Alexandre Jollivet

On inverse problems in electromagnetic field in classical mechanics at fixed energy

In this paper, we consider inverse scattering and inverse boundary value problems at sufficiently large and fixed energy for the multidimensional relativistic and nonrelativistic Newton equations in a static external electromagnetic field (V, B), V ∈ C 2 , B ∈ C 1 in classical mechanics. Developing the approach going back to Gerver-Nadirashvili 1983's work on an inverse problem of mechanics, we obtain, in particular, theorems of uniqueness.

where x = x(t) is a C 1 function with values in Ω, ṗ = dp dt , ẋ = dx dt , and V ∈ C 2 ( Ω, R) (i.e. there exists Ṽ ∈ C 2 (R n , R) such that Ṽ restricted to Ω is equal to V), B ∈ F mag ( Ω) where F mag ( Ω) is the family of magnetic fields on Ω, i.e.

F mag ( Ω) = {B ′ ∈ C 1 ( Ω, A n (R)) | B ′ = (B ′ i,k ), ∂ ∂x i B ′ k,l (x)+ ∂ ∂x l B ′ i,k (x)+ ∂ ∂x k B ′ l,i (x
) = 0, x ∈ D, i, k, l = 1 . . . n} and A n (R) denotes the space of n × n real antisymmetric matrices.

By V C 2 ,Ω we denote the supremum of the set {|∂ j x V (x)| | x ∈ Ω, j = (j 1 , .., j n ) ∈ (N ∪ {0}) n , n i=1 j i ≤ 2} and by B C 1 ,Ω we denote the supremum of the set {|∂ j x B i,k (x)| | x ∈ Ω, i, k = 1 . . . n, j = (j 1 , .., j n ) ∈ (N ∪ {0}) n , n i=1 j i ≤ 1}. The equation (1.1) is an equation for x = x(t) and is the equation of motion in R n of a relativistic particle of mass m = 1 and charge e = 1 in an external electromagnetic field described by V and B (see [E] and, for example, Section 17 of [LL]). In this equation x is the position of the particle, p is its impulse, t is the time and c is the speed of light.

For the equation (1.1) the energy

E = c 2 1 + |p(t)| 2 c 2 + V (x(t)) (1.2)
is an integral of motion. We denote by B c the euclidean open ball whose radius is c and whose centre is 0.

In this paper we consider the equation (1.1) in two situations. We study equation (1.1) when Ω = D where D is a bounded strictly convex in the strong sense open domain of R n , n ≥ 2, with C 2 boundary.

(1.3)

And we study equation (1.1) when

Ω = R n and |∂ j 1 x V (x)| ≤ β |j 1 | (1 + |x|) -α-|j 1 | , x ∈ R n , n ≥ 2, |∂ j 2 x B i,k (x)| ≤ β |j 2 |+1 (1 + |x|) -α-1-|j 2 | , x ∈ R n ,
(1.4)

for |j 1 | ≤ 2, |j 2 | ≤ 1, i, k = 1 . . . n and some α > 1 (here j is the multiindex j ∈ (N ∪ {0}) n , |j| = n i=1 j i and β |j| are positive real constants). For the equation (1.1) under condition (1.3), we consider boundary data. For equation (1.1) under condition (1.4), we consider scattering data.

1.2 Boundary data. For the equation (1.1) under condition (1.3), one can prove that at sufficiently large energy E (i.e. E > E( V C 2 ,D , B C 1 ,D , D)), the solutions x at energy E have the following properties (see Properties 2.1 and 2.2 in Section 2 and see Section 6):

for each solution x(t) there are t 1 , t 2 ∈ R, t 1 < t 2 , such that x ∈ C 3 ([t 1 , t 2 ], R n ), x(t 1 ), x(t 2 ) ∈ ∂D, x(t) ∈ D for t ∈]t 1 , t 2 [, x(s 1 ) = x(s 2 ) for s 1 , s 2 ∈ [t 1 , t 2 ], s 1 = s 2 ;

(1.5) for any two distinct points q 0 , q ∈ D, there is one and only one solution x(t) = x(t, E, q 0 , q) such that x(0) = q 0 , x(s) = q for some s > 0.

(1.6) Let (q 0 , q) be two distinct points of ∂D. By s V,B (E, q 0 , q) we denote the time at which x(t, E, q 0 , q) reaches q from q 0 . By k 0,V,B (E, q 0 , q) we denote the velocity vector ẋ(0, E, q 0 , q). By k V,B (E, q 0 , q) we denote the velocity vector ẋ(s V,B (E, q 0 , q), E, q 0 , q). We consider k 0,V,B (E, q 0 , q), k V,B (E, q 0 , q), q 0 , q ∈ ∂D, q 0 = q, as the boundary value data.

Remark 1.1. For q 0 , q ∈ ∂D, q 0 = q, the trajectory of x(t, E, q 0 , q) and the trajectory of x(t, E, q, q 0 ) are distinct, in general.

Note that in the present paper we always assume that the aforementioned real constant E( V C 2 ,D , B C 1 ,D , D), considered as function of V C 2 ,D and B C 1 ,D , satisfies 1.3 Scattering data. For the equation (1.1) under condition (1.4), the following is valid (see [Y]): for any (v -, x -) ∈ B c × R n , v -= 0, the equation (1.1) has a unique solution x ∈ C 2 (R, R n ) such that

E(λ 1 , λ 2 , D) ≤ E(λ ′ 1 , λ ′ 2 , D) if λ 1 ≤ λ ′ 1 and λ 2 ≤ λ ′ 2 , (1.7) for λ 1 , λ 2 , λ ′ 1 , λ ′ 2 ∈ [0, +∞[.
x(t) = v -t + x -+ y -(t), (1.8)
where ẏ-(t) → 0, y -(t) → 0, as t → -∞; in addition for almost any

(v -, x -) ∈ B c × R n , v -= 0,
x(t) = v + t + x + + y + (t), (1.9)

where v + = 0, |v + | < c, v + = a(v -, x -), x + = b(v -, x -), ẏ+ (t) → 0, y + (t) → 0, as t → +∞.

For an energy E > c 2 , the map S E : S E × R n → S E × R n (where

S E = {v ∈ B c | |v| = c 1 -c 2 E 2 })
given by the formulas 1.4 Inverse scattering and boundary value problems. In the present paper, we consider the following inverse boundary value problem at fixed energy for the equation (1.1) under condition (1.3):

v + = a(v -, x -), x + = b(v -, x -), ( 1
Problem 1 : given k V,B (E, q 0 , q), k 0,V,B (E, q 0 , q) for all q 0 , q ∈ ∂D, q 0 = q, at fixed sufficiently large energy E, find V and B.

The main results of the present work include the following theorem of uniqueness for Problem 1.

Theorem 1.1. At fixed E > E( V C 2 ,D , B C 1 ,D , D), the boundary data k V,B (E, q 0 , q), (q 0 , q) ∈ ∂D × ∂D, q 0 = q, uniquely determine V, B.

At fixed E > E( V C 2 ,D , B C 1 ,D , D), the boundary data k 0,V,B (E, q 0 , q), (q 0 , q) ∈ ∂D × ∂D, q 0 = q, uniquely determine V, B.

Theorem 1.1 follows from Theorem 3.1 given in Section 3. In the present paper, we also consider the following inverse scattering problem at fixed energy for the equation (1.1) under condition (1.4):

Problem 2 : given S E at fixed energy E, find V and B.

The main results of the present work include the following theorem of uniqueness for Problem 2.

Theorem 1.2. Let λ ∈ R + and let D be a bounded strictly convex in the strong sense open domain of

R n with C 2 boundary. Let V 1 , V 2 ∈ C 2 0 (R n , R), B 1 , B 2 ∈ C 1 0 (R n , A n (R)) ∩ F mag (R n ), max( V 1 C 2 ,D , V 2 C 2 ,D , B 1 C 1 ,D , B 2 C 1 ,D ) ≤ λ, and supp(V 1 ) ∪ supp(V 2 ) ∪supp(B 1 ) ∪ supp(B 2 ) ⊆ D. Let S µ E
be the scattering map at fixed energy E subordinate to (V µ , B µ ) for µ = 1, 2. Then there exists a nonnegative real constant E(λ, D) such that for any E > E(λ, D), (V 1 , B 1 ) ≡ (V 2 , B 2 ) if and only if S 1 E ≡ S 2 E . Theorem 1.2 follows from Theorem 1.1, (1.7) and Proposition 2.1 of Section 2.

Remark 1.2. Note that for V ∈ C 2 0 (R n , R), if E < c 2 + sup{V (x) | x ∈ R n } then S E does not determine uniquely V.

Remark 1.3. Theorems 1.1 and 1.2 give uniqueness results. In this paper we do not prove and do not obtain stability results for Problem 1 and for Problem 2.

1.5 Historical remarks. An inverse boundary value problem at fixed energy and at high energies was studied in [GN] for the multidimensional nonrelativistic Newton equation (without magnetic field) in a bounded open strictly convex domain. In [GN] results of uniqueness and stability for the inverse boundary value problem at fixed energy are derived from results for the problem of determining an isotropic Riemannian metric from its hodograph (for this geometrical problem, see [MR], [B] and [BG]).

Novikov [N2] studied inverse scattering for nonrelativistic multidimensional Newton equation (without magnetic field). Novikov [N2] gave, in particular, a connection between the inverse scattering problem at fixed energy and Gerver-Nadirashvili's inverse boundary value problem at fixed energy. Developing the approach of [GN] and [N2], the author [J3] studied an inverse boundary problem and inverse scattering problem for the multidimensional relativistic Newton equation (without magnetic field) at fixed energy. In [J3] results of uniqueness and stability are obtained.

Inverse scattering at high energies for the relativistic multidimensional Newton equation was studied by the author (see [J1], [J2]).

As regards analogs of Theorems 1.1, 1.2 and Proposition 2.1 for the case B ≡ 0 for nonrelativistic quantum mechanics see [N1], [NSU], [N3] and further references therein. As regards an analog of Theorem 1.2 for the case B ≡ 0 for relativistic quantum mechanics see [I]. As regards analogs of Theorems 1.1, 1.2 for the case B ≡ 0 for nonrelativistic quantum mechanics, see [ER], [NaSuU] and further references given therein.

As regards results given in the literature on inverse scattering in quantum mechanics at high energy limit see references given in [J2].

1.6 Structure of the paper. The paper is organized as follows. In Section 2, we give some properties of boundary data and scattering data and we connect the inverse scattering problem at fixed energy to the inverse boundary value problem at fixed energy. In Section 3, we give, actually, a proof of Theorem 1.1 (based on Theorem 3.1 formulated in Section 3). Section 4 is devoted to the proof of Lemma 3.1 and Theorem 3.1 formulated in Section 3. Section 5 is devoted to the proof of Lemma 2.1 and Proposition 3.1 formulated in Section 2 and in Section 3. Section 6 is devoted to the proof of Properties (2.1) and (2.2). In Section 7, we give results similar to Theorems 1.1, 1.2 for the nonrelativistic case.

Acknowledgement. This work was fulfilled in the framework of Ph. D. thesis research under the direction of R.G. Novikov. for each solution x(t) there are

Scattering data and boundary value data

t 1 , t 2 ∈ R, t 1 < t 2 , such that x ∈ C 3 ([t 1 , t 2 ], R n ), x(t 1 ), x(t 2 ) ∈ ∂D, x(t) ∈ D for t ∈]t 1 , t 2 [, x(s 1 ) = x(s 2 ) for s 1 , s 2 ∈ [t 1 , t 2 ], s 1 = s 2 , ẋ(t 1 ) • N(x(t 1 )) < 0 and ẋ(t 2 ) • N(x(t 2 )) > 0, where N(x(t i ))
is the unit outward normal vector of ∂D at x(t i ) for i = 1, 2;

(2.1) for any two points q 0 , q ∈ D, q = q 0 , there is one and only one solution x(t) = x(t, E, q 0 , q) such that x(0) = q 0 , x(s) = q for some s > 0; ẋ(0, E, q 0 , q) ∈ C 1 (( D × D)\ Ḡ, R n ), where Ḡ is the diagonal in D × D;

(2.2) where • denotes the usual scalar product on R n (and where by " ẋ(0, E, q 0 , q)

∈ C 1 (( D × D)\ Ḡ, R n )" we mean that ẋ(0, E, q 0 , q) is the restriction to ( D × D)\ Ḡ of a function which belongs to C 1 ((R n × R n )\∆) where ∆ is the diag- onal of R n × R n ). Remark 2.1. If B ∈ C 1 ( D, A n (R)) and B ∈ F mag ( D) (where A n (R)
denotes the space of n × n real antisymmetric matrices), then at fixed energy

E > E( V C 2 ,D , B C 1 ,D , D) solutions x(t) of equation (1.1) under condition (1.
3) also have properties (2.1), (2.2) (see Section 6).

We remind that the aforementioned real constant

E( V C 2 ,D , B C 1 ,D , D), considered as function of V C 2 ,D and B C 1 ,D , satisfies (1.7). In addi- tion, real constant E( V C 2 ,D , B C 1 ,D , D) has the following property: for any C 2 continuation Ṽ of V on R n , and for any B ∈ C 1 (R n , A n (R)) such that B ≡ B on D, one has E( Ṽ C 2 ,Dx 0 ,ε , B C 1 ,Dx 0 ,ε , D x 0 ,ε ) → E( V C 2 ,D , B C 1 ,D , D), as ε → 0, (2.3) where D x 0 ,ε = {x 0 + (1 + ε)(x -x 0 ) | x ∈ D} for any x 0 ∈ D and ε > 0 (note that D x 0 ,ε is a bounded, open, strictly convex (in the strong sense) domain of R n with C 2 boundary). Let E > E( V C 2 ,D , B C 1 ,D , D).
Consider the solution x(t, E, q 0 , q) from (2.2) for q 0 , q ∈ D, q 0 = q. We define vectors k V,B (E, q 0 , q) and k 0,V,B (E, q 0 , q) by k V,B (E, q 0 , q) = ẋ(s V,B (E, q 0 , q), E, q 0 , q), k 0,V,B (E, q 0 , q) = ẋ(0, E, q 0 , q), where we define s = s V,B (E, q 0 , q) as the root of the equation x(s, E, q 0 , q) = q, s > 0.

For q 0 = q ∈ D, we put s V,B (E, q 0 , q) = 0.

Note that

|k 0,V,B (E, q 0 , q)| = c 1 -E-V (q 0 ) c 2 -2 , |k V,B (E, q 0 , q)| = c 1 -E-V (q) c 2 -2 , (2.4) for (q, q 0 ) ∈ ( D × D)\ Ḡ.
Using Properties (2.1) and (2.2), we obtain

Lemma 2.1. At fixed E > E( V C 2 ,D , B C 1 ,D , D), we have that s V,B (E, q 0 , q) ∈ C( D × D, R), s V,B (E, q 0 , q) ∈ C 1 (( D × D)\ Ḡ, R) and k V,B (E, q 0 , q) ∈ C 1 (( D × D)\ Ḡ, R n ).
We consider k V,B (E, q 0 , q), k 0,V,B (E, q 0 , q), q 0 , q ∈ ∂D, q 0 = q as the boundary value data.

Remark 2.2. Note that if x(t) is solution of (1.1) under condition (1.3), then x(-t) is solution of (1.1) with B replaced by -B ∈ F mag ( D) under condition (1.3). Hence the following equalities are valid:

at fixed E > E( V C 2 ,D , B C 1 ,D , D), k 0,V,B (E, q 0 , q) = -k V,-B (E, q, q 0 ), (2.5) k V,B (E, q 0 , q) = -k 0,V,-B (E, q, q 0 ), (2.6) s V,B (E, q 0 , q) = s V,-B (E, q, q 0 ),
(2.7) for q 0 , q ∈ D, q 0 = q.

2.2 Properties of the scattering operator. For equation (1.1) under condition (1.4), the following is valid (see [Y]): for any

(v -, x -) ∈ B c × R n , v -= 0, the equation (1.1) under condition (1.4) has a unique solution x ∈ C 2 (R, R n ) such that x(t) = v -t + x -+ y -(t), (2.8) where ẏ-(t) → 0, y -(t) → 0, as t → -∞; in addition for almost any (v -, x -) ∈ B c × R n , v -= 0, x(t) = v + t + x + + y + (t), (2.9)
where

v + = 0, |v + | < c, v + = a(v -, x -), x + = b(v -, x -), ẏ+ (t) → 0, y + (t) → 0, as t → +∞. The map S : B c × R n → B c × R n
given by the formulas

v + = a(v -, x -), x + = b(v -, x -) (2.
(v -, x -) ∈ D(S), then v -= 0 and a(v -, x -) = 0).
The map S has the following simple properties (see [Y]): (S) is continuous and preserves the element of volume; for any

D(S) is an open set of B c × R n and Mes((B c × R n )\D(S)) = 0 for the Lebesgue measure on B c × R n induced by the Lebesgue measure on R n × R n ; the map S : D(S) → R
(v, x) ∈ D(S), a(v, x) 2 = v 2 .
For E > c 2 , the map S restricted to

Σ E = {(v -, x -) ∈ B c × R n | |v -| = c 1 - c 2 E 2 }
is the scattering operator at fixed energy E and is denoted by S E .

We will use the fact that the map S is uniquely determined by its restriction to M(S) = D(S) ∩ M, where

M = {(v -, x -) ∈ B c × R n |v -= 0, v -x -= 0}.
This observation is based on the fact that if x(t) satisfies (1.1), then x(t + t 0 ) also satisfies (1.1) for any t 0 ∈ R. In particular, the map S at fixed energy E is uniquely determined by its restriction to M E (S) = D(S) ∩ M E , where

M E = Σ E ∩ M.

