied displacement δ < 0. At the interface y = 0, the contact is assumed to obey Coulomb's law of friction with a constant friction coefficient. The celerities of dilatation and shear waves are first introduced:

c 1 = (λ + 2μ)/ρ, c 2 = μ/ρ, τ = c 2 /c 1 ,
for the top layer and for the bottom layer (superscript *), to write the governing equations for the displacements of the top layer wte 1 + u(xwt, y, t) and of the bottom layer u * (x, y, t) under the form:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (1 -τ 2 ( w c2 ) 2 )u x,xx + τ 2 u x,yy + (1 -τ 2 )u y,xy = τ 2 (u x,tt -2 w c2 u x,tx ), u y,yy + τ 2 (1 -( w c2 ) 2 )u y,xx + (1 -τ 2 )u x,xy = τ 2 (u y,tt -2 w c2 u y,tx ), u * x,xx + τ * 2 u * x,yy + (1 -τ * 2 )u * y,xy = τ * 2 u * x,tt , u * y,yy + τ * 2 u * y,xx + (1 -τ * 2 )u * x,xy = τ * 2 u * y,tt . (19)
Boundary and interface conditions are

u(x -wt, h, t) = 0, u * (x, -h * , t) = 0, u y (x -wt, 0, t) = u * y (x, 0, t), σ yy (x, 0, t) = σ * yy (x, 0, t), σ xy (x, 0, t) = σ * xy (x, 0, t), fσ yy (x, 0, t) = -σ xy (x, 0, t).
The stability of the steady sliding solution can be obtained by a linearization of the dynamic equation under the assumption of sliding perturbed motions near the steady sliding state. These motions are searched for under the form of slip waves of wave-length L = 1/k: u(xwt, y, t) = e 2πkct e 2ikπ(x-wt) X(y), u * (x, y, t) = e 2πkct e 2ikπx X * (y).

The condition of existence of non null displacement modes (X, X * ) requires that the pair c and k must be a root of the following equation:

F (c, k) = ρc 2 2 (A(p, q, kh))(iB(p * , q * , kh * ) + f C(p * , q * , kh * )) + ρ * c * 2 2 (A(p * , q * , kh * ))(iB(p, q, kh) -f C(p, q, kh)) = 0, (20)
where p, q, A, B, C are appropriate functions, [23]

p 2 = 1 + ( c -iw c 2 ) 2 , q 2 = 1 + τ 2 ( c -iw c 2 ) 2 ,
A(p, q, kh) = -4pq(1 + p 2 ) + pq(4 + (1 + p 2 ) 2 ) cosh (2πpkh) cosh (2πqkh)

-((1 + p 2 ) 2 + 4p 2 q 2 ) sinh (2πpkh) sinh (2πqkh), B(p, q, kh) = q(1 -p 2 )(sinh (2πpkh) cosh (2πqkh) -pq cosh (2πpkh) sinh (2πqkh)), C(p, q, kh) = pq(3 + p 2 ) -pq(3 + p 2 ) cosh (2πpkh) cosh (2πqkh) + (2p 2 q 2 + (1 + p 2 )) sinh (2πpkh) sinh (2πqkh).
The case of a rigid top layer is obtained when

c 2 → +∞ F (c, k) = iB(p * q * , kh * ) + f C(p * , q * , kh * ) = 0. ( 21
)
For an elastic half-plane compressed into a moving rigid half-plane, cf. Martins et al. [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF], the results are:

F (c) = iq * (1 -p * 2 ) + f (1 + p * 2 -2p * q * ) = 0.
In the case of two elastic half-planes, cf. Adams [START_REF] Adams | Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction[END_REF], this equation can be written as:

F(c)= ρc 2 2 ((1 + p 2 ) 2 -4pq)(iq * (1 -p * 2 )+ f(1 + p * 2 -2p * q * )) + ρ * c * 2 2 ((1 + p * 2 ) 2 -4p * q * )(iq(1 -p 2 ) -f (1 + p 2 -2pq)) = 0. ( 23
)
It has been established in each case that there exists a critical value f d ≥ 0 such that the steady sliding solution is unstable for f ≥ f d . For example, f d = 0 occurs for the system of two layers of finite depths while the possibility f d > 0 may happen in the sliding contact of half-spaces, cf. [2, 18 -20, 34].

