On Inverse Problems in Electromagnetic Field in Classical Mechanics at Fixed

Volume 17, Number 2, 2007

On Inverse Problems in Electromagnetic Field in Classical Mechanics at Fixed

Energy

By Alexandre Jollivet

ABSTRACT. In this article, 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 ∈C²,B∈C¹in 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.

1. Introduction 1.1. Relativistic Newton equation

Consider the Newton-Einstein equation in a static electromagnetic field in an open subset ofRⁿ,n≥2,

˙

p= −∇V (x)+¹_cB(x)x ,˙ p= ^x˙

1−^{| ˙x}_c^|₂²

, (1.1)

wherex =x(t )is aC¹function with values in,p˙ = ^dp_dt,x˙ = ^dx_dt, andV ∈ C²(,¯ R)(i.e., there existsV˜ ∈ C²(Rⁿ,R)such thatV˜ restricted to¯ is equal to V),B ∈ Fmag()¯ where Fmag()¯ is the family of magnetic fields on, i.e.,¯ Fmag()¯ = {B ∈ C¹(, A