2.3

Relation between scattering data and boundary value data. Assume that

V ∈ C 2 0 ( D, R), B ∈ C 1 0 ( D, A n (R)), B ∈ F mag ( D).
(2.11)

We consider equation (1.1) under condition (1.3) and equation (1.1) under condition (1.4). We shall connect the boundary value data k V,B (E, q 0 , q), k 0,V,B (E, q 0 , q) for E > E( V C 2 ,D , B C 1 ,D , D) and (q, q 0 ) ∈ (∂D×∂D)\∂G, to the scattering data a, b.

Proposition 2.1. Let E > E( V C 2 ,D , B C 1 ,D , D).
Under condition (2.11), the following statement is valid: s V,B (E, q 0 , q), k V,B (E, q 0 , q), k 0,V,B (E, q 0 , q) given for all (q, q 0 ) ∈ (∂D × ∂D)\∂G, are determined uniquely by the scattering data a(v -, x -), b(v -, x -) given for all (v -, x -) ∈ M E (S). The converse statement holds: s V,B (E, q 0 , q), k V,B (E, q 0 , q), k 0,V,B (E, q 0 , q) given for all (q, q 0 ) ∈ (∂D×∂D)\∂G, determine uniquely the scattering data a(v -, x -), b(v -, x -) for all (v -, x -) ∈ M E (S).

Proof of Proposition 2.1. First of all we introduce functions χ, τ -and τ + dependent on D.

For (v, x) ∈ R n \{0} × R n , χ(v, x) denotes the nonnegative number of points contained in the intersection of ∂D with the straight line parametrized by

R → R n , t → tv +x. As D is a strictly convex open subset of R n , χ(v, x) ≤ 2 for all v, x ∈ R n , v = 0. Let (v, x) ∈ R n \{0} × R n . Assume that χ(v, x) ≥ 1. The real τ -(v, x) denotes the smallest real number t such that τ -(v, x)v + x ∈ ∂D, and the real τ + (v, x) denotes the greatest real number t such that τ + (v, x)v + x ∈ ∂D (if χ(v, x) = 1 then τ -(v, x) = τ + (v, x)).
Direct statement. Let (q 0 , q) ∈ (∂D × ∂D)\∂G. Under conditions (2.11) and from (2.1) and (2.2), it follows that there exists an unique (v -, x -) ∈ M E (S) such that

χ(v -, x -) = 2, q 0 = x -+ τ -(v -, x -)v -, q = b(v -, x -) + τ + (a(v -, x -), b(v -, x -))a(v -, x -). In addition, s V,B (E, q 0 , q) = τ + (a(v -, x -), b(v -, x -))-τ -(v -, x -) and k V,B (E, q 0 , q) = a(v -, x -) and k 0,V,B (E, q 0 , q) = v -. Converse statement. Let (v -, x -) ∈ M E (S). Under conditions (2.11), if χ(v -, x -) ≤ 1 then (a(v -, x -), b(v -, x -)) = (v -, x -).
Assume that χ(v -, x -) = 2. Let

q 0 = x -+ τ -(v -, x -)v -.
From (2.1) and (2.2) it follows that there is one and only one solution of the equation k 0,V,B (E, q 0 , q) = v -, q ∈ ∂D, q = q 0 .

(2.12)

We denote by q(v -, x -) the unique solution of (2.12). Hence we obtain

a(v -, x -) = k V,B (E, q 0 , q(v -, x -)), b(v -, x -) = q(v -, x -) -k V,B (E, q 0 , q(v -, x -))(s V,B (E, q 0 , q(v -, x -)) +τ -(v -, x -)).
Proposition 2.1 is proved.

For a more complete discussion about connection between scattering data and boundary value data, see [N2] considering the nonrelativistic Newton equation (without magnetic field).

Inverse boundary value problem

In this Section, Problem 1 of Introduction is studied.

3.1 Notations. For x ∈ D, and for E > V (x) + c 2 , we define

r V,E (x) = c E -V (x) c 2 2 -1. At E > E( V C 2 ,D , B C 1 ,D ,
D) for q 0 , q ∈ D × D, q 0 = q, we define the vectors kV,B (E, q 0 , q) and k0,V,B (E, q 0 , q) by kV,B (E, q 0 , q) = k V,B (E,q 0 ,q)

1- |k V,B (E,q 0 ,q)| 2 c 2 , k0,V,B (E, q 0 , q) = k 0,V,B (E,q 0 ,q) 1- |k 0,V,B (E,q 0 ,q)| 2 c 2 . (3.1)
and kV,B (E, q 0 , q) = ( k1 V,B (E, q 0 , q), . . . , kn V,B (E, q 0 , q)), k0,V,B (E, q 0 , q) = ( k1 0,V,B (E, q 0 , q), . . . , kn 0,V,B (E, q 0 , q)). Note that from (2.4), it follows that

| kV,B (E, q 0 , q)| = r V,E (q), | k0,V,B (E, q 0 , q)| = r V,E (q 0 ). (3.2)
For B ∈ F mag ( D), let F pot (D, B) be the set of C 1 magnetic potentials for the magnetic field B, i.e.

F pot (D, B) := {A ∈ C 1 ( D, R n ) | B i,k (x) = ∂ ∂x i A k (x) -∂ ∂x k A i (x), x ∈ D, i, k = 1 . . . n}.
(3.3) (The set F pot (D, B) is not empty: take, for example, A(x) = -1 0 sB(x 0 + s(xx 0 ))(xx 0 )ds, for x ∈ D and some fixed point x 0 of D.)

3.2 Hamiltonian mechanics. Let A ∈ F pot (D, B). The equation (1.1) in D is the Euler-Lagrange equation for the Lagrangian L defined by L( ẋ, x) = -c 2 1 - ẋ2 c 2 + c -1 A(x) • ẋ -V (x), ẋ ∈ B c
and x ∈ D, where • denotes the usual scalar product on R n . The Hamiltonian H associated to the Lagrangian L by Legendre's transform (with respect to ẋ) is

H(P, x) = c 2 (1 + c -2 |P - c -1 A(x)| 2 )
1/2 +V (x) where P ∈ R n and x ∈ D. Then equation (1.1) in D is equivalent to the Hamilton's equation ẋ = ∂H ∂P (P, x), Ṗ = -∂H ∂x (P, x),

(3.4) for P ∈ R n , x ∈ D.
For a solution x(t) of equation (1.1) in D, we define the impulse vector

P (t) = ẋ(t) 1 -| ẋ(t)| 2 c 2 + c -1 A(x(t)).
Further for q 0 , q ∈ D, q 0 = q, and t ∈ [0, s(E, q 0 , q)], we consider

P (t, E, q 0 , q) = ẋ(t, E, q 0 , q) 1 -| ẋ(t,E,q 0 ,q)| 2 c 2 + c -1 A(x(t, E, q 0 , q)), (3.5)
where x(., E, q 0 , q) is the solution given by (2.2). From Maupertuis's principle (see [A]), it follows that if x(t), t ∈ [t 1 , t 2 ], is a solution of (1.1) in D with energy E, then x(t) is a critical point of the functional A(y) =

t 2 t 1 [r V,E (y(t))| ẏ(t)| + c -1 A(y(t)) • ẏ(t)
] dt defined on the set of the functions y ∈ C 1 ([t 1 , t 2 ], D), with boundary conditions y(t 1 ) = x(t 1 ) and y(t 2 ) = x(t 2 ). Note that for q 0 , q ∈ D, q 0 = q, functional A taken along the trajectory of the solution x(., E, q 0 , q) given by (2.2) is equal to the reduced action S 0 V,A,E (q 0 , q) from q 0 to q at fixed energy E for (3.4), where

S 0 V,A,E (q 0 , q) = 0, if q 0 = q,
s(E,q 0 ,q) 0 P (s, E, q 0 , q) • ẋ(s, E, q 0 , q)ds, if q 0 = q, (3.6) for q 0 , q ∈ D.

3.3 Properties of S 0 V,A,E at fixed and sufficiently large energy E. The following Propositions 3.1, 3.2 give properties of S 0 V,A,E at fixed and sufficiently large energy E.

Proposition 3.1. Let E > E( V C 2 ,D , B C 1 ,D , D).
The following statements are valid:

S 0 V,A,E ∈ C( D × D, R), (3.7) S 0 V,A,E ∈ C 2 (( D × D)\ Ḡ, R), (3.8) ∂S 0 V,A,E ∂x i (ζ, x) = ki V,B (E, ζ, x) + c -1 A i (x), (3.9) ∂S 0 V,A,E ∂ζ i (ζ, x) = -ki 0,V,B (E, ζ, x) -c -1 A i (ζ), (3.10) ∂ 2 S 0 V,A,E ∂ζ i ∂x j (ζ, x) = - ∂ ki 0,V,B ∂x j (E, ζ, x) = ∂ kj V,B ∂ζ i (E, ζ, x), (3.11) for (ζ, x) ∈ ( D × D)\ Ḡ, ζ = (ζ 1 , .., ζ n ), x = (x 1 , .
., x n ), and i, j = 1 . . . n. In addition,

max(| ∂S 0 V,A,E ∂x i (ζ, x)|, | ∂S 0 V,A,E ∂ζ i (ζ, x)|) ≤ M 1 , (3.12) | ∂ 2 S 0 V,A,E ∂ζ i ∂x j (ζ, x)| ≤ M 2 |ζ -x| , (3.13) for (ζ, x) ∈ ( D × D)\ Ḡ, ζ = (ζ 1 , .., ζ n ), x = (x 1 , .
., x n ), and i, j = 1 . . . n, and where M 1 and M 2 depend on V, B and D.

Proposition 3.1 is proved in Section 5. Equalities (3.9) and (3.10) are known formulas of classical Hamiltonian mechanics (see Section 46 and further Sections of [A]). (3.14) has the following properties:

Proposition 3.2. Let E > E( V C 2 ,D , B C 1 ,D , D). The map ν V,B,E : ∂D × D → S n-1 , defined by ν V,B,E (ζ, x) = - k V,B (E, ζ, x) |k V,B (E, ζ, x)| , for (ζ, x) ∈ ∂D × D,
ν V,B,E ∈ C 1 (∂D × D, S n-1 ), the map ν V,B,E,x : ∂D → S n-1 , ζ → ν V,B,E (ζ, x), is a C 1 orientation preserving diffeomorphism from ∂D onto S n-1 (3.15)
for x ∈ D (where we choose the canonical orientation of S n-1 and the orientation of ∂D given by the canonical orientation of R n and the unit outward normal vector).

Proposition 3.2 follows from (1.1), (1.2) and properties (2.1), (2.2).

Remark 3.1. Taking account of (3.9) and (3.10), we obtain the following formulas: at

E > E( V C 2 ,D , B C 1 ,D , D), for any x, ζ ∈ D, x = ζ, B i,j (x) = -c ∂ kj V,B ∂x i (E, ζ, x) - ∂ ki V,B ∂x j (E, ζ, x) , (3.16)
B i,j (x) = -c ∂ kj 0,V,B ∂x i (E, x, ζ) - ∂ ki 0,V,B
∂x j (E, x, ζ) .

(3.17)

3.4 Results of uniqueness. We denote by ω 0,V,B the n -1 differential form on ∂D × D obtained in the following manner:

-for x ∈ D, let ω V,B,x be the pull-back of ω 0 by ν V,B,E,x where ω 0 denotes the canonical orientation form on S n-1 (i.e.

ω 0 (ζ)(v 1 , .., v n-1 ) = det(ζ, v 1 , .., v n-1 ), for ζ ∈ S n-1 and v 1 , .., v n-1 ∈ T ζ S n-1 ), -for (ζ, x) ∈ ∂D × D and for any v 1 , .., v n-1 ∈ T (ζ,x) (∂D × D), ω 0,V,B (ζ, x)(v 1 , .., v n-1 ) = ω V,B,x (ζ)(σ ′ (ζ,x) (v 1 ), .., σ ′ (ζ,x) (v n-1 )),
where σ :

∂D × D → ∂D, (ζ ′ , x ′ ) → ζ ′ , and σ ′ (ζ,x) denotes the derivative (linear part) of σ at (ζ, x).
From smoothness of ν V,B,E , σ and ω 0 , it follows that ω 0,V,B is a continuous n -1 form on ∂D × D.

Now let λ ∈ R + and V 1 , V 2 ∈ C 2 ( D, R), B 1 , B 2 ∈ F mag ( D), such that max( V 1 C 2 ,D , V 2 C 2 ,D , B 1 C 1 ,D , B 2 C 1 ,D ) ≤ λ. For µ = 1, 2, let A µ ∈ F pot (D, B µ ). Let E > E(λ, λ, D) where E(λ, λ, D) is defined in (1.7). Consider β 1 , β 2 the differential one forms defined on (∂D × D)\ Ḡ by β µ (ζ, x) = n j=1 kj Vµ,Bµ (E, ζ, x)dx j , (3.18) for (ζ, x) ∈ (∂D × D)\ Ḡ, x = (x 1 , . . . , x n ) and µ = 1, 2. Consider the differential forms Φ 0 on (∂D× D)\ Ḡ and Φ 1 on (∂D× D)\ Ḡ defined by Φ 0 (ζ, x) = -(-1) n(n+1) 2 (β 2 -β 1 )(ζ, x) ∧ d ζ (S 0 V 2 ,A 2 ,E -S 0 V 1 ,A 1 ,E )(ζ, x) ∧ p+q=n-2 (dd ζ S 0 V 1 ,A 1 ,E (ζ, x)) p ∧ (dd ζ S 0 V 2 ,A 2 ,E (ζ, x)) q , (3.19) for (ζ, x) ∈ (∂D × D)\ Ḡ, where d = d ζ + d x , Φ 1 (ζ, x) = -(-1) n(n-1) 2 β 1 (ζ, x) ∧ (dd ζ S 0 V 1 ,A 1 ,E (ζ, x)) n-1 +β 2 (ζ, x) ∧ (dd ζ S 0 V 2 ,A 2 ,E (ζ, x)) n-1 -β 1 (ζ, x) (3.20) ∧(dd ζ S 0 V 2 ,A 2 ,E (ζ, x)) n-1 -β 2 (ζ, x) ∧ (dd ζ S 0 V 1 ,A 1 ,E (ζ, x)) n-1 , for (ζ, x) ∈ (∂D × D)\ Ḡ, where d = d ζ + d x . Consider the C 2 map incl : (∂D × ∂D)\∂G → (∂D × D)\ Ḡ, (ζ, x) → (ζ, x).
From (3.8), (3.12) and (3.13), it follows that Φ 0 is continuous on (∂D × D)\ Ḡ and incl * (Φ 0 ) is integrable on ∂D ×∂D and Φ 1 is continuous on (∂D × D)\ Ḡ and integrable on ∂D × D (where incl * (Φ 0 ) is the pull-back of the differential form Φ 0 by the inclusion map incl).

Lemma 3.1. Let λ ∈ R + and E > E(λ, λ, D). Let V 1 , V 2 ∈ C 2 ( D, R), B 1 , B 2 ∈ F mag ( D) such that max( V 1 C 2 ,D , V 2 C 2 ,D , B 1 C 1 ,D , B 2 C 1 ,D ) ≤ λ. For µ = 1, 2, let A µ ∈ F pot (D, B µ ).
The following equalities are valid:

∂D×∂D incl * (Φ 0 ) = ∂D× D Φ 1 ; (3.21) 1 (n -1)! Φ 1 (ζ, x) = (r V 1 ,E (x) n ω 0,V 1 ,B 1 (ζ, x) + r V 2 ,E (x) n ω 0,V 2 ,B 2 (ζ, x) -kV 1 ,B 1 (E, ζ, x) • kV 2 ,B 2 (E, ζ, x) (3.22) × r V 1 ,E (x) n-2 ω 0,V 1 ,B 1 (ζ, x) + r V 2 ,E (x) n-2 ×ω 0,V 2 ,B 2 (ζ, x))) ∧ dx 1 ∧ .. ∧ dx n , for (ζ, x) ∈ ∂D × D.
Equality (3.21) follows from regularization and Stokes' formula. Proof of Lemma 3.1 is given in Section 4.

Taking account of Lemma 3.1, Proposition 3.2 and Remark 3.1, we obtain the following Theorem of uniqueness.

Theorem 3.1. Let λ ∈ R + and E > E(λ, λ, D). Let V 1 , V 2 ∈ C 2 ( D, R), B 1 , B 2 ∈ F mag ( D) such that max( V 1 C 2 ,D , V 2 C 2 ,D , B 1 C 1 ,D , B 2 C 1 ,D ) ≤ λ. For µ = 1, 2, let A µ ∈ F pot (D, B µ ).
The following estimate is valid:

D (r V 1 ,E (x) -r V 2 ,E (x)) r V 1 ,E (x) n-1 -r V 2 ,E (x) n-1 dx ≤ Γ( n 2 ) 2π n 2 (n -1)! ∂D×∂D incl * (Φ 0 ). (3.23)
In addition, the following statements are valid:

if k V 1 ,B 1 (E, ζ, x) = k V 2 ,B 2 (E, ζ, x) for ζ, x ∈ ∂D, ζ = x, then V 1 ≡ V 2 and B 1 ≡ B 2 on D; if k 0,V 1 ,B 1 (E, ζ, x) = k 0,V 2 ,B 2 (E, ζ, x) for ζ, x ∈ ∂D, ζ = x, then V 1 ≡ V 2 and B 1 ≡ B 2 on D. Proof of Theorem 3.1 is given in Section 4. If B 1 ≡ 0, B 2 ≡ 0 and V 1 , V 2 ,
and D are smoother than C 2 , then Lemma 3.1 and Theorem 3.1 follow from results of [B] and [GN].

4 Proof of Theorem 3.1 and Lemma 3.1

Using Lemma 2.1, (2.4), Propositions 3.1, 3.2 and Lemma 3.1, we first prove Theorem 3.1.