Stick-Slip-Separation Waves

The fact that the steady sliding solution is unstable leads to the study of possible dynamic bifurcations of the sliding contact of solids. In the spirit of Hopf bifurcation [START_REF] Troger | Nonlinear Stability and Bifurcation Theory[END_REF], a periodic response can be expected as an alternative stable response. This possibility has been explored in the example of two coaxial cylinders, [START_REF] Moirot | Etude de la Stabilité d'un Équilibre en Présence du Frottement de Coulomb[END_REF][START_REF] Moirot | An example of stick-slip waves[END_REF][START_REF] Oueslati | Transition vers une onde glissementadhérence-décollement sous contact frottant de Coulomb[END_REF].

The mechanical response in plane strain of a brake-like system composed of an elastic tube, of internal radius R and external radius R * , in frictional contact on its inner surface with a rotating rigid cylinder of radius R + Δ and of angular rotation Ω has been discussed, cf. Fig. 3. The mismatch Δ ≥ 0 is a load parameter controlling the normal contact pressures. This model problem enables us to exhibit the existence of nontrivial periodic solutions in the form of stick-slip or stick-slip-separation waves propagating on the contact surface. The governing equations of the system follow from the kinetic relations, the fundamental law, the linear elastic constitutive equations, and the boundary unilateral contact conditions with Coulomb's friction:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¯ = (∇ū) s , Divσ = γ ü, σ = ν (1 + ν)(1 -2ν) T r(¯ )I + 1 1 + ν ¯ , ūr (ξ, θ, t) = ūθ (ξ, θ, t) = 0, σrr (1, θ, t) = -p(θ, t), σrθ (1, θ, t) = -q(θ, t), ūr ≥ δ, p ≥ 0, p(ū r -δ) = 0, |q| ≤ f p, q(1 -uθ ) -f p|1 -uθ | = 0, ( 24
)
where non-dimensional variables are introduced

ū = u R , σ = σ E , r = r R , γ = ρR 2 Ω 2 E , ξ = R * R , δ = Δ R , t = Ωt, u = dū d t .
The steady sliding solution is given by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ūer = δ 1 ξ 2 -1 ξ 2 r -r , ūeθ = δf 1 ξ 2 -1 ξ 2 r -r 1 + 1 ξ 2 (1 -2ν)
,

p e = δ 1 ξ 2 -1 1 1 + ν ξ 2 + 1 1 -2ν > 0, q e = f p e . ( 25
)
Since closed form dynamical solutions cannot be generated, two complementary approaches has been followed. The first approach is semi-analytical after a reduction to a simpler system of equations. The second approach consists of a numerical simulation by the finite element method and appropriate timeintegrations.

An interesting simplification to the problem is obtained when the displacement is sought in the form

ūr = U (θ, t)F (r), ūθ = V (θ, t)F (r), F(r) = 1 ξ 2 -1 ξ 2 r -r . ( 26
)
In this approximation, the following local equations are obtained from the virtual work equation

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Ü -bU -dV + gU = P, V -aV + dU + hV = Q, P ≥ 0, U -δ ≥ 0, P(U -δ) = 0, |Q| ≤ f P, Q(1 -V ) -f P |1 -V | = 0, (27)
where denotes the derivative with respect to θ and a, b, g, h, d are material and geometry constants. All of them are positive except for the coupling coefficient d.