Proof of Theorem 3.1. From (3.22) and Proposition 3.2 and definition of ω 0,Vµ,Bµ , µ = 1, 2, it follows that

1 (n -1)! ∂D× D Φ 1 = (4.1) D r V 1 ,E (x) n S n-1 1 + w • kV 2 ,B 2 (E, ζ 1,x (w), x) r V 1 ,E (x) dσ(w)dx + D r V 2 ,E (x) n S n-1 1 + kV 1 ,B 1 (E, ζ 2,x (w), x) • w r V 2 ,E (x) dσ(w)dx,
where dσ is the canonical measure on S n-1 , and where • denotes the usual scalar product on R n , and where for x ∈ D and w ∈ S n-1 and µ = 1, 2, ζ µ,x (w) denotes the unique point ζ of ∂D such that w = ν Vµ,Bµ,E,x (ζ). Hence using Cauchy-Bunyakovski-Schwarz inequality and (3.2) and the equality

S n-1 dσ(w) = 2π n 2 Γ( n 2 ) , we obtain 1 (n -1)! ∂D× D Φ 1 ≥ 2π n 2 Γ( n 2 ) D (r V 1 ,E (x) -r V 2 ,E (x))(r V 1 ,E (x) n-1 -r V 2 ,E (x) n-1 )dx. (4.2) Estimate (4.2) and equality (3.21) prove (3.23). Now assume that k V 1 ,B 1 (E, ζ, x) = k V 2 ,B 2 (E, ζ, x) for ζ, x ∈ ∂D, ζ = x. Then from (3.1) and (3.18), it follows that the one form incl * (β 2 -β 1 )(ζ, x) is null for any ζ, x ∈ ∂D, ζ = x. Hence from (3.19), it follows that the 2n -2 form incl * (Φ 0 )(ζ, x) is null for any ζ, x ∈ ∂D, ζ = x. Thus using (3.23), we obtain D (r V 1 ,E (x)-r V 2 ,E (x))(r V 1 ,E (x) n-1 -r V 2 ,E (x) n-1
)dx ≤ 0, and as n ≥ 2, this latter inequality implies that

r V 1 ,E ≡ r V 2 ,E on D. (4.3) Thus V 1 ≡ V 2 . Using (4.3) and the equality | kV i ,B i (E, ζ, x)| = r V i ,E (x) for i = 1, 2,
x ∈ D and ζ ∈ ∂D, and using (4.1), we obtain that

1 (n -1)! ∂D× D Φ 1 = 1 2 2 i=1 D r V i ,E (x) n-2 S n-1 kV 1 ,B 1 (E, ζ i,x (w), x) -kV 2 ,B 2 (E, ζ i,x (w), x) 2 dσ(w)dx.
As ∂D× D Φ 1 = 0 (due to (3.21)), we obtain that for any x ∈ D, and any

w ∈ S n-1 , kV 1 ,B 1 (E, ζ 1,x (w), x) = kV 2 ,B 2 (E, ζ 1,x (w), x). At fixed x ∈ D, we know that ζ 1,x is onto ∂D. Hence the following equality is valid kV 1 ,B 1 (E, ζ, x) = kV 2 ,B 2 (E, ζ, x), ζ ∈ ∂D, x ∈ D. (4.4)
From (4.4) and (3.16), it follows that

B 1 ≡ B 2 on D. Now assume that k 0,V 1 ,B 1 (E, ζ, x) = k 0,V 2 ,B 2 (E, ζ, x) for ζ, x ∈ ∂D, ζ = x. Then using (2.5) and replacing B i by -B i , i = 1, 2, in the proof, we obtain (V 1 , B 1 ) ≡ (V 2 , B 2 ).
Using Lemma 2.1, (2.4), Propositions 3.1, 3.2, we prove Lemma 3.1.

Proof of Lemma 3.1. We first prove (3.22). Let U be an open subset of R n-1 and φ : U → ∂D such that φ is a C 2 parametrization of ∂D. Let φ 0 : U ×D → ∂D × D, (ζ, x) → (φ(ζ), x).
We work in coordinates given by (U × D, φ 0 ). Let µ, µ ′ = 1, 2. On one hand from definition of ω 0,Vµ,Bµ , definition of ν Vµ,Bµ,E,x and (3.2), we obtain

ω 0,Vµ,Bµ (ζ, x) ∧ dx 1 ∧ . . . ∧ dx n = (-1) n r Vµ,E (x) -n (4.5) ×det kVµ,Bµ (E, ζ, x), ∂ kVµ,Bµ ∂ζ 1 (E, ζ, x), . . . , ∂ kVµ,Bµ ∂ζ n-1 (E, ζ, x) dζ 1 ∧ . . . ∧ dζ n-1 ∧ dx 1 ∧ . . . ∧ dx n for ζ ∈ U and x ∈ D and µ = 1, 2.
On the other hand straightforward calculations give (dd

ζ S 0 Vµ,Aµ,E (ζ, x) = m 1 =1..n m 2 =1..n-1 ∂ 2 S 0 Vµ,Aµ,E ∂xm 1 ∂ζm 2 (ζ, x)dx m 1 ∧ dζ m 2 ) β µ ′ (ζ, x) ∧ (dd ζ S 0 Vµ,Aµ,E (ζ, x)) n-1 = (-1) (n-1)(n-2) 2 (n -1)! (4.6) ×det kV µ ′ ,B µ ′ (E, ζ, x), ∂ 2 S 0 Vµ ,Aµ,E ∂ζ 1 ∂x (ζ, x), . . . , ∂ 2 S 0 Vµ,Aµ,E ∂ζ n-1 ∂x (ζ, x) dζ 1 ∧ . . . ∧ dζ n-1 ∧ dx 1 ∧ . . . ∧ dx n , for ζ ∈ U, x ∈ D. Note that due to (3.2) and Proposition 3.2, kVµ,Bµ (E, ζ, x) is orthogonal to ∂ ∂ζm kVµ,Bµ (E, ζ, x), m = 1 . . . n -1, and that ( kVµ,Bµ (E, ζ, x), ∂ ∂ζ 1 kVµ,Bµ (E, ζ, x), . . . , ∂ ∂ζ n-1 kVµ,Bµ (E, ζ, x)
) is a basis of R n . Hence from (4.5), (4.6) and (3.9), we obtain Definition (3.20) and equality (4.7) proves (3.22). We sketch the proof of (3.21). Let ε ∈]0, 1[ and x 0 ∈ D. We consider

β µ ′ (ζ, x) ∧ (dd ζ S 0 Vµ,Aµ,E (ζ, x)) n-1 = -(-1) n(n-1) 2 (n -1)!r Vµ,E (x) n (4.7) × kV µ ′ ,B µ ′ (E, ζ, x) • kVµ,Bµ (E, ζ, x) r Vµ,E (x) 2 ω 0,Vµ,Bµ (ζ, x) ∧ dx 1 ∧ . . . ∧ dx n , for ζ ∈ U, x ∈ D.
D ε = {x 0 + ε(x -x 0 ) | x ∈ D}. (4.8)
As D is a strictly convex (in the strong sense) open domain of R n , with C 2 boundary, it follows that D ε is also a strictly convex (in the strong sense) open domain of R n , with C 2 boundary, and in addition, as 0

< ε < 1, Dε ⊆ D, (4.9) dist(∂D, ∂D ε ) = (1 -ε)dist(∂D, x 0 ). (4.10) where dist(∂D, ∂D ε ) = inf{|x -y| | x ∈ D ε , y ∈ ∂D} and dist(∂D, x 0 ) = inf{|y -x 0 | | y ∈ ∂D} > 0.
For M and N two finite-dimensional oriented C 2 manifold (with or without boundary), we consider on M × N the differential product structure induced by the already given differential structures of M and N and we consider the orientation of M × N given by the already fixed orientation of M and of N . The orientation of ∂D ε is given by the unit outward normal vector and ∂D × Dε is a C 2 manifold with boundary

∂D × ∂D ε (which is a C 2 manifold without boundary). Let incl ε ∈ C 2 (∂D × ∂D ε , ∂D × Dε ) be defined by incl ε (ζ, x) = (ζ, x), (ζ, x) ∈ ∂D × ∂D ε .
Here we omit the details of the proof of the following statement:

∂D× Dε Φ 1 → ∂D× D Φ 1 and ∂D×∂Dε incl * ε (Φ 0 ) → ∂D×∂D incl * (Φ 0 ) as ε → 1 -.
These statements follow from (3.12), (3.13) and (3.9). We shall prove that

∂D× Dε Φ 1 = ∂D×∂Dε incl * ε (Φ 0 ). (4.11) For µ = 1, 2, let S 0µ ∈ C 2 ((R n × R n )\{(x, x) | x ∈ R n }, R) such that S 0µ (ζ, x) = S 0 Vµ,Aµ,E (ζ, x), (ζ, x) ∈ ( D × D)\ Ḡ (from (3.8), it follows that such a function S 0µ exists). Let δ ε = dist(∂D, ∂D ε ). Let W 1,δε be the open subset ∂D + B(0, δε 2 ) = {x + y | x ∈ ∂D, y ∈ R n , |y| < δε 2 } and let W 2,δε be the open subset D ε +B(0, δε 2 ) = {x+y | x ∈ D ε , y ∈ R n , |y| < δε 2 }. Note that W 1,δε is an open neighborhood of ∂D which does not intersect W 2,δε
which is an open neighborhood of Dε . Hence S 0µ ∈ C 2 (W 1,δε × W 2,δε , R) and there exists a sequence of functions (S 0µ,m ) such that

S 0µ,m ∈ C 3 (W 1,δε × W 2,δε , R), (4.12) sup (x,y)∈∂D× Dε α=(α 1 ,α 2 )∈N 2 |α|=α 1 +α 2 ≤2 | ∂ |α| (S 0µ,m -S 0µ ) ∂x α 1 ∂y α 2 (x, y)| → 0, as m → +∞. (4.13) Fix m ∈ N. Let µ = 1, 2.
We define the differential one-form, β µ m on (∂D × Dε ) by

β µ m (ζ, x) = d x S 0µ,m (ζ, x) - 1 c n j=1 A j µ (x)dx j , (4.14) for (ζ, x) ∈ ∂D × Dε and x = (x 1 , . . . , x n ) and A µ (x) = (A 1 µ (x), . . . , A n µ (x)) and where d = d ζ + d x is the De Rham differential operator on ∂D × Dε .
We define the continuous differential 2n -2 form Φ 0,m on ∂D × Dε by

Φ 0,m (ζ, x) = -(-1) n(n+1) 2 (β 2 m -β 1 m )(ζ, x) ∧ d ζ (S 0 2,m -S 0 1,m )(ζ, x) ∧ p+q=n-2 (dd ζ S 0 1,m (ζ, x)) p ∧ (dd ζ S 0 2,m (ζ, x)) q , (4.15) for ζ ∈ ∂D, x ∈ Dε , where d = d ζ +d x is the De Rham differential operator on ∂D × Dε . We define the continuous differential 2n -1 form Φ 1,m on ∂D × Dε by Φ 1,m (ζ, x) = -(-1) n(n-1) 2 β 1 m (ζ, x) ∧ (dd ζ S 0 1,m (ζ, x)) n-1 + β 2 m (ζ, x) ∧(dd ζ S 0 2,m (ζ, x)) n-1 -β 1 m (ζ, x) ∧ (dd ζ S 0 2,m (ζ, x)) n-1 -β 2 m (ζ, x) ∧ (dd ζ S 0 1,m (ζ, x)) n-1 , (4.16) for (ζ, x) ∈ ∂D × Dε .
From (4.14)-(4.16), (3.9), (4.13), it follows that

∂D× Dε Φ 1,m → ∂D× Dε Φ 1 , as m → +∞, (4.17) ∂D×∂Dε incl * ε (Φ 0,m ) → ∂D×∂Dε incl * ε (Φ 0 ), as m → +∞. (4.18)
If we prove that ∂D× Dε Φ 1,m = ∂D×∂Dε incl * ε (Φ 0,m ), then formula (4.11) will follow from (4.17) and (4.18).

From (4.12), it follows that

dd ζ S 0µ,m is a C 1 form on ∂D × Dε , (4.19) d(dd ζ S 0µ,m ) = 0, (4.20)
where d is the De Rham differential operator on ∂D × Dε . From (4.14), it follows that

dβ µ m (ζ, x) = -dd ζ S 0µ,m (ζ, x) - 1 c 1≤j 1 <j 2 ≤n B µ j 1 ,j 2 (x)dx j 1 ∧ dx j 2 , (4.21) for (ζ, x) ∈ ∂D × Dε and x = (x 1 , . . . , x n ) (B µ j 1 ,j 2 (x) denotes the elements of B µ (x)). From (4.19)-(4.21), it follows that Φ 0,m is C 1 on ∂D × Dε and that dΦ 0,m (ζ, x) = (-1) n-1 Φ 1,m (ζ, x) + ω(ζ, x), (4.22) ω(ζ, x) = (-1) n(n+1) 2 1 c 1≤j 1 <j 2 ≤n (B 2 j 1 ,j 2 (x) -B 1 j 1 ,j 2 (x))dx j 1 ∧ dx j 2 ∧d ζ (S 0 2,m -S 0 1,m )(ζ, x) ∧ p+q=n-2 (dd ζ S 0 1,m (ζ, x)) p ∧ (dd ζ S 0 2,m (ζ, x)) q ,
for (ζ, x) ∈ ∂D × Dε and x = (x 1 , . . . , x n ). Note that as B µ , µ = 1, 2, is continuously differentiable on D, then the 2-form defined on ∂D × Dε by

1≤j 1 <j 2 ≤n (B 2 j 1 ,j 2 (x) -B 1 j 1 ,j 2 (x))dx j 1 ∧ dx j 2 , (ζ, x) ∈ ∂D × Dε is also C 1 . Note that ( Dε is a n-dimensional C 2 manifold and (4.20)) ω(ζ, x) = dω(ζ, x), (4.23)
where

ω(ζ, x) = (-1) n(n+1) 2 (S 0 2,m -S 0 1,m )(ζ, x) c 1≤j 1 <j 2 ≤n (B 2 j 1 ,j 2 (x) -B 1 j 1 ,j 2 (x)) dx j 1 ∧ dx j 2 ∧ p+q=n-2 (dd ζ S 0 1,m (ζ, x)) p ∧ (dd ζ S 0 2,m (ζ, x)) q , (4.24) for (ζ, x) ∈ ∂D × Dε and x = (x 1 , . . . , x n ). Since ∂D ε is a (n -1)-dimensional C 2 manifold, it follows that incl * ε ω(ζ, x) = 0 (4.25) for (ζ, x) ∈ ∂D × ∂D ε .
Φ 1,m = ∂D×∂Dε incl * ε (Φ 0,m ).
5 Proof of Lemma 2.1 and Proposition 3.1

5.1 Continuation of (V, B) and notations.

Let Ṽ ∈ C 2 (R n , R) be such that Ṽ ≡ V on D and Ṽ C 2 ,R n < ∞. Let B ∈ C 1 (R n , A n (R)) (where A n (R)
denotes the space of real antisymmetric matrices) such that B ≡ B on D and B C 1 ,R n < ∞. Let ψ be the flow for the differential system ẋ = p

1 + p 2 c 2 , (5.1) ṗ = -∇ Ṽ (x) + 1 c B(x) p 1 + p 2 c 2 , for x ∈ R n and p ∈ R n (it means that a solution of (5.1), (x(t), p(t)), t ∈ ]t -, t + [, which passes through (x 0 , p 0 ) ∈ R n × R n at time t = 0, is written as (x(t), p(t)) = ψ(t, x 0 , p 0 ) for t ∈]t -, t + [).
For equation (5.1), the energy

E = c 2 1 + |p(t)| 2 c 2 + Ṽ (x(t)) (5.2)
is an integral of motion.

Under the conditions on Ṽ and B, ψ is defined on

R × R n × R n and ψ ∈ C 1 (R × R n × R n , R n × R n ),
and a solution x(t), t ∈]t -, t + [, of (1.1) which starts at q 0 ∈ D at time 0 with velocity v is written as

x(t) = ψ 1 (t, x, v 1-v 2 c 2 ), t ∈]t -, t + [ (we write ψ = (ψ 1 , ψ 2 ) where ψ i = (ψ 1 i , . . . , ψ n i ) ∈ C 1 (R × R n × R n , R n ), i = 1, 2).
For v ∈ R n and x ∈ R n , we define the vector

F (x, v) of R n by F (x, v) = -∇ Ṽ (x) + 1 c B(x)v. (5.3) For x ∈ R n and E > c 2 + Ṽ (x), we denote by r Ṽ ,E (x) the positive real number r Ṽ ,E (x) = c E -Ṽ (x) c 2 2 -1, (5.4)
and we denote by S n-1

x,E the following sphere of R n of center 0

S n-1 x,E = {p ∈ R n ||p| = r Ṽ ,E (x)}.
(5.5) 5.2 Growth estimates for a function g. Consider the function g : R n → B c defined by

g(x) = x 1 + |x| 2 c 2
(5.6) where x ∈ R n . This function was considered for example in [J1].

We remind that g has the following simple properties:

|∇g i (x)| 2 ≤ 1 1 + |x| 2 c 2 , (5.7) |g(x) -g(y)| ≤ √ n sup ε∈[0,1] 1 1 + |εx+(1-ε)y| 2 c 2 |x -y|, (5.8) |∇g i (x) -∇g i (y)| ≤ 3 √ n c sup ε∈[0,1] 1 1 + |εx+(1-ε)y| 2 c 2
|x -y|, (5.9) for x, y ∈ R n , i = 1 . . . n, and g = (g 1 , . . . , g n ). The function g is an infinitely smooth diffeomorphism from R n onto B c , and its inverse is given by g

-1 (x) = x 1- |x| 2 c 2
, for x ∈ B c .

5.3

Proof of Lemma 2.1. For q 0 , q ∈ D, q 0 = q, let t +,q 0 ,q = sup{t > 0 | ψ 1 (t, q 0 , k0,V,B (E, q 0 , q))∈ D}. From Properties (2.1) and (2.2), it follows that k 0,V,B (E, ., .) is continuous on ( D × D)\ Ḡ and for q 0 , q ∈ D, q 0 = q, and any s 1 , s 2 ∈ [0, t +,q 0 ,q [, s 1 = s 2 , ψ 1 (s 1 , q 0 , k0,V,B (E, q 0 , q)) = ψ 1 (s 2 , q 0 , k0,V,B (E, q 0 , q)) and N(q + ) • ∂ ∂t ψ 1 (t, q 0 , k0,V,B (E, q 0 , q)) |t=t +,q 0 ,q is positive, where q + = ψ 1 (t +,q 0 ,q , q 0 , k0,V,B (E, q 0 , q)) (and where • denotes the usual scalar product on R n ). Using also continuity of ψ 1 , one obtains that s V,B (E, q 0 , q) is continuous on ( D × D)\ Ḡ. Then we obtain that s V,B (E, q 0 , q) ∈ C 1 (( D × D)\ Ḡ, R) by applying implicit function theorem on maps (E, ., .) on (R n × R n )\∆ (such a continuation exists thanks to (2.2), and note that for any q 0 , q ∈ D, q = q 0 , kV,B (E, q 0 , q) = 0). Note that k V,B (E, q 0 , q) = g(ψ 2 (s V,B (E, q 0 , q), q 0 , k0,V,B (E, q 0 , q))), q 0 , q ∈ D, q 0 = q. It remains to prove that s V,B (E, q 0 , q) is continuous on G = {(q ′ , q ′ ) | q ′ ∈ D}. Let q 0 = q ∈ D. Let (q 0,m ) and (q m ) be two sequences of points of D such that q 0,m = q 0 for all m and q 0,m goes to q 0 and q m goes to q = q 0 as m → +∞. Let R = lim sup m→+∞ s V,B (E, q 0,m , q m ) ∈ [0, +∞]. We shall prove that R = 0. Assume that R > 0. Note that by conservation of en-

m i : R × ((R n × R n )\∆), (t, x, y) → y i -ψ i 1 (t, x, k0,V,B (E, x, y)), i = 1 . . . n, where ∆ = {(x, x) | x ∈ R n } and k0,V,B (E, ., .) is a C 1 continuation of k0,V,B
ergy | k0,V,B (E, q 0,m , q m )| ≤ c E+ V ∞ c 2 2 -1. Using definition of R and compactness of the closed ball of R n whose radius is c E+ V ∞ c 2 2 -1 and
whose centre is 0, we obtain that there exist subsequences of q 0,m and q m (respectively still denoted by q 0,m and q m ) such that lim m→+∞ s V,B (E, q 0,m , q m ) = R, (5.10) k0,V,B (E, q 0,m , q m ) converges to some k ∈ R n .

(5.11)

Using conservation of energy, we obtain that

|k| = c E -V (q 0 ) c 2 2 -1.
(5.12)

Using (5.11) and (5.10) and continuity of ψ 1 , we obtain that

ψ 1 (t, q 0 , k) = lim m→+∞ ψ 1 (t, q 0,m , k0,V,B (E, q 0,m , q m )), for all t ∈ [0, R[. (5.13)
For all m and t ∈ [0, s V,B (E, q 0,m , q m )[, ψ 1 (t, q 0,m , k0,V,B (E, q 0,m , q m )) ∈ D. Hence using (5.13), we obtain that (5.14) In addition, ψ 1 (0, q 0 , k) = q 0 ∈ D.

ψ 1 (t, q 0 , k) ∈ D, t ∈ [0, R[.
(5.15)

Then R = +∞ (otherwise this would contradict (2.1), in particular the fact that the solution of (1.1) under condition (1.3) with energy E, which starts at time 0 at q 0 = ψ 1 (0, q 0 , k), reaches the boundary ∂D at a time t + > 0 and satisfies the estimate ∂ψ 1 ∂t (t + , q 0 , k) • N(ψ 1 (t + , q 0 , k)) > 0) . Using continuity of ψ 1 and lim m→+∞ q 0,m = q 0 , lim m→+∞ q m = q 0 , lim m→+∞ s V,B (E, q 0,m , q m ) = R, lim m→+∞ k0,V,B (E, q 0,m , q m ) = k and the definition of s V,B (E, q 0,m , q m ), we obtain that

ψ 1 (R, q 0 , k) = lim m→+∞ ψ 1 (s V,B (E, q 0,m , q m ), q 0 , k0,V,B (E, q 0,m , q m )) = lim m→+∞ q m = q 0 .
(5.16) Properties (5.16), (5.15), (5.14) and (2.1) imply R = 0, which contradicts the assumption R > 0. Finally we proved that s V,B (E, ., .

) ∈ C(( D × D)\∂G, R). Let x 0 ∈ D. From (2.
3), it follows that for sufficiently small positive ε, E is greater than

E( Ṽ C 2 ,Dx 0 ,ε , B C 1 ,Dx 0 ,ε , D x 0 ,ε ) where D x 0 ,ε = {x 0 + (1 + ε)(x -x 0 ) | x ∈ D}.
Hence one obtains that solutions of energy E for equation (5.1) in D x 0 ,ε also have properties (2.1) and (2.2); and replacing V , B, and D by Ṽ , B and D x 0 ,ε above in the proof, one obtains that s Ṽ , B (E, ., .) is continuous on ( Dx 0 ,ε × Dx 0 ,ε )\{(q, q) | q ∈ ∂D x 0 ,ε } (s Ṽ , B (E, q ′ 0 , q ′ ), (q ′ 0 , q ′ ) ∈ Dx 0 ,ε × Dx 0 ,ε , are defined as s V,B (E, q 0 , q), (q 0 , q) ∈ D × D, are defined in Subsection 2.1). Now, using also D ⊆ D x 0 ,ε and the equality s V,B (E, q 0 , q) = s Ṽ , B (E, q 0 , q) for q 0 , q ∈ D, one obtains s V,B (E, ., .) ∈ C( D × D, R) (the equality s V,B (E, q 0 , q) = s Ṽ , B (E, q 0 , q) for q 0 , q ∈ D, follows from the fact that if (x(t), p(t)) is solution of (5.1) in D, then (x(t), p(t)) is also solution of (5.1) in D x 0 ,ε ).

Lemma 2.1 is proved.

Proof of Proposition

3.1. From Lemma 2.1, ψ ∈ C 1 (R×R n ×R n , R n ×R n ), A ∈ C 1 ( D, R n ), it follows that S 0 V,A,E ∈ C( D × D, R) and S 0 V,A,E ∈ C 1 (( D × D)\ Ḡ, R).
Equalities (3.9) and (3.10) are known equalities (see Section 46 and further Sections of [A]). Statements (3.8), (3.11), (3.12) follow from (3.9) and (3.10). We shall prove (3.13). We omit indices V,B for s V,B , k0,V,B and kV,B where k0 , k are defined by (3.1). Using the equality yx = s(E,x,y) 0 (E, x, y), for all x, y ∈ D, y = x.

∂ψ 1 ∂t (t, x, k0 (E, x, y))dt and estimate | ∂ψ 1 ∂t (t, x, k0 (E, x, y))| ≤ c, we obtain |y -x| ≤ cs
(5.17)

Derivating equality ψ 1 (s(E, x, y), x, k0 (E, x, y)) = y with respect to y i , we obtain that

e i = ( ∂ψ j 1 ∂ k (s(E, x, y), x, k0 (E, x, y)) • ∂ k0 ∂y i (E, x, y)) j=1..n (5.18) + ∂s ∂y i (E, x, y) ∂ψ 1 ∂s (s(E, x, y), x, k0 (E, x, y)),
for any x, y ∈ D, x = y and where (e 1 , . . . , e n ) is the canonical basis of R n (and where • denotes the usual scalar product on R n ). For t ∈ R, x ∈ D, and k ∈ R n and j = 1..n the following equality is valid:

ψ j 1 (t, x, k) = x j + tg j (k)+ t 0 g j (k + s 0 F (ψ 1 (σ, x, k), g(ψ 2 (σ, x, k)))dσ) -g j (k) ds. Hence we obtain that for t ∈ R, x ∈ D, k = (k 1 , . . . , k n ) ∈ R n , and j = 1..n, ∂ψ j 1 ∂k l (t, x, k) = t ∂g j ∂k l (k) + t 0 ∂g j ∂k l (k + s 0 F (ψ 1 (σ, x, k), g(ψ 2 (σ, x, k))) dσ) - ∂g j ∂k l (k) ds + t 0 ∇g j (k + s 0 F (ψ 1 (σ, x, k), g(ψ 2 (σ, x, k)))dσ) • s 0 ∂ ∂k l F (ψ 1 , g(ψ 2 )) |(σ,x,k) dσds, (5.19) for any l = 1..n. Define R = sup (x ′ ,y ′ )∈ D s(E, x ′ , y ′ ), M 3 = sup t∈[0,R],x ′ ∈ D,l=1...n |k|≤c sup x ′ ∈ D E-V (x ′ ) c 2 2 -1 | ∂ ∂k l F (ψ 1 , g(ψ 2 )) |(t,x ′ ,k) |, M 4 = max(M 3 , √ n V C 2 ,D + n B C 1 ,D ).
Then using (5.19) and growth properties of g, we obtain that where (5.22) for i = 1..n. From (3.2), (5.21) and (5.22), it follows that

| ∂ψ j 1 ∂k l (s(E, x, y), x, k0 (E, x, y)) -s(E, x, y) ∂g j ∂k l ( k0 (E, x, y))| ≤ M 4 s(E, x, y) 2 (1 + 3 √ n c ), ( 5
M 5 = √ n V C 2 ,D + n B C 1 ,D . Using (3.2), we obtain that k0 (E, x, y) • ∂ k0 ∂y i (E, x, y) = 1 2 ∂| k0 | 2 ∂y i (E, x, y) = 0,
| k(E, x, y) | k(E, x, y)| • ∂ k0 ∂y i (E, x, y)| ≤ 1 | k(E, x, y)| |( k(E, x, y) -k0 (E, x, y)) • ∂ k0 ∂y i (E, x, y)| + 1 | k(E, x, y)| | k0 (E, x, y) • ∂ k0 ∂y i (E, x, y)| ≤ c -1 inf x ′ ∈ D E -V (x ′ ) c 2 2 -1 -1 2 M 5 s(E, x, y)| ∂ k0 ∂y i (E, x, y)|, (5.23) for i = 1..n. Let (v 1 , . . . , v n-1 ) be an orthonormal family of R n such that ( k(E,x,y) | k(E,x,y)| , v 1 , . . . , v n-1
) is an orthonormal basis of R n . Note that using definition of g and (5.22), we obtain

(∇g j ( k0 (E, x, y)) • ∂ k0 ∂y i (E, x, y)) j=1..n = (5.24) 1 + | k0 (E, x, y)| 2 c 2 -1/2 ∂ k0 ∂y i (E, x, y), i = 1..n.
Hence using (5.24), (5.20) and (5.18) (and k(E, x, y) • v h = 0), we obtain

s(E, x, y)| ∂ k0 ∂y i (E, x, y) • v h | = 1 + | k0 (E, x, y)| 2 c 2 s(E, x, y) ×|(∇g j ( k0 (E, x, y)) • ∂ k0 ∂y i (E, x, y)) j=1..n • v h | ≤ 1 + | k0 (E, x, y)| 2 c 2 nM 4 s(E, x, y) 2 (1 + 3 c )| ∂ k0 ∂y i (E, x, y)| + |( ∂ψ j 1 ∂k (s(E, x, y), x, k0 (E, x, y)) • ∂ k0 ∂y i (E, x, y)) j=1..n • v h | = 1 + | k0 (E, x, y)| 2 c 2 nM 4 s(E, x, y) 2 (1 + 3 c )| ∂ k0 ∂y i (E, x, y)| + |v h i | ≤ sup x ′ ∈ D E -V (x ′ ) c 2 (nM 4 (1 + 3 c )s(E, x, y) 2 | ∂ k0 ∂y i (E, x, y)| + 1), (5.25) for i = 1 . . . n. Let M 6 = c -1 (c -2 inf x ′ ∈ D(E -V (x ′ ))) 2 -1 -1 2 M 5 + c -2 × sup x ′ ∈ D(E -V (x ′ ))(nM 4 (1 + 3 c ) + 1)
. From (5.23) and (5.25), it follows that

s(E, x, y)| ∂ k0 ∂y i (E, x, y)| ≤ √ nM 6 (1 + s(E, x, y)(s(E, x, y)| ∂ k0 ∂y i (E, x, y)|)),
(5.26) for i = 1 . . . n.

Using uniform continuity of s(E, ., .) on D × D, we obtain that there exists some η > 0 such that if x, y ∈ D, |x -y| < η, then √ nM 6 s(E, x, y) ≤ 1 2 . Then, using (5.26), we obtain that s(E, x, y)| ∂ k0

∂y i (E, x, y)| ≤ 2 √ nM 6 , for
x, y ∈ D, |x -y| < η and i = 1..n. Now using the continuous differentiability of k0 (E, ., .) on ( D × D)\ Ḡ, we obtain that

| ∂ k0 ∂y i (E, x, y)| ≤ M ′ i s(E,
x,y) for x, y ∈ D, x = y and where

M ′ i = max(2M 6 √ n, R sup x ′ ,y ′ ∈ D,|x ′ -y ′ |≥η | ∂ k0 ∂y i (E, x, y)|). Putting M 2 = sup i=1..n cM ′
i and using (5.17) and (3.11), we obtain (3.13).

6 Proof of Properties (2.1) and (2.2)

In this Section we first consider solutions x(t) of (5.1) in an open bounded subset Ω of R n (see Subsection 6.1) and we give properties of these solutions at fixed and sufficiently large energy (see Subsections 6.3 and 6.4) (Ω should be thought as an open neighborhood of D). Using these properties we prove Properties 2.1 and 2.2 (see Subsection 6.5). Subsections 6.6, 6.7, 6.8, 6.9 are devoted to the proof of Propositions 6.1, 6.2, 6.3, 6.4 formulated in Subsection 6.4. We keep notations of Subsections 5.1, 5.2.

Additional notations.

Let Ω be a bounded open subset of R n with frontier ∂Ω. We define a positive number δ(Ω) by

δ(Ω) = sup x∈Ω |x|. (6.1)
We consider the following equation in Ω :

ṗ = F (x, ẋ), p = ẋ 1 -| ẋ| 2 c 2 , x ∈ Ω, p ∈ R n . (6.2)
where the force F (x, ẋ) = -∇ Ṽ (x) + 1 c B(x) ẋ is defined by (5.3). For the equation (6.2), the energy

E = c 2 1+|p(t)| 2 c 2 + Ṽ (x(t)
) is an integral of motion.

Note that if x(t), t ∈]t -, t + [, is a solution of (6.2) which starts at x 0 ∈ Ω at time 0 with velocity v then x(t) = ψ 1 (t, x 0 , g -1 (v)), t ∈]t -, t + [, where the function g is defined by (5.6) and where ψ = (ψ 1 , ψ 2 ) is the flow of the differential system (5.1). We obtain, in particular, x(t) → ψ 1 (t ± , x 0 , g -1 (v)), as t → t ± , and ẋ(t) → g(ψ 2 (t ± , x 0 , g -1 (v))), as t → t ± .

We denote by Λ the open subset of R × Ω × R n where the flow of the differential system (6.2) is defined, i.e.

Λ = {(t, x, p) ∈ R × Ω × R n | ∀s ∈ [0, t] ψ 1 (s, x, p) ∈ Ω},
where ψ = (ψ 1 , ψ 2 ) is the flow of the differential system (5.1).