Finally, only the non-dimensional displacements on the contact surface U (θ, t) and V (θ, t) and the non-dimensional reactions P (θ, t) and Q(θ, t) remain as unknowns in the reduced equations. The steady sliding solution, given by U e = δ, V e = δf g/h, P = P e and Q e = f P e , is unstable for the reduced system. Indeed, under the assumption of sliding motions, a small perturbed motion is described by U = U e , V = V e + V * , P = P e + P * and

Q = Q e + Q * . It follows that V * -aV * + f dV * + hV * = 0. ( 28
)
If a general solution is sought in the form V * = e s te ikθ , then -s 2 = ak 2 + h + ikf d. When f = 0, it follows that s = ±iω k with ω 2 k = ak 2 + h. Thus two harmonic waves propagating in opposite senses of the form cos (kθ ± ω k t + ϕ) are obtained as in classical elasticity. When f > 0 and d > 0, then s = ±(s rk + is ik ), s rk > 0, s ik < 0, thus a general solution of the difference V * of the form V * = e ±s rk t cos (kθ ± s ik t + ϕ) is obtained and represents two waves propagating in opposite senses: an exploding wave in the sense of the implied rotation, and a damping wave propagating in the opposite direction. If f > 0 and d < 0, the exploding wave propagates in the opposite sense since the previous expression of s is still valid with s rk > 0 and s ik > 0.

It is expected that in some particular situations, there is a dynamic bifurcation of Poincaré-Andronov-Hopf's type. This means that the perturbed motion may evolve to a periodic response. This transition has been observed numerically in many examples of the literature, cf. for example [START_REF] Oestreich | Bifurcation and stability analysis for a non-smooth friction oscillator[END_REF] or [START_REF] Vola | Friction and instability of steady sliding squeal of a glass/rubber contact[END_REF]. To explore this idea, a periodic solution has been sought in the form of a wave propagating at constant velocity:

U = U (φ), V = V (φ), φ = θ -ct , ( 29
)
where c is the non-dimensional wave velocity, U and V are periodic functions of period T = 2π/k. The physical velocity of the wave is thus c = |c|RΩ and the associated dynamic response is periodic of frequency |c|kΩ. The propagation occurs in the sense of the rotation when c > 0. According to the regime of contact, a slip wave, a stick-slip wave, a slip-separation wave or a stick-slipseparation wave can be discussed. The governing equations of such a wave follow from [START_REF] Moreau | On unilateral Constraints, Friction and Plasticity[END_REF]:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (c 2 -b)U -dV + gU = P, (c 2 -a)V + dU + hV = Q, P ≥ 0, U ≥ δ, P (U -δ) = 0, |Q| ≤ f P, Q(1 -V ) -f P |1 -V | = 0. (30)
The existence of stick-slip waves is obtained when the load is sufficiently strong or when the rotation is slow. For example, for ξ = 1.25 and f = 1, stickpositive slip solutions are obtained for 8 ≤ k ≤ 12. It is found that c must have the sign of d. These waves propagate in the sense of the previous exploding perturbed motions, thus opposite to the rotation of the cylinder when d < 0, with a frequency and a celerity independent of the rotation velocity Ω. The celerity is close to the celerities of dilatation and shear waves in the solid while the frequency is inversely proportional to the radius R. For example, for E/ρ = 1000 m/s, ξ = 1.25, f = 1, R = 1 m and Ω = 100 rad/s, the results obtained concerning the mode-8 wave are Ψ = 0.839, c = 1255 m/s and the associated frequency is 10045 Hz. If ξ = 2, f = 0.3, R = 0.5 m, Ω = 10 rad/s, a frequency 8240 Hz and a celerity 1030 m/s are obtained for k = 4 as shown in Fig. 4. For ξ = 1.15 for example, d is positive and the propagation goes in the rotation direction. The limiting case ξ → 1 can be interpreted as the sliding motion of a rigid plate on an elastic layer or of a rigid half-space on an elastic half-space [START_REF] Adams | Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction[END_REF][START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF]. The obtained solution is a wave with an oscillation about the steady sliding response. The amplitude of the wave is linearly proportional to the rotation Ω. It also increases with the friction coefficient f and decreases with the mismatch. Thus, for vanishing rotations, the steady sliding solution is recovered as the limit of the dynamic response. The stick-slip solution can no longer be available if the rotation is strong enough since the associated pressure may become negative. In the same spirit, for a small mismatch, the pressure may become negative under the assumption of a stick-slip regime everywhere. This means that the possibility of separation is not excluded when the mismatch is not strong enough or if the rotation or the friction coefficient is sufficiently high.