For E > c 2 + sup x∈Ω Ṽ (x), we denote by V E the following smooth 2n -1dimensional submanifold of R 2n

V E = {(x, p) ∈ Ω × R n | |p| = r Ṽ ,E (x)}, (6.3)
where r Ṽ ,E (x) is defined by (5.4). For E > c 2 + sup x∈Ω Ṽ (x), we also consider the map

ϕ E ∈ C 1 (Λ ∩ (]0, +∞[×V E ) , Ω × Ω), defined by ϕ E (t, x, p) = (x, ψ 1 (t, x, p)), for (t, x, p) ∈ Λ ∩ (]0, +∞[×V E ) .
(6.4) 6.2 Estimates for the force F . We define the nonnegative real number β( Ṽ , B, Ω) by

β( Ṽ , B, Ω) = max     sup x∈Ω α∈(N∪{0}) n |α|≤2 |∂ α x Ṽ (x)|, sup x∈Ω α ′ ∈(N∪{0}) n |α ′ |≤1 |∂ α ′ x Bi,j (x)|     .
(6.5)

The following estimates are valid:

|F (x, v)| ≤ nβ( Ṽ , B, Ω)( 1 c |v| + 1), (6.6) |F (x, v) -F (x ′ , v ′ )| ≤ nβ( Ṽ , B, Ω) |x -x ′ | 1 + |v ′ | c + 1 c |v ′ -v| , (6.7) for x, x ′ ∈ Ω and v, v ′ ∈ R n .
6.3 Some constants. For x ∈ Ω and E > c 2 + sup x ′ ∈Ω V (x ′ ), we define the following real constants

C 1 = 2c 2 + sup x ′ ∈Ω Ṽ (x ′ ) + 8|x ′ | |∇ Ṽ (x ′ )| + i,j=1...n | Bi,j (x ′ )| , (6.8) C 2 = c 2 1 + 800n 2 β( Ṽ , B, Ω) 2 δ(Ω) 2 c 4 + sup x ′ ∈Ω
Ṽ (x ′ ), (6.9)

C 3 = C 4 1 + 5δ(Ω)
C 5 e 5δ(Ω)C 5 (6.10)

C 4 = 1 + 10 √ 2n 3/2 δ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x) (6.11) × 10n 3/2 δ(Ω) 2 β( Ṽ , B, Ω) E -Ṽ (x) 5 + 600n 3/2 δ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) + 24n 1/2 + 20 √ 2n 2 δ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x) + 40n 3/2 δ(Ω)β( Ṽ , B, Ω) c 2 E-Ṽ (x) c 2 2 -1 , C 5 = 1 + 10 √ 2n 3/2 δ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x)
(6.12)

× 20n 3/2 β( Ṽ , B, Ω)δ(Ω) E -Ṽ (x) 1 + 120n 3/2 β( Ṽ , B, Ω)δ(Ω) E -Ṽ (x) , C 6 = inf x ′ ∈Ω 1 - c 2 E -Ṽ (x ′ ) 2 - 20n 2 β( Ṽ , B, Ω)δ(Ω) E -Ṽ (x ′ )
, (6.13)

C 7 = inf x ′ ∈Ω c 1 - c 2 E -Ṽ (x ′ ) 2 - 5c(n + 1) 1/2 n 2 δ(Ω) E -Ṽ (x ′ ) (6.14) ×β( Ṽ , B, Ω)e 10n 3/2 β( Ṽ , B,Ω)δ(Ω) E-Ṽ (x ′ )   1+2cn 1/2 e 10n 3/2 β( Ṽ , B,Ω)δ(Ω) E-Ṽ (x ′ )   × cr Ṽ ,E (x ′ ) E -Ṽ (x ′ ) 12 √ n + 1 + 10 √ nδ(Ω) .
Now assume that Ω is a bounded strictly convex (in the strong sense) open domain of R n with C 2 boundary. Let χ Ω be a C 2 defining function for Ω, i.e. Ω = χ -1

Ω (] -∞, 0[) and ∂Ω = χ -1 Ω ({0}) and for all x ∈ ∂Ω ∇χ Ω (x) = 0 and the Hessian matrix Hessχ Ω (x) of χ Ω at x is a symmetric positive definite matrix. For E > c 2 + sup x∈Ω Ṽ (x), we define the real constant C 8 (E, Ṽ , B, Ω) by

C 8 = C 10 (Ω)   1 - c 2 E -sup y∈Ω Ṽ (y) 2   - 4nC 9 (Ω)β( Ṽ , B, Ω) E -sup y∈Ω Ṽ (y)
, (6.15)

where C 9 (Ω) and C 10 (Ω) are the two positive real numbers defined by

C 9 (Ω) = sup x∈∂Ω |∇χ Ω (x)|, (6.16) C 10 (Ω) = inf x∈∂Ω v∈S n-1 |Hessχ Ω (x)(v, v)|.
(6.17)

Note that from (6.10)-( 6.15), it follows that

sup x∈Ω C 3 (E, x, Ṽ , B, Ω) → 0, as E → +∞, C 6 (E, Ṽ , B, Ω) → 1 > 0, as E → +∞, C 7 (E, Ṽ , B, Ω) → c > 0, as E → +∞, C 8 (E, Ṽ , B, Ω) → C 10 (Ω) > 0, as E → +∞. (6.18)
When Ω is strictly convex in the strong sense with C 2 boundary, then one can relate an upper bound for the real constant E( Ṽ C 2 ,Ω , B C 1 ,Ω , Ω) (mentioned in Subsection 2.1) with constants C 1 , C 2 , sup x∈Ω C 3 , C 6 , C 7 and C 8 (see Subsections 6.4 and 6.5).

Remark 6.1. We remind that Ṽ is a C 1 continuation of V on R n and that B ∈ C 1 (R n , A n (R)) is such that B ≡ B on D. Note that from (6.8)-( 6 6.4 Properties of the first component of the flow of (6.2) at fixed and sufficiently large energy E. The following Proposition 6.1 gives an upper bound for living time for solutions of (6.2) with energy E when E is sufficiently large.

Proposition 6.1. Let E ≥ C 1 ( Ṽ , B, Ω), (6.19)
where C 1 is defined by (6.8). Let x :]t -, t + [→ Ω be a solution of (6.2) with energy E, where t ± ∈ R ∪ {±∞}. Then the following statement holds: t -, t + are finite and they satisfy the following estimate

|t + -t -| ≤ 5δ(Ω) c , (6.20)
where δ(Ω) is defined by (6.1).

A proof of Proposition 6.1 is given in Subsection 6.6.

For E ≥ C 1 ( Ṽ , B, Ω) (C 1 is defined by (6.8)) and for (x, p) ∈ V E , we define the real numbers t +,x,p and t -,x,p by t +,x,p = sup{t > 0 | (t, x, p) ∈ Λ}, (6.21) t -,x,p = inf{t < 0 | (t, x, p) ∈ Λ}.

(6.22)

The following Proposition 6.2 gives, in particular, a one-to-one property of the map ϕ E defined by (6.4). Proposition 6.2. Let

E ≥ max(C 1 ( Ṽ , B, Ω), C 2 ( Ṽ , B, Ω)), (6.23)
where constants C 1 and C 2 are defined by (6.8) and (6.9). Let x ∈ Ω and let p 1 , p 2 ∈ S n-1

x,E (defined by (5.5)). Then the following estimate is valid:

||ψ 1 (t 1 , x, p 1 ) -ψ 1 (t 2 , x, p 2 )| -|t 1 v 1 -t 2 v 2 || ≤ C 3 |t 1 v 1 -t 2 v 2 |, (6.24) for (t 1 , t 2 ) ∈ [0, t +,x,p 1 [×[0, t +,x,p 2 [, where v i = p i 1- p 2 i c 2
, i = 1, 2, and where

C 3 = C 3 (E,
x, Ṽ , B, Ω) is defined by (6.10).

A proof of Proposition 6.2 is given in Subsection 6.7. We remind that sup x∈Ω C 3 (E, x, Ṽ , B, Ω) → 0, as E → +∞ (see (6.18)).

(6.25)

Taking account of (6.25) and the equality ψ 1 (0, x, p) = x for any (x, p) ∈ V E and taking account of Proposition 6.2, we obtain that at fixed and sufficiently large energy E the map ϕ E defined by (6.4) is one-to-one and its range is included in (Ω × Ω)\{(x, x) | x ∈ Ω}.

The following Proposition 6.3 is proved in Subsection 6.8.

Proposition 6.3. Assume that (6.26) where C 1 , C 6 and C 7 are defined by (6.8), (6.13) and (6.14). Then the map ϕ E defined by (6.4) is a local C 1 diffeomorphism at any point (t, x, p) ∈ Λ ∩ (]0, +∞[×V E ) . Now assume that Ω is a bounded strictly convex (in the strong sense) open domain of R n with C 2 boundary. Let χ Ω be a C 2 defining function for Ω. For E > c 2 + sup x∈Ω Ṽ (x), real constant C 8 (E, Ṽ , B, Ω) is defined by (6.15) with respect to χ Ω .

E ≥ C 1 ( Ṽ , B, Ω), E ≥ c 2 1 + 400n 2 β( Ṽ , B,Ω) 2 δ(Ω) 2 c 4 + sup x∈Ω Ṽ (x), min(C 6 (E, Ṽ , B, Ω), C 7 (E, Ṽ , B, Ω)) > 0,
The following Proposition 6.4 gives a surjectivity property of the map ϕ E defined by (6.4) at fixed and sufficiently large energy E.

Proposition 6.4. Let E ≥ C 1 ( Ṽ , B, Ω), E ≥ c 2 1 + 400n 2 β( Ṽ , B,Ω) 2 δ(Ω) 2 c 4 + sup x∈Ω Ṽ (x), min(C 6 (E, Ṽ , B, Ω), C 7 (E, Ṽ , B, Ω), C 8 (E, Ṽ , B, Ω)) > 0. (6.27)
where C 1 , C 6 , C 7 and C 8 are defined by (6.8), (6.13), (6.14) and (6.15).

Then (Ω × Ω)\{(x, x) | x ∈ Ω} is included in the range of the map ϕ E defined by (6.4).

A proof of Proposition 6.4 is given in Subsection 6.9. Taking account of Propositions 6.2, 6.3, 6.4, we obtain, in particular, that at fixed and sufficiently large energy E the map ϕ E defined by ( 6

.4) is a C 1 diffeomorphism from Λ ∩ (]0, +∞[×V E ) onto (Ω × Ω)\{(x, x) | x ∈ Ω}.
Now we are ready to prove Properties 2.1 and 2.2.

ε = {x 0 + (1 + ε)(x ′ -x 0 ) | x ′ ∈ D}.
Then Ω ε is also a bounded strictly convex in the strong sense open domain of R n with C 2 boundary and the map (6.29) where C 1 , C 2 , C 3 , C 6 , C 7 and C 8 are defined by (6.8), (6.9), (6.10), (6.13), (6.14) and (6.15) (taking account of (6.18), we obtain that if E is sufficiently large, then (6.29) is satisfied). Let ε > 0. We denote by

χ Ωε ∈ C 2 (R n , R) defined by χ Ωε (x) = χ D (x 0 + x-x 0 1+ε ), x ∈ R n , is a C 2 defining function for Ω ε . In addition, note that x ∈ ∂Ω ε ⇔ x 0 + x-x 0 1+ε ∈ ∂D, ∇χ Ωε (x) = (1 + ε) -1 ∇χ D (x 0 + x-x 0 1+ε ), x ∈ R n , Hessχ Ωε (x) = (1 + ε) -2 Hessχ D (x 0 + x-x 0 1+ε ), x ∈ R n , sup x∈Ωε inf{|x -y| | y ∈ D} = ε sup{|x -x 0 | | x ∈ D} -→ ε→0 + 0. (6.28) Note also that Ω ε 2 ⊆ Ω ε 1 if 0 < ε 2 < ε 1 . Let E > c 2 + sup x∈D V (x). Assume that E > max(C 1 ( Ṽ , B, D), C 2 ( Ṽ , B, D)), sup x∈Ω C 3 (E, x, Ṽ , B, D) < 1, min(C 6 (E, Ṽ , B, D), C 7 (E, Ṽ , B, D), C 8 (E, Ṽ , B, D)) > 0,
Λ ε the open subset of R × Ω ε × R n defined by Λ ε = {(t, x, p) ∈ R × Ω × R n | ∀s ∈ [0, t] ψ 1 (s, x, p) ∈ Ω ε }, and we denote by V E,ε the following smooth 2n -1-dimensional submanifold of R 2n V E,ε = {(x, p) ∈ Ω ε × R n | |p| = r Ṽ ,E (x)}. From (6.28) and continuity of ∂ α x Ṽ and ∂ α ′ x B for α, α ′ ∈ (N ∪ {0}) n , |α| ≤ 2, |α ′ | ≤ 1
, and from (6.8)-( 6.15), it follows that

C i ( Ṽ , B, Ω ε ) → C i ( Ṽ , B, D), as ε → 0 + , for i = 1, 2, sup x∈Ωε C 3 (E, x, Ṽ , B, Ω ε ) → sup x∈D C 3 (E, x, Ṽ , B, D), as ε → 0 + , C i (E, Ṽ , B, Ω ε ) → C i (E, Ṽ , B, D), as ε → 0 + , for i = 6, 7, 8.
Taking also account of (6.29) and Propositions 6.2, 6.3, 6.4, we obtain that there exists ε 0 > 0 such that (6.30) Let q 0 , q ∈ D, q 0 = q. Let ε 1 ∈]0, ε 0 [. From (6.30), it follows that there exists an unique p ε 1 ∈ S n-1 q 0 ,E and an unique positive real number t ε 1 such that q = ψ 1 (t ε 1 , q 0 , p ε 1 ) and (t ε 1 , q 0 , p ε

ϕ ε E : Λ ε ∩ (]0, +∞[×V E,ε ) → Ω ε × Ω ε , (t, x, p) → (x, ψ 1 (t, x, p)), is a C 1 diffeomorphism from Λ ε ∩ (]0, +∞[×V E,ε ) onto (Ω ε × Ω ε )\{(x, x) | x ∈ Ω ε } for any ε ∈]0, ε 0 [.
1 ) ∈ Λ ε 1 . Consider the function m ∈ C 2 (R, R), defined by m(t) = χ D (ψ 1 (t, q 0 , p ε 1 )), t ∈ R. Derivating twice m, we obtain m(t) = Hessχ D (ψ 1 (t, q 0 , p ε 1 ))(g(ψ 2 (t, q 0 , p ε 1 )), g(ψ 2 (t, q 0 , p ε 1 ))) (6.31) + 1 + |ψ 2 (t, q 0 , p ε 1 )| 2 c 2 -1/2 ∇χ D (ψ 1 (t, q 0 , p ε 1 )) • F (ψ 1 (t, q 0 , p ε 1 ), g(ψ 2 (t, q 0 , p ε 1 ))) - ψ 2 (t, q 0 , p ε 1 ) • F (ψ 1 (t, q 0 , p ε 1 ), g(ψ 2 (t, q 0 , p ε 1 ))) c 2 1 + |ψ 2 (t,q 0 ,pε 1 )| 2 c 2 3/2 ×∇χ D (ψ 1 (t, q 0 , p ε 1 )) • ψ 2 (t, q 0 , p ε 1 ),
for t ∈ R, where g is the function defined by (5.6) and • denotes the usual scalar product on R n (we used (5.1)). In addition, note that using the fact that χ D is a C 2 defining function of D, we obtain that for t ∈ R

ψ 1 (t, q 0 , p ε 1 ) ∈ D ⇔ m(t) < 0, ψ 1 (t, q 0 , p ε 1 ) ∈ ∂D ⇔ m(t) = 0.
Assume that there exists some s ∈]0,

t ε 1 [ such that ψ 1 (s, q 0 , p ε 1 ) ∈ D (i.e. m(s) ≥ 0). Let s 0 = sup{s ′ ∈ [0, s] | ψ 1 (s ′ , x, p ε 1 ) ∈ D}. Hence ψ 1 (s 0 , q 0 , p ε 1 ) ∈ ∂D, (i.e. m(s 0 ) = 0), (6.32) m(t) ≤ 0, for t ∈ [0, s 0 ].
(6.33)

From (6.32), (6.31), (5.2), and the estimates (6.6), |g(ψ 2 (t, q 0 , p ε 1 ))| < c, and definition (6.15), it follows that m(s 0 ) ≥ c 2 C 8 (E, Ṽ , B, D) > 0 (we used (6.29)). From (6.33) and from the estimate m(s 0 ) > 0 and Taylor expansion of

m at s 0 (m(t) = ṁ(s 0 )(t -s 0 ) + 1 2 m(s 0 )(t -s 0 ) 2 + o((t -s 0 ) 2 ), t ∈ R) it follows that ṁ(s 0 ) > 0.
Using also the equality m(s 0 ) = 0, we obtain that there exists ε ′ > 0 such that

s 0 + ε ′ < t ε 1 and m(s 0 + ε ′ ) > 0 which implies that ψ 1 (s 0 + ε ′ , q 0 , p ε 1 ) ∈ D. Then, due to sup z∈Ωε inf{|z -z ′ | | z ′ ∈ D} → 0 as ε → 0, there exists ε 2 ∈]0, ε 1 [ such that ψ 1 (s 0 + ε ′ , q 0 , p ε 1 )
∈ Ω ε 2 and using also (6.30), we obtain that there exists (p ε 2 , t ε 2 ) ∈ S n-1 q 0 ,E ×]0, +∞[ such that (p ε 2 , t ε 2 ) = (p ε 1 , t ε 1 ) and (t ε 2 , q 0 , p ε 2 ) ∈ Λ ε 2 and q = ψ 1 (t ε 2 , q 0 , p ε 2 ), which contradicts unicity of (p ε 1 , t ε 1 ).

We finally proved that ψ 1 (s, q 0 , p ε 1 ) ∈ D for all s ∈]0, t ε 1 [. (6.34) Now consider (6.36) (using Proposition 6.1 and (6.29), we obtain that t 2 and t 1 are real numbers that satisfy t 2t 1 ≤ 5δ(D) c ). Then for i = 1, 2, from (6.31), (5.2), (6.15) and ( 6.29), it follows that m(t i ) ≥ c 2 C 8 (E, Ṽ , B, D) > 0 . For all t ∈]t 1 , t 2 [, ψ 1 (t, q 0 , p) ∈ D. Hence m(t) < 0, t ∈]t 1 , t 2 [. This latter estimate and the estimate m(t i ) > 0 and Taylor expansion of (6.37) where N(t i ) = ∇χ(ψ 1 (t,q 0 ,pε 1 ))

t 2 = sup{t ∈]0, +∞[ | ψ 1 (s, q 0 , p ε 1 ) ∈ D for all s ∈]0, t]}, (6.35) t 1 = inf{t ∈ R | ψ 1 (s, q 0 , p ε 1 ) ∈ D for all s ∈ [t, t ε 1 [}
m at t i (m(t) = ṁ(t i )(t -t i ) + 1 2 m(t i )(t -t i ) 2 + o((t -t i ) 2 ), t ∈ R) for i = 1, 2, imply that ṁ(t 2 ) > 0 and ṁ(t 1 ) < 0, i.e. ∂ψ 1 ∂t (t, q 0 , p ε 1 ) |t=t 1 • N(t 1 ) < 0, ∂ψ 1 ∂t (t, q 0 , p ε 1 ) |t=t 2 • N(t 2 ) > 0,
|∇χ(ψ1(t,q0,pε 1 ))| , i = 1, 2. Statement (6.34) with (6.37) and (6.30) (with "ε = ε 1 ") proves (2.1) and (2.2).

6.6 Proof of Proposition 6.1. We denote

ẋ(t) 1-ẋ(t) 2 c 2 by p(t) for t ∈]t -, t + [. Let I(t) = 1 2 |x(t)| 2 , for t ∈]t -, t + [.
Derivating twice I and using (6.2), we obtain , and from (6.38) and ( 5.2), it follows that

Ï(t) = p(t) 2 1 + p(t) 2 c 2 + 1 1 + p(t) 2 c 2 F   x(t), p(t) 1 + p(t) 2 c 2   • x(t) (6.38) - p(t) • x(t) c 2 (1 + p(t) 2 c 2 ) 3/2 p(t) • F   x(t), p(t) 1 + p(t) 2 c 2   for t ∈]t -, t + [, where • denotes the usual scalar product in R n . From the estimate |p(t)| 1+ p(t) 2 c 2 < c, t ∈]t -, t + [
Ï(t) ≥ c 2 1 - 1 ( E-Ṽ (x(t)) c 2 ) 2 -2 |x(t)|(|∇ Ṽ (x(t))|+ i,j=1...n | Bi,j (x(t))|) E-Ṽ (x(t)) c 2 , for t ∈]t -,
(s) = x(s) • ẋ(s) ≥ -|x(s)|| ẋ(s)| ≥ -cδ(Ω)
. Using (6.1), we obtain

I(t) -I(s) = 1 2 (|x(t)| 2 - |x(s)| 2 ) ≤ δ(Ω) 2 .
From (6.40) and the two latter inequalities, it follows that 0

≥ -δ(Ω) 2 -cδ(Ω)(t -s) + c 2 4 (t -s) 2 , which implies that t -s ≤ δ(Ω) c (2 √ 2 + 2) < 5δ(Ω) c (the roots of -δ(Ω) 2 -cδ(Ω)X + c 2 4 X 2 are (δ(Ω)/c)(2 ± 2 √ 2)
). As t → t + and s → t -, the latter inequality proves (6.20). Proposition 6.1 is proved.

6.7 Proof of Proposition 6.2. Throughout this Subsection, we denote by γ x,p i (t) the point of R n defined by

γ x,p i (t) = ψ 1 (t, x, p i ),
for any t ∈ R and i = 1, 2, where ψ = (ψ 1 , ψ 2 ) is the flow of the differential system (5.1).

From (5.1), it follows that

γ x,p i (t) = x + tv i + t i 0 g(p i + σ 0 F (γ x,p i (τ ), γx,p i (τ ))dτ ) -g(p i )) dσ (6.41)
for t ∈ [0, t +,x,p i [ and i = 1, 2, where t +,x,p i is defined by (6.21) for i = 1, 2. From (6.41), it follows that

|t 1 v 1 -t 2 v 2 |-∆(t 1 , t 2 ) ≤ |γ x,p 1 (t 1 )-γ x,p 2 (t 2 )| ≤ |t 1 v 1 -t 2 v 2 |+∆(t 1 , t 2 ) (6.42)
where

∆(t 1 , t 2 ) = t 1 0 g(p 1 + σ 0 F (γ x,p 1 (τ ), γx,p 1 (τ ))dτ ) -g(p 1 ) dσ (6.43) - t 2 0 g(p 2 + σ 0 F (γ x,p 2 (τ ), γx,p 2 (τ ))dτ ) -g(p 2 ) dσ , for t 1 ∈ [0, t +,x,p 1 [ and t 2 ∈ [0, t +,x,p 2 [. We shall look for an upper bound of ∆(t 1 , t 2 ), t 1 ∈ [0, t +,x,p 1 [ and t 2 ∈ [0, t +,x,p 2 [, t 2 ≤ t 1 .
First case: v 1 • v 2 ≤ 0. Using (5.8), we obtain that

t i 0 g(p i + σ 0 F (γ x,p i (τ ), γx,p i (τ ))dτ ) -g(p i ) dσ (6.44) ≤ √ n t i 0 sup ε∈[0,1] 1 + c -2 p i + ε σ 0 F (γ x,p i (s), γx,p i (s))ds 2 -1/2 × σ 0 |F (γ x,p i (s), γx,p i (s))|dsdσ,
for i = 1, 2. From (6.6) and (6.20) and from the estimate |g(ψ 2 (s,

x, p i ))| ≤ c, it follows that σ 0 |F (γ x,p i (s), γx,p i (s))|ds ≤ 10nδ(Ω)β( Ṽ , B, Ω) c . (6.45) for σ ∈ [0, t i ], i = 1, 2. Using p i ∈ S n-1
x,E and (6.45) and (6.23), we obtain (6.46) for ε ∈ [0, 1] and σ ∈ [0, t i ]. From (6.45), (6.46) and (6.44), it follows that

p i + ε σ 0 F (γ x,p i (s), γx,p i (s))ds ≥ 1 2 r Ṽ ,E (x),
t i 0 g(p i + σ 0 F (γ x,p i (s), γx,p i (s))ds) -g(p i ) dσ ≤ t i 20n 3/2 cδ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) , (6.47
)

for i = 1, 2. Using v 1 • v 2 ≤ 0, we obtain that |v i |s i ≤ |s 1 v 1 -s 2 v 2 | for i = 1, 2
and for s 1 ≥ 0, s 2 ≥ 0. Using this latter inequality and (6.43) and (6.47) and

equality |v i | = c 1 -E-Ṽ (x) c 2 -2
, we obtain

∆(t 1 , t 2 ) ≤ 40c -1 n 3/2 δ(Ω)β( Ṽ , B, Ω)r Ṽ ,E (x) -1 |t 1 v 1 -t 2 v 2 |. Second case: v 1 • v 2 ≥ 0. From (6.43), it follows that ∆(t 1 , t 2 ) ≤ ∆ 1 (t 1 , t 2 ) + ∆ 2 (t 1 , t 2 ), (6.48)
where

∆ 1 (t 1 , t 2 ) = t 1 t 2 g(p 1 + σ 0 F (γ x,p 1 (τ ), γx,p 1 (τ ))dτ ) -g(p 1 ) dσ (6.49) ∆ 2 (t 1 , t 2 ) = t 2 0 g(p 1 + σ 0 F (γ x,p 1 (τ ), γx,p 1 (τ ))dτ ) -g(p 1
)dσ (6.50)

-g(p 2 + σ 0 F (γ x,p 2 (τ ), γx,p 2 (τ ))dτ ) -g(p 2 ) dσ .
An upper bound for ∆ 1 (t 1 , t 2 ). Using (6.49) and (5.8), we obtain that

∆ 1 (t 1 , t 2 ) ≤ √ n(t 1 -t 2 ) t 1 t 2 |F (γ x,p 1 (s), γx,p 1 (s))|ds (6.51) × sup ε∈[0,1] σ∈[0,t 1 ] 1 + c -2 p 1 + ε σ 0 F (γ x,p 1 (s), γx,p 1 (s))ds 2 -1/2 .
In the same manner than in the first case (v 1 • v 2 ≤ 0), we obtain

∆ 1 (t 1 , t 2 ) ≤ (t 1 -t 2 ) 20n 3/2 cδ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) . (6.52) Note that from |v 1 | = |v 2 | and t i ≥ 0, i = 1, 2, it follows that |v 1 |(t 1 -t 2 ) ≤ |t 1 v 1 -t 2 v 2 |.
Using this latter inequality with (6.52), we obtain

∆ 1 (t 1 , t 2 ) ≤ 20n 3/2 δ(Ω)β( Ṽ , B, Ω) cr Ṽ ,E (x) |t 1 v 1 -t 2 v 2 | (6.53) (we use the equality |v 1 | = c 1 -E-Ṽ (x) c 2 -2
).

An upper bound for ∆ 2 (t 1 , t 2 ). Note that

g j (p i + σ 0 F (γ x,p i (τ ), γx,p i (τ ))dτ ) -g j (p i ) = 1 0 ∇g j (p i + ε σ 0 F (γ x,p i (τ ), γx,p i (τ ))dτ )• σ 0 F (γ x,p i (s), γx,p i (s))ds dε for i = 1, 2 and j = 1..n, where g = (g 1 , . . . , g n ). Hence ∆ j 2 (t 1 , t 2 ) ≤ t 2 0 ∆ 2,1,j (σ)dσ + t 2 0 ∆ 2,2,j (σ)dσ, (6.54) where ∆ 2 (t 1 , t 2 ) = (∆ 1 2 (t 1 , t 2 ), . . . , ∆ n 2 (t 1 , t 2 )) and ∆ 2,1,j (σ) = 1 0 ∇g j (p 1 + ε σ 0 F (γ x,p 1 (τ ), γx,p 1 (τ ))dτ ) -∇g j (6.55) (p 2 + ε σ 0 F (γ x,p 2 (τ ), γx,p 2 (τ ))dτ ) • σ 0 F (γ x,p 1 (s), γx,p 1 (s))dsdε , ∆ 2,2,j (σ) = 1 0 ∇g j (p 2 + ε σ 0 F (γ x,p 2 (τ ), γx,p 2 (τ ))dτ )• (6.56) σ 0 [F (γ x,p 1 (s), γx,p 1 (s)) -F (γ x,p 2 (s), γx,p 2 (s))] dsdε
for σ ∈ [0, t 2 ] and j = 1 . . . n.

We first look for an upper bound for ∆ 2,1,j (σ). Since v 1 • v 2 ≥ 0, we obtain p 1 • p 2 ≥ 0. Using this latter inequality and the equality p 2 1 = p 2 2 , we obtain that

|µp 1 + (1 -µ)p 2 | = µ 2 p 2 1 + (1 -µ) 2 p 2 2 + 2µ(1 -µ)p 1 • p 2 ≥ µ 2 p 2 1 + (1 -µ) 2 p 2 2 ≥ 1 √ 2 |p 1 | = r Ṽ ,E (x) √ 2 , (6.57)
for any µ ∈ [0, 1]. From (6.45) and (6.57) and (6.23), it follows that

µp 1 + (1 -µ)p 2 + µǫ σ 0 F (γ x,p 1 (τ ), γx,p 1 (τ ))dτ + (1 -µ)ǫ × σ 0 F (γ x,p 2 (τ ), γx,p 2 (τ ))dτ ≥ |µp 1 + (1 -µ)p 2 | - 10nδ(Ω)β( Ṽ , B, Ω) c ≥ 1 2 √ 2 r Ṽ ,E (x), (6.58) for µ, ǫ ∈ [0, 1]. From (5.1), it follows that γx,p i (σ) = g(p i + σ 0 F (γ x,p i (τ ), γx,p i (τ ))dτ )
for σ ∈ [0, t i ], i = 1, 2. Using this latter equality and (5.8), we obtain

| γx,p 1 (σ) -γx,p 2 (σ)| ≤ (6.59) √ n sup µ∈[0,1] 1 + c -2 |µp 1 + (1 -µ)p 2 + µ σ 0 F (γ x,p 1 (τ ), γx,p 1 (τ ))dτ + (1 -µ) σ 0 F (γ x,p 2 (τ ), γx,p 2 (τ ))dτ | 2 -1/2 ×|p 1 -p 2 + σ 0 (F (γ x,p 1 (τ ), γx,p 1 (τ )) -F (γ x,p 2 (τ ), γx,p 2 (τ ))) dτ | for σ ∈ [0, t 2 ].
Note that from (6.7) and the inequality | γx,p 2 (τ )| ≤ c, it follows that

|F (γ x,p 1 (τ ), γx,p 1 (τ )) -F (γ x,p 2 (τ ), γx,p 2 (τ ))| ≤ (6.60) nβ( Ṽ , B, Ω)(2|γ x,p 1 (τ ) -γ x,p 2 (τ )| + 1 c | γx,p 1 (τ ) -γx,p 2 (τ )|), for τ ∈ [0, t 2 ].
Using (6.58)-(6.60), we obtain

| γx,p 1 (σ) -γx,p 2 (σ)| ≤ 2 3/2 √ nc 2 E -Ṽ (x) |p 1 -p 2 | + nβ( Ṽ , B, Ω) (6.61) 2 σ 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ + 1 c σ 0 | γx,p 1 (τ ) -γx,p 2 (τ )|dτ , for σ ∈ [0, t 2 ].
We shall use the following Gronwall's lemma.

Gronwall's lemma. Let a > 0 and let φ ∈ C([0, a], [0, +∞[) be a continuous map and let

A, B ∈ [0, +∞[ be such that φ(t) ≤ A + B t 0 φ(s)ds for all t ∈ [0, a]. Then φ(t) ≤ Ae Bt for all t ∈ [0, a].
Taking account of (6.61), Gronwall's lemma and (6.20), we obtain that

| γx,p 1 (σ) -γx,p 2 (σ)| ≤ 2 3/2 √ nc 2 E -Ṽ (x) |p 1 -p 2 | + 2nβ( Ṽ , B, Ω) (6.62) σ 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x) , for σ ∈ [0, t 2 ].
From (5.9) and (6.55), it follows that

∆ 2,1,j (σ) ≤ (6.63) 3 √ n c 1 0 sup µ∈[0,1] 1 + 1 c 2 µp 1 + (1 -µ)p 2 + µǫ σ 0 F (γ x,p 1 (τ ), γx,p 1 (τ ))dτ + (1 -µ)ǫ σ 0 F (γ x,p 2 (τ ), γx,p 2 (τ ))dτ 2 -1 ×|p 1 -p 2 + ε σ 0 (F (γ x,p 1 (τ ), γx,p 1 (τ )) -F (γ x,p 2 (τ ), γx,p 2 (τ ))) dτ | × σ 0 |F (γ x,p 1 (s), γx,p 1 (s))|dsdε, for σ ∈ [0, t 2 ].
From (6.6) and the estimate

| γx,p 1 (s)| ≤ c, it follows that σ 0 |F (γ x,p 1 (s), γx,p 1 (s))|ds ≤ 2nβ( Ṽ , B, Ω)σ. (6.64) for σ ∈ [0, t 1 ].
From (6.63), (6.58), (6.64) and (6.60) and (6.62), it follows that

∆ 2,1,j (σ) ≤ 48n 3/2 c 3 β( Ṽ , B, Ω) (E -Ṽ (x)) 2 |p 1 -p 2 |σ + nβ( Ṽ , B, Ω)σ (6.65) 2 σ 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ + 2 3/2 √ nc E -Ṽ (x) |p 1 -p 2 |σ + 2nβ( Ṽ , B, Ω) σ 0 s 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ ds × e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x) for σ ∈ [0, t 2 ]. From |v 1 | = |v 2 | and t 2 ≤ t 1 , it follows that |v 1 -v 2 |σ ≤ |v 1 -v 2 |t 2 ≤ |t 1 v 1 -t 2 v 2 |, for σ ∈ [0, t 2 ]
. Note that using these latter estimates and p i ∈ S n-1

x,E , i = 1, 2, we obtain

|p 1 -p 2 |σ ≤ E -Ṽ (x) c 2 |t 1 v 1 -t 2 v 2 |, (6.66) for σ ∈ [0, t 2 ].
Using (6.66), the estimate

σ 0 s 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ ds ≤ σ σ 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ and σ ≤ 5δ(Ω) c
(due to (6.20)) and (6.65), we obtain

∆ 2,1,j (σ) ≤ 1 + 10 √ 2n 3/2 δ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x) × 48n 3/2 cδ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) |t 1 v 1 -t 2 v 2 | (6.67) + 480n 5/2 c 2 δ(Ω)β( Ṽ , B, Ω) 2 (E -Ṽ (x)) 2 σ 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ for σ ∈ [0, t 2 ].
We look for an upper bound for ∆ 2,2,j (σ), σ ∈ [0, t 2 ], defined by (6.56). Using (6.56), (5.7), and (6.46), and using (6.60), (6.62) and (6.66), we obtain

∆ 2,2,j (σ) ≤ 1 + 10 √ 2n 3/2 δ(Ω)β( Ṽ , B, Ω) E -Ṽ (x) e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x) × 4nc 2 β( Ṽ , B, Ω) E -Ṽ (x) σ 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ (6.68) + 2 5/2 n 3/2 cβ( Ṽ , B, Ω) E -Ṽ (x) e 10 √ 2δ(Ω)n 3/2 β( Ṽ , B,Ω) E-Ṽ (x) |t 1 v 1 -t 2 v 2 |, for σ ∈ [0, t 2 ].
Note also that from (6.42), it follows that

σ 0 |γ x,p 1 (τ )-γ x,p 2 (τ )|dτ ≤ σ 0 ∆ (τ, τ )dτ + σ 2 2 |v 1 -v 2 |, for σ ∈ [0, t 2 ]
. Hence using also σ ≤ 5δ(Ω) c (due to (6.20)) and σ|v

1 -v 2 | ≤ |t 1 v 1 -t 2 v 2 |, we obtain σ 0 |γ x,p 1 (τ ) -γ x,p 2 (τ )|dτ ≤ σ 0 ∆(τ, τ )dτ + 5δ(Ω) 2c |t 1 v 1 -t 2 v 2 |, (6.69) for σ ∈ [0, t 2 ]. Note that t 2 ≤ 5δ(Ω) c
and note that from positiveness of ∆ it follows that

t 2 0 σ 0 ∆(τ, τ )dτ dσ ≤ t 2 t 2
0 ∆(τ, τ )dτ. Hence using (6.69), we obtain

t 2 0 σ 0 |γ x,p 1 (τ )-γ x,p 2 (τ )|dτ dσ ≤ 5δ(Ω) c t 2 0 ∆(τ, τ )dτ + 25δ(Ω) 2 2c 2 |t 1 v 1 -t 2 v 2 |, (6.70) for σ ∈ [0, t 2 ].
Combining (6.48), (6.53), (6.67), (6.68) and (6.70), we obtain

∆(t 1 , t 2 ) ≤ C 4 (E, x, Ṽ , B, Ω)|t 1 v 1 -t 2 v 2 | + cC 5 (E, x, Ṽ , B, Ω) t 2 0 ∆(τ,
-v 2 |σ ≤ |t 1 v 1 -t 2 v 2 |, σ ≤ t 2 , give in particular ∆(σ, σ) ≤ C 4 |t 1 v 1 -t 2 v 2 | + cC 5 σ 0 ∆(τ, τ )dτ (6.72)
for σ ∈ [0, t 2 ]. Using (6.72) and using Gronwall's lemma (formulated above) and σ ≤ 5δ(Ω) c , we obtain

∆(σ, σ) ≤ C 4 e 5δ(Ω)C 5 |t 1 v 1 -t 2 v 2 |, (6.73) for σ ∈ [0, t 2 ].
Using (6.73) and (6.71) and t 2 ≤ 5δ(Ω) c , we obtain ∆(t

1 , t 2 ) ≤ C 3 (E, x, Ṽ , B, Ω)|t 1 v 1 -t 2 v 2 |, for t 1 ∈ [0, t +,x,p 1 [ and t 2 ∈ [0, t +,x,p 2 [, t 1 ≥ t 2 .
Proposition 6.2 is proved.