A numerical simulation with an explicit scheme using Lagrange multipliers [START_REF] Baillet | Tribologie de l'interface fibre/matrice, approche théorique et expérimentale[END_REF][START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF] has been performed. The case ξ = 2 and f = 0.3 has been considered. Starting from a motionless initial state, the mismatch displacement is then increased linearly from 0 to its final value. A cyclic limit response is then obtained for large time. The numerical simulation leads to a stick-slip wave in mode 4 without forcing and the obtained response is close to the analytical solution of the reduced approach. It was also checked that a stick-slip-separation wave is effectively obtained when the mismatch is small enough or when the friction is high enough. For example, when Ω = 50 rad/s, δ = 0.001 and f = 0.7, the limit cycle results as a stick-slip-separation wave. The result for radial displacements is shown in Fig. 5.

In fact, it is well known that a periodic response does not systematically result from the flutter instability of the steady sliding solution. It has been observed in various examples of discrete or continuous systems that the response may be quasi-periodic or non-periodic or chaotic, [START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF][START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF][START_REF] Popp | Stick-slip vibrations and chaos[END_REF]. Periodic responses prevail in the example of coaxial cylinders because of the special geometry of the system. Periodic solutions under the form of stick-slip waves have been also obtained by Adams in the sliding contact of two elastic half-spaces [START_REF] Adams | Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip[END_REF]. The generation of dynamic stick-slip-separation waves on the contact surface can be compared to Shallamach waves in the sliding contact of rubber in statics, cf. [3, 5, 17, 19, 20, 31, 35, 36, 39 -41].

Friction-Induced Vibrations and Noises

Figure 6. Some unstable modes of the system pad-disk in an automotive disk brake [START_REF] Moirot | Etude de la Stabilité d'un Équilibre en Présence du Frottement de Coulomb[END_REF][START_REF] Moirot | Some problems of friction-induced vibrations and instabilities[END_REF] It is well known that the presence of friction induces mechanical vibrations and noise emittences in the sliding contact of solids. For example, the creaking noise of a door, the unsteady motion with fits and starts of a windscreenwiper can be interpreted as stick-slip motions resulting from the instability of the steady sliding solution. In particular, the phenomena of squeal [START_REF] Crolla | Brake noise and vibration: state of art[END_REF] have been interpreted in the literature following this approach. The squeals of band brakes in washing machines have been discussed in [START_REF] Nakai | Band brake squeal[END_REF], the squeals of automotive disk brakes has been examined in [START_REF] Moirot | Etude de la Stabilité d'un Équilibre en Présence du Frottement de Coulomb[END_REF][START_REF] Moirot | Brake squeal: a problem of flutter instability of the steady sliding solution[END_REF]. cf. Fig. 6. In the same spirit, the squeal of a system glass-rubber in finite deformation is considered in [START_REF] Vola | Friction and instability of steady sliding squeal of a glass/rubber contact[END_REF].

Figure 1 .

 1 Figure 1. A solid in unilateral contact with an obstacle

Figure 2 .

 2 Figure 2. Sliding contact of two elastic layers

Figure 3 .

 3 Figure 3. The problem of coaxial cylinders in frictional contact.

Figure 4 .

 4 Figure 4. Semi-analytical approach: an example of stick-slip wave in mode 4 with ξ = 2, f = 0.3, Ω = 10 rad/s, R = 0.25 m and δ = 0.005. It is found that Ψ = 0.644, c = 1030 m/s. Phase diagram and variations of V/V e , P/P e and Q/Q e in [0, 2π/k].

Figure 5 .

 5 Figure 5. A mode-4 stick-slip-separation wave, obtained by numerical simulations for ξ = 2, Ω = 50 rad/s, f = 0.7, δ = 0.001. The isovalues of the radial displacement (mm): separation, slip and stick nodes on the contact surface are given respectively in red, green, blue.