6.8 Proof of Proposition 6.3. We shall work in coordinates. We consider the following infinitely smooth parametrizations of S n-1 , φ i,± : B n-1 (0, 1) → S n-1 , i = 1 . . . n, defined by

φ i,± (w) = (6.74)        w 1 , . . . , w i-1 , ± 1 -n-1 l=1 w l 2 , w i , . . . , w n-1 , if 1 ≤ i ≤ n -1, w 1 , . . . , w n-1 , ± 1 -n-1 l=1 w l 2 , if i = n
for w = (w 1 , . . . , w n-1 ) ∈ B n-1 (0, 1) and where B n-1 (0, 1) denotes the unit Euclidean open ball of R n-1 of center 0. Let (t 0 , x 0 , p 0 ) ∈ Λ∩(]0, +∞[×V E ), p 0 = (p 1 0 , . . . , p n 0 ). Then (t, U is an open subset of B n-1 (0, 1), (6.75)

x 0 , p 0 ) ∈ Λ for all t ∈ [0, t 0 ]. As Λ is an open subset of R×R n ×R n , there exists ε > 0 such that {(t, x, p) ∈ R×R n ×R n | -ε < t < t 0 +ε, max(|x-x 0 |, |p-p 0 |) < ε} ⊆ Λ.
|p k 0 | ≥ n -1/2 |p 0 |, (6.76) if ± p k 0 > 0 then φ(w) = φ k,± (
w) for all w ∈ U, (6.77) (t, x, r Ṽ ,E (x)φ(w)) ∈ Λ, for (w, t, x) ∈ U×]ε, t 0 + ε[×B(x 0 , ε). (6.78) (6.79) where ψ = (ψ 1 , ψ 2 ) is the flow of the differential system (5.1). Let w 0 ∈ U be such that φ(w 0 ) = p 0 |p 0 | . We shall prove that ( ∂Q ∂t (t 0 , x 0 , w 0 ), ∂Q ∂w 1 (t 0 , x 0 , w 0 ), . . . ,

Consider Q ∈ C 1 (] -ε, t 0 + ε[×B(x 0 , ε) × U, Ω) defined by Q(t, x, w) = ψ 1 (t, x, r Ṽ ,E (x)φ(w)), (t, x, w) ∈] -ε, t 0 + ε[×B(x 0 , ε) × U,
∂Q ∂w n-1 (t 0 , x 0 , w 0 )) is a basis of R n .
Note that from (6.2), it follows that

Q(t, x, w) = x + tg(r Ṽ ,E (x)φ(w)) + t 0 g r Ṽ ,E (x)φ(w) (6.80) + σ 0 F (Q(s, x, w), ∂Q ∂s (s, x, w))ds -g(r Ṽ ,E (x)φ(w)) dσ,
for w ∈ U and (t, x) ∈]ε, t 0 + ε[×B(x 0 , ε) (where g, r Ṽ ,E and F are defined by (5.6), (5.4) and (5.3)).

We shall prove (6.82). Using (6.80) we obtain ∂Q ∂t (t, x, w) = g(r Ṽ ,E (x)φ(w)) + g r Ṽ ,E (x)φ(w) (6.81)

+ t 0 F (Q(s, x, w), ∂Q ∂s (s, x, w))ds -g(r Ṽ ,E (x)φ(w)) ,
for w ∈ U and (t, x) ∈]ε, t 0 + ε[×B(x 0 , ε). Combining (6.81), (6.26), (5.8), (6.6) and estimates (6.82) for w ∈ U and (t, x) ∈ [0, t 0 + ε[×B(x 0 , ε).

| ∂Q ∂t (t, x, w)| ≤ c, t ≤ 5δ(Ω) c , it follows that ∂Q ∂t (t, x, w) -g(r Ṽ ,E (x)φ(w)) ≤ 4n 3/2 c 2 β( Ṽ , B, Ω) E -Ṽ (x) t,
We shall prove (6.93

). Let i = 1 . . . n -1. Let X i ∈ C(] -ε, t 0 + ε[×B(x 0 , ε) × U, R n ) be defined by X j i (s, x, w) = n l=1 ∂F j ∂x ′ l (x ′ , ∂Q ∂s (s, x, w)) |x ′ =Q(s,x,w) ∂Q l ∂w i (s, x, w) (6.83) + ∂F j ∂y ′ l (Q(s, x, w), y ′ ) |y ′ = ∂Q ∂s (s,x,w)
∂ Ql ∂w i (s, x, w) , for j = 1 . . . n, (s, x, w) ∈]ε, t 0 + ε[×B(x 0 , ε) × U, and where

X i = (X 1 i , . . . , X n i ) and Q ∈ C 1 (] -ε, t 0 + ε[×B(x 0 , ε) × U, R) is defined by Q(s, x, w) = g(ψ 2 (s, x, r Ṽ ,E (x)φ(w))) = ∂Q ∂s (s, x, w), (6.84) for (s, x, w) ∈] -ε, t 0 + ε[×B(x 0 , ε) × U.
From (6.5), and (5.3), it follows that

|X i (σ, x, w)| ≤ β( Ṽ , B, Ω)n 2 √ n ∂Q ∂w i (σ, x, w) + c -1 ∂ Q ∂w i (σ, x, w) , (6.85) for (σ, x, w) ∈] -ε, t 0 + ε[×B(x 0 , ε) × U.
We shall estimate Q. Note that from (6.2), it follows that Ql (s, x, w) = g l (r Ṽ ,E (x)φ(w) + s 0 F (Q(σ, x, r Ṽ ,E (x)φ(w)), Q(σ, x, r Ṽ ,E (x)φ(w)))dσ), for (s, x, w) ∈]ε, t 0 + ε[×B(x 0 , ε) × U and l = 1 . . . n. From this latter equality and (6.83), it follows that

∂ Ql ∂w i (s, x, w) = ∇g l r Ṽ ,E (x)φ(w) + s 0 F (Q(σ, x, r Ṽ ,E (x)φ(w)), Q(σ, x, r Ṽ ,E (x)φ(w)))dσ • r Ṽ ,E (x) ∂φ ∂w i (w) + s 0 X i (σ, x, w)dσ , for (s, x, w) ∈] -ε, t 0 + ε[×B(x 0 , ε) × U. Hence ∂ Qj ∂w i (s, x, w) -r Ṽ ,E (x)∇g j (r Ṽ ,E (x)φ(w)) • ∂φ ∂w i (w) ≤ (6.86) r Ṽ ,E (x) ∇g j r Ṽ ,E (x)φ(w) + s 0 F (Q(σ, x, r Ṽ ,E (x)φ(w)), Q(σ, x, r Ṽ ,E (x)φ(w)))dσ -∇g j r Ṽ ,E (x)φ(w) • ∂φ ∂w i (w) + ∇g j r Ṽ ,E (x)φ(w) + s 0 F (Q(σ, x, r Ṽ ,E (x)φ(w)), Q(σ, x, r Ṽ ,E (x) φ(w)))dσ) • s 0 X i (σ, x, w)dσ ,
for j = 1 . . . n and (s, x, w) ∈]ε, t 0 + ε[×B(x 0 , ε) × U. We estimate the second term of the sum on the right-hand side of (6.86) by using (5.7), (6.6), (6.26) and s ≤ 5δ(Ω) c , and (6.85). We estimate the first term of the sum on the right-hand side of (6.86) by using (5.9) and (6.6), (6.26) and s ≤ 5δ(Ω) c , and the estimate

s 0 F (Q(σ, x, r Ṽ ,E (x)φ(w)), Q(σ, x, r Ṽ ,E (x)φ(w)))dσ ≤ 2nβ( Ṽ , B, Ω)s, for (s, x, w) ∈] -ε, t 0 + ε[×B(x 0 , ε) × U. We obtain ∂ Q ∂w i (s, x, w) - r Ṽ ,E (x) E-Ṽ (x) c 2 ∂φ ∂w i (w) ≤ 2n √ nc 3 β( Ṽ , B, Ω) 12 √ n + 1 × r Ṽ ,E (x) E -Ṽ (x) 2 | ∂φ ∂w i (w)|s + 2c 2 β( Ṽ , B, Ω)n √ n E -Ṽ (x) s 0 ∂Q ∂w i (σ, x, w) dσ ×2 √ n + c -1 s 0 ∂ Q ∂w i (σ, x, w) - r Ṽ ,E (x) E-Ṽ (x) c 2 ∂φ ∂w i (w) dσ   , for (s, x, w) ∈ [0, t 0 + ε[×B(x 0 , ε) × U (note that E-Ṽ (x) c 2 -1 ∂φ ∂w i (w) = ∇g j (r Ṽ ,E (x)φ(w)) • ∂φ ∂w i (w) j=1...n
). From Gronwall's lemma (formulated in Subsection 6.7) and t 0 +ε ≤ 5δ (Ω) c , it follows that

∂ Q ∂w i (s, x, w) - r Ṽ ,E (x) E-Ṽ (x) c 2 ∂φ ∂w i (w) ≤ 2c 2 n 3/2 β( Ṽ , B, Ω) E -Ṽ (x) e 10δ(Ω)β( Ṽ , B,Ω)n 3/2 E-Ṽ (x) (6.87) × c 12 √ n + 1 r Ṽ ,E (x) E -Ṽ (x) ∂φ ∂w i (w) s + 2 √ n s 0 ∂Q ∂w i (σ, x, w) dσ , for (s, x, w) ∈ [0, t 0 + ε[×B(x 0 , ε) × U. From (6.84), it follows that Q(s, x, w) = x + s 0 Q(σ, x, w)dσ, for (s, x, w) ∈ [0, t 0 + ε[×B(x 0 , ε) × U. Hence ∂Q ∂w i (s, x, w) = s 0 ∂ Q ∂w i (σ, x, w)dσ, for (s, x, w) ∈ [0, t 0 + ε[×B(x 0 , ε) × U.
This latter equality and (6.87) imply

∂Q ∂w i (s, x, w) - r Ṽ ,E (x) E-Ṽ (x) c 2 ∂φ ∂w i (w)s ≤ 2c 2 n 3/2 β( Ṽ , B, Ω) E -Ṽ (x) (6.88) e 10δ(Ω)β( Ṽ , B,Ω)n 3/2 E-Ṽ (x) c 12 √ n + 1 r Ṽ ,E (x) E -Ṽ (x) ∂φ ∂w i (w) s 2 2 +2 √ n s 0 τ 0 ∂Q ∂w i (σ, x, w) dσdτ , for (s, x, w) ∈ [0, t 0 + ε[×B(x 0 , ε) × U. Note that s 0 τ 0 ∂Q ∂w i (σ, x, w) dσdτ ≤ s s 0 ∂Q ∂w i (σ, x, w) dσ (6.89) ≤ 5δ(Ω) c s 0 ∂Q ∂w i (σ, x, w) -σ r Ṽ ,E (x) E-Ṽ (x) c 2 ∂φ ∂w i (w) dσ + 5r Ṽ ,E (x)δ(Ω) c E-Ṽ (x) c 2 ∂φ ∂w i (w) s 2 2 , for (s, x, w) ∈ [0, t 0 + ε[×B(x 0 , ε) × U (we used that t 0 + ε ≤ 5δ(Ω) c ). Let C ′ 3 (E, x, Ṽ , B, Ω) = 20cn 2 β( Ṽ , B, Ω)δ(Ω)e 10δ(Ω)β( Ṽ , B,Ω)n 3/2 E-Ṽ (x) (6.90) C ′ 4 (E, x, Ṽ , B, Ω) = 2c 2 n 3/2 β( Ṽ , B, Ω)e 10δ(Ω)β( Ṽ , B,Ω)n 3/2 E-Ṽ (x) (6.91) ×   c 12 √ n + 1 r Ṽ ,E (x) E -Ṽ (x) + 10r Ṽ ,E (x) √ nδ(Ω) c E-Ṽ (x) c 2   , C ′ 5 (E, x, Ṽ , B, Ω) = C ′ 4 (E, x, Ṽ , B, Ω)e C ′ 3 (E,x, Ṽ , B,Ω) E-Ṽ (x)
, (6.92) for x ∈ Ω. From (6.88)-(6.92) and Gronwall's lemma (formulated in Subsection 6.7), it follows that

∂Q ∂w i (s, x, w) - r Ṽ ,E (x) E-Ṽ (x) c 2 ∂φ ∂w i (w)s ≤ C ′ 5 (E, x, Ṽ , B, Ω) E -Ṽ (x) ∂φ ∂w i (w) s 2 2 (6.93) for (s, x, w) ∈ [0, t 0 + ε[×B(x 0 , ε) × U.
Now we assume without loss of generality that the integer k in (6.76) is n, and p n 0 > 0. We remind that w 0 ∈ U is defined by φ(w 0 ) = p 0 |p 0 | . We shall prove (6.99).

From (6.77), it follows that (6.95) for l = 1 . . . n -1 and where (e 1 , . . . , e n ) is the canonical basis of R n and w 0 = (w 1 0 , . . . , w n-1 0 ) (for (6.95), we used the estimate

∂φ ∂w l (w 0 ) = e l - w l 0 1 -|w 0 | 2 e n , (6.94) | ∂φ ∂w l (w 0 )| ≤ √ 1 + n,
p n 0 |p 0 | ≥ n -1/2 which implies that 1 -|w 0 | 2 ≥ 1
n and we used |w l 0 | ≤ |w 0 | < 1, l = 1 . . . n -1 and we used (6.94)). In addition, using (6.94), we obtain

| n-1 l=1 µ l ∂φ ∂w l (w 0 )| ≥ |(µ 1 , . . . , µ n-1 )| , (6.96) for all (µ 1 , . . . , µ n-1 ) ∈ R n-1 .
Using the fact that φ(w 0 ) ∈ S n-1 is orthogonal to ∂φ ∂w l (w 0 ), l = 1 . . . n-1, and using (6.96), we obtain

|µ 1 φ(w 0 ) + n-1 l=1 µ l+1 ∂φ ∂w l (w 0 )| = µ 2 1 + | n-1 l=1 µ l+1 ∂φ ∂w l (w 0 )| 2 ≥ n -1/2 n l=1 |µ l |, (6.97) for all (µ 1 , . . . , µ n ) ∈ R n . Note that λ 1 ∂Q ∂t (t 0 , x 0 , w 0 ) + n-1 l=1 λ l+1 ∂Q ∂w l (t 0 , x 0 , w 0 ) ≥ (6.98) λ 1 g(r Ṽ ,E (x 0 )φ(w 0 )) + n-1 l=1 λ l+1 c 2 r Ṽ ,E (x 0 )t 0 E -Ṽ (x 0 ) ∂φ ∂w l (t 0 , x 0 , w 0 ) -|λ 1 | ∂Q ∂t (t 0 , x 0 , w 0 ) -g(r Ṽ ,E (x 0 )φ(w 0 )) - n-1 l=1 |λ l+1 | ∂Q ∂w l (t 0 , x 0 , w 0 ) - c 2 r Ṽ ,E (x 0 )t 0 E -Ṽ (x 0 ) ∂φ ∂w l (t 0 , x 0 , w 0 ) , for (λ 1 , . . . , λ n ) ∈ R n .
We estimate the first term on the right-hand side of (6.98) by using (6.97)

(note that g(r Ṽ ,E (x 0 )φ(w 0 )) = c 2 r Ṽ ,E (x 0 ) E-Ṽ (x 0 ) φ(w 0 )).
We estimate the second term and third term on the right-hand side of (6.98) by using (6.82) and (6.93) and (6.95). Using also the estimate t 0 ≤ 5δ(Ω) c , we finally obtain

λ 1 ∂Q ∂t (t 0 , x 0 , w 0 ) + n-1 l=1 λ l+1 ∂Q ∂w l (t 0 , x 0 , w 0 ) ≥ |λ 1 | cC 6 √ n + t 0 n-1 l=1 |λ l+1 | C 7 √ n , ( 6
.99) for (λ 1 , . . . , λ n ) ∈ R n . This latter inequality and (6.26) and t 0 > 0 imply that the family ∂Q ∂t (t 0 , x 0 , w 0 ), ∂Q ∂w 1 (t 0 , x 0 , w 0 ) , . . . , ∂Q ∂w n-1 (t 0 , x 0 , w 0 ) is free. Then using inverse function theorem, it follows that ϕ E is a local C 1 diffeomorphism at (t 0 , x 0 , p 0 ). Proposition 6.3 is proved.

6.9 Proof of Proposition 6.4. Before proving Proposition 6.4, we shall first prove the following Lemma 6.1.

Lemma 6.1. Assume that

E ≥ C 1 ( Ṽ , B, Ω), C 8 (E, Ṽ , B, Ω) > 0, (6.100)
where C 1 and C 8 are defined by (6.8) and (6.15). Let x ∈ Ω and p ∈ S n-1 x,E . Then ψ 2 (t +,x,p , x, p) • N(ψ 1 (t +,x,p , x, p)) > 0, (6.101)

where t +,x,p is defined by (6.21) and ψ = (ψ 1 , ψ 2 ) is the flow of the differential system (5.1), and where N(y) denotes the unit outward normal vector of ∂D at y ∈ ∂D (• denotes the usual scalar product on R n ).

Proof of Lemma 6.1. Consider the function m ∈ C 2 (R, R) defined by m(t) = χ Ω (ψ 1 (t, x, p)), t ∈ R, (6.102)

where χ Ω is a C 2 defining function for Ω (see definition of C 8 , (6.15)). Derivating twice (6.102) and using (5.1), we obtain m(t) = Hessχ Ω (ψ 1 (t, x, p))(g(ψ 2 (t, x, p)), g(ψ 2 (t, x, p))) (6.103)

+ 1 + |ψ 2 (t, x, p)| 2 c 2 -1/2 ∇χ Ω (ψ 1 (t, x, p)) • F (ψ 1 (t, x, p), g(ψ 2 (t, x, p))) - ψ 2 (t, x, p) • F (ψ 1 (t, x, p), g(ψ 2 (t, x, p))) c 2 1 + |ψ 2 (t,x,p)| 2 c 2 3/2 ×∇χ Ω (ψ 1 (t, x, p)) • ψ 2 (t,
(m(t) = ṁ(t +,x,p )(t -t +,x,p ) + 1 2 m(t +,x,p )(t -t +,x,p ) 2 + o((t -t +,x,p ) 2 ), t ∈ R) imply that ṁ(t +,x,p ) > 0.
Lemma 6.1 is proved. Now we are ready to prove Proposition 6.4. Let x ∈ Ω. As n ≥ 2, the set Ω\{x} is connected and we shall prove that the set

A x = {y ∈ Ω\{x} | there exists p ∈ S n-1
x,E and t > 0 such that (t, x, p) ∈ Λ and ψ 1 (t, x, p) = y} is a closed and open nonempty subset of Ω\{x} (where ψ = (ψ 1 , ψ 2 ) is the differential flow of (5.1)). Then we will have A x = Ω\{x}, which will prove Proposition 6.4.

Note that A x is nonempty since for p ∈ S n-1

x,E , (0, x, p) ∈ Λ and ∂ψ 1 ∂t (t, x, p) |t=0 = g(p) = 0. Hence there exists ε > 0 such that (ε, x, p) ∈ Λ and ψ 1 (ε, x, p) = ψ 1 (0, x, p) = x.

Note also that A

x is an open subset of Ω\{x}. Let y ∈ A x . Then there exists p ∈ S n-1

x,E and t > 0 such that (t, x, p) ∈ Λ and ψ 1 (t, x, p) = y. From (6.27) and Proposition 6.3, it follows, in particular, that there exists an open neighborhood U ⊆ Ω\{x} of y such that U ⊆ A x .

It remains to prove that A x is a closed subset of Ω\{x}. Consider a sequence (y k ) of points of Ω\{x} which converges to some y ∈ Ω\{x} as k → +∞. For each k, there exists p k ∈ S n-1

x,E and t k > 0 such that (t k , x, p k ) ∈ Λ and ψ 1 (t k , x, p k ) = y k . (6.104)

From Proposition 6.1, it follows that t k ∈ [0, 5δ(Ω) c ] for all k. Using compactness of [0, 5δ(Ω) c ] and compactness of S n-1

x,E , we can assume that (t k ) converges to some t ∈ [0, 5δ(Ω) c ] and that (p k ) converges to some p ∈ S n-1 x,E . Using (6.104) and continuity of ψ 1 , we obtain

y = lim k→+∞ y k = lim k→+∞ ψ 1 (t k , x, p k ) = ψ 1 (t, x, p).
(6.105) Note that t > 0 since y = x. Let s ∈ [0, t[. Then using that t k → t as k → +∞, we obtain that there exists a rank N s such that s < t k for k ≥ N s . Hence (s, x, p k ) ∈ Λ for k ≥ N s and, in particular, ψ 1 (s, x, p k ) ∈ Ω for k ≥ N s .

Hence we obtain that .106) Using Lemma 6.1 with (6.105) (y ∈ Ω) and (6.106), we obtain that (t, x, p) ∈ Λ ∩ ]0, +∞[×{x} × S n-1

ψ 1 (s, x, p) = lim k→+∞ ψ 1 (s, x, p k ) ∈ Ω, for s ∈ [0, t[. ( 6
x,E and ψ 1 (t, x, p) = y. Hence y ∈ A x . Proposition 6.4 is proved.

7 The nonrelativistic case The equation (7.1) is an equation for x = x(t) and is the equation of motion in R n of a nonrelativistic particle of mass m = 1 and charge e = 1 in an external electromagnetic field described by V and B. In this equation x is the position of the particle, ẋ is its velocity, t is the time.

For the equation ( 7.1) the energy

E = 1 2 | ẋ(t)| 2 + V (x(t)) (7.2)
is an integral of motion. V,B (E, q 0 , q), k nr 0,V,B (E, q 0 , q), k nr V,B (E, q 0 , q), as were defined s V,B (E, q 0 , q), k 0,V,B (E, q 0 , q), k V,B (E, q 0 , q), in Section 2 for any q 0 , q ∈ D, q 0 = q. Further one can consider the following nonrelativistic version of Problem 1 formulated in Introduction Problem 1 ′ : given k nr V,B (E, q 0 , q), k nr 0,V,B (E, q 0 , q) for all q 0 , q ∈ ∂D, q 0 = q, at fixed sufficiently large energy E, find V and B.

The following uniqueness theorem holds Theorem 7.1. At fixed E > E nr ( V C 2 ,D , B C 1 ,D , D), the boundary data k nr V,B (E, q 0 , q), (q 0 , q) ∈ ∂D × ∂D, q 0 = q, uniquely determine V, B. At fixed E > E nr ( V C 2 ,D , B C 1 ,D , D), the boundary data k nr 0,V,B (E, q 0 , q), (q 0 , q) ∈ ∂D × ∂D, q 0 = q, uniquely determine V, B.

Theorem 7.1 is proved in Subsection 7.6.

7.3 Inverse scattering problem. We consider equation (7.1) under condition (1.4).

The following is valid (see, for example, [LT] where classical scattering of particles in a long-range magnetic field is studied, and see [S] where classical scattering of particles in a short-range electric field is studied): for any (v -, x -) ∈ R n × R n , v -= 0, the equation (7.1) has a unique solution x ∈ C 2 (R, R n ) such that x(t) = v -t + x -+ y -(t), (7.3) where ẏ-(t) → 0, y -(t) → 0, as t → -∞; in addition for almost any (v -, x -) ∈ R n × R n , v -= 0,

x(t) = v + t + x + + y + (t), (7.4)

where v + = 0, v + = a nr (v -, x -), x + = b nr (v -, x -), ẏ+ (t) → 0, y + (t) → 0, as t → +∞.

For an energy E > 0, the map S

. Let V 1 , V 2 ∈ C 2 0 (R n , R), B 1 , B 2 ∈ C 1 0 (R n , A n (R)) ∩ F mag (R n ) max( V 1 C 2 ,D , V 2 C 2 ,D , B 1 C 1 ,D , B 2 C 1 ,D ) ≤ λ, and supp(V 1 ) ∪ supp(V 2 ) ∪supp(B 1 ) ∪ supp(B 2 ) ⊆ D. Let S µ
E be the (nonrelativistic) scattering map at fixed energy E subordinate to (V µ , B µ ) for µ = 1, 2. Then there exists a nonnegative real constant E nr (λ, D) such that for any E > E nr (λ, D), (V 1 , B 1 ) ≡ (V 2 , B 2 ) if and only if S 1 E ≡ S 2 E .

7.4 Classical Hamiltonian mechanics. For x ∈ D and for E > V (x), we define r nr V,E (x) = 2(E -V (x)). For a solution x(t) of equation ( 7.1) in D, we define the impulse vector P nr (t) = ẋ(t) + A(x(t)).

Further for q 0 , q ∈ D, q 0 = q, and t ∈ [0, s nr (E, q 0 , q)], we consider P nr (t, E, q 0 , q) = ẋnr (t, E, q 0 , q) + A(x nr (t, E, q 0 , q)), (7.7) where x nr (., E, q 0 , q) is the solution given by (2.2) in the nonrelativistic case. From Maupertuis's principle (see [A]), it follows that if x(t), t ∈ [t 1 , t 2 ], is a solution of (7.1) in D with energy E, then x(t) is a critical point of the functional A(y) = t 2 t 1 r nr V,E (y(t))| ẏ(t)|+ A(y(t)) • ẏ(t)] dt defined on the set of the functions y ∈ C 1 ([t 1 , t 2 ], D), with boundary conditions y(t 1 ) = x(t 1 ) and y(t 2 ) = x(t 2 ). Note that for q 0 , q ∈ D, q 0 = q, functional A taken along the trajectory of the solution x nr (., E, q 0 , q) given by (2.2) is equal to the reduced action S nr 0 V,A,E (q 0 , q) from q 0 to q at fixed energy E for (7.6), where S nr 0 V,A,E (q 0 , q) = 0, if q 0 = q, s nr (E,q 0 ,q) 0 P nr (s, E, q 0 , q) • ẋnr (s, E, q 0 , q)ds, if q 0 = q, (7.8) for q 0 , q ∈ D.

7.5 Properties of the reduced action at a fixed and sufficiently large energy.

The reduced action at fixed and sufficiently large energy for (7.6) has the same properties that those given in Proposition 3.1, 3.2 for the reduced action at fixed and sufficiently large energy for the relativistic case.

Let (7.17) for x ∈ D (where we choose the canonical orientation of S n-1 and the orientation of ∂D given by the canonical orientation of R n and the unit outward normal vector).

Remark 7.1. Equalities (7.11) and (7.12) are known formulas of classical Hamiltonian mechanics (see Section 46 and further Sections of [A]).

Remark 7.2. Taking account of (7.11) and (7.12), we obtain the fol-

  Relativistic Newton equation. Consider the Newton-Einstein equation in a static electromagnetic field in an open subset Ω of R n , n ≥ 2, ṗ = -∇V (x) + 1 c B(x) ẋ,

  .10) is called the scattering map at fixed energy E for the equation (1.1) under condition (1.4). By D(S E ) we denote the domain of definition of S E . The data a(v -, x -), b(v -, x -) for (v -, x -) ∈ D(S E ) are called the scattering data at fixed energy E for the equation (1.1) under condition (1.4).

2. 1

 1 Properties of the boundary value data. Let D be a bounded strictly convex open subset of R n , n ≥ 2, with C 2 boundary. At fixed sufficiently large energy E (i.e. E > E( V C 2 ,D , B C 1 ,D , D) ≥ c 2 + sup x∈ D V (x)) solutions x(t) of the equation (1.1) under condition (1.3) have the following properties (see Section 6):

  10) is called the scattering map for the equation (1.1) under condition (1.4). The functions a(v -, x -), b(v -, x -) are called the scattering data for the equation (1.1) under condition (1.4). By D(S) we denote the domain of definition of S; by R(S) we denote the range of S (by definition, if

  .20) for x, y ∈ D, x = y and j, l = 1..n. where k0 is defined by (3.1). Let x, y ∈ D, x = y. Using the identity k(E, x, y) = k0 (E, x, y) + s(E,x,y) 0 -∇V (ψ 1 (s, x, k0 (E, x, y))) 1 (s, x, k0 (E, x, y)))g(ψ 2 (s, x, k0 (E, x, y))) ds, we obtain the following estimate | k(E, x, y) -k0 (E, x, y)| ≤ M 5 s(E, x, y), (5.21)

  .15) it follows that C 1 ( Ṽ , B, D), C 2 ( Ṽ , B, D), sup x∈D C 3 (E, x, Ṽ , B, D), C 6 (E, Ṽ , B, D), C 7 (E, Ṽ , B, D) and C 8 (E, Ṽ , B, D) depend only on (V, B) and D.

6. 5

 5 Final part of the proof of Properties (2.1) and (2.2). Let χ D be a C 2 defining function for D, i.e. χ D ∈ C 2 (R n , R) and D = χ -1 D (] -∞, 0[), and ∂D = χ -1 D ({0}), and for all x ∈ ∂D ∇χ D (x) = 0 and the Hessian matrix Hessχ D (x) of χ D at x is a symmetric definite positive matrix. Let x 0 ∈ D. For ε > 0, we define the open neighborhood Ω ε of D by Ω

  We denote by B(x 0 , ε) the Euclidean open ball of R n of center x 0 and radius ε. Let (U, φ) be an infinitely smooth parametrization of an open neighborhood of p 0 |p 0 | in S n-1 , and k = 1 . . . n such that

7. 1

 1 Nonrelativistic Newton equation in electromagnetic field. Consider the classical nonrelativistic Newton equation in a static electromagnetic field in an open subset Ω of R n , n ≥ 2, ẍ = -∇V (x) + B(x) ẋ, (7.1) where x = x(t) is a C 1 function with values in Ω, ẋ = dx dt , and V ∈ C 2 ( Ω, R), B ∈ F mag ( Ω).

7. 2

 2 Inverse boundary problem. Consider equation (7.1) under condition (1.3). One can prove that at sufficiently large energyE (i.e. E > E nr ( V C 2 ,D , B C 1 ,D , D) ≥ sup x∈D V (x) where real constant E nr ( V C 2 ,D , B C 1 ,D , D)also has properties (2.3) and (1.7)), the solutions x of energy E have properties (2.1) and (2.2) (the proof is obtained by slight modifications of proofs of Section 6). Then at fixed energy E > E nr ( V C 2 ,D , B C 1 ,D , D), one can define s nr

  nr E : S nr E × R n → S nr E × R n (where S nr E = {v ∈ R n | |v| = √ 2E})given by the formulasv + = a nr (v -, x -), x + = b nr (v -, x -), (7.5) is called the scattering map at fixed energy E for the equation (7.1) under condition (1.4). By D(S nr E ) we denote the domain of definition of S nr E . The data a nr (v -, x -), b nr (v -, x -) for (v -, x -) ∈ D(S nr E ) are called the scattering data at fixed energy E for the equation (7.1) under condition (1.4). We consider the following inverse scattering problem at fixed energy for the equation (7.1) under condition (1.4): Problem 2 ′ : given S nr E at fixed energy E, find V and B. From Theorem 7.1 and property (1.7), we obtain Theorem 7.2. Let λ ∈ R + and let D be a bounded strictly convex (in the strong sense) open domain of R n with C 2 boundary

  Let A ∈ F pot(D, B). The equation (7.1) in D is the Euler-Lagrange equation for the Lagrangian L nr defined byL nr ( ẋ, x) = 1 2 | ẋ| 2 +A(x)• ẋ-V (x), ẋ ∈ R nand x ∈ D, where • denotes the usual scalar product on R n . The Hamiltonian H nr associated to the Lagrangian L nr by Legendre's transform (with respect to ẋ) is H nr (P, x) = 1 2 |P -A(x)| 2 + V (x) where P ∈ R n and x ∈ D. Then equation (1.1) in D is equivalent to the Hamilton's equation ẋ = ∂H nr ∂P (P, x), Ṗ = -∂H nr ∂x (P, x), (7.6) for P ∈ R n , x ∈ D.

  E > E nr ( V C 2 ,D , B C 1 ,D , D).The reduced action S nr 0 V,A,E at fixed 50 energy E has the following properties:S nr 0 V,A,E ∈ C( D × D, x) = k nr ,i V,B (E, ζ, x) + A i (xx) = -k nr ,i 0,V,B (E, ζ, x) -A i (ζ, x) ∈ ( D × D)\ Ḡ, ζ = (ζ 1 , .., ζ n ), x = (x 1 , .., x n ), and i, j = 1 . . . n. In addition, x) ∈ ( D × D)\ Ḡ, ζ = (ζ 1 , .., ζ n ), x = (x 1 , .., x n ), and i, j = 1 . . . n, and where M 1 and M 2 depend on V, B and D.In addition, the map ν V,B,E : ∂D × D → S n-1 , defined byν nr V,B,E (ζ, x) = -k nr V,B (E, ζ, x) |k nr V,B (E, ζ, x)| , for (ζ, x) ∈ ∂D × D,(7.16) has the following properties:ν nr V,B,E ∈ C 1 (∂D × D, S n-1 ), the map ν nr V,B,E,x : ∂D → S n-1 , ζ → ν nr V,B,E (ζ, x), is a C 1 orientation preserving diffeomorphism from ∂D onto S n-1

  ∈ [0, t +,x,p 1 [ and t 2 ∈ [0, t +,x,p 2 [, t 1 ≥ t 2 and where C 4 and C 5 are defined by (6.11) and (6.12). Let t 1 ∈ [0, t +,x,p 1 [ and t 2 ∈ [0, t +,x,p 2 [, t 1 ≥ t 2 . Estimates (6.71) and |v 1

	for t 1
	τ )dτ,
	(6.71)

  +,x,p [, ψ 1 (t, x, p) ∈ Ω. Hence m(t) < 0, t ∈ [0, t +,x,p [. This estimate and the estimate m(t +,x,p ) > 0 and Taylor expansion of m at t +,x,p

	x, p),
	for t ∈ R and where g is the function defined by (5.6). From (6.103), conservation of energy and (6.6) and |g(ψ 2 (t, x, p))| < c, it follows that m(t +,x,p ) ≥ c 2 C 8 (E, Ṽ , B, Ω) > 0 (we used (6.100)). For all t ∈ [0, t

∂x j (E, x, ζ).

7.6 Proof of Theorem 7.1. We define the n -1 differential form ω nr 0,V,B on ∂D × D as was defined the n -

Let E > E nr (λ, λ, D) where E nr (λ, λ, D) is defined by the nonrelativistic formulation of (1.7). Consider β nr ,1 , β nr ,2 the differential one forms defined on (∂D × D)\ Ḡ by (7.18) for (ζ, x) ∈ (∂D × D)\ Ḡ, x = (x 1 , . . . , x n ) and µ = 1, 2. Define the differential forms Φ nr 0 and Φ nr as were defined Φ 0 and Φ in Subsection 3.3 (replace β µ , S 0 Vµ ,Aµ,E by β nr ,µ , S nr 0 Vµ,Aµ,E , µ = 1, 2). Then Lemma 3.1 and Theorem 3.1 remain valid by replacing Φ 0 , Φ, r Vµ,E , ω 0,Vµ,Bµ and kVµ,Bµ by Φ nr 0 , Φ nr , r nr Vµ,E , ω nr 0,Vµ,Bµ and k nr Vµ,Bµ , and the proof of these results are obtained by slight modifications of proof of Lemma 3.1 and Theorem 3.1. Hence Theorem 7.1 follows from the nonrelativistic formulation of Theorem 3.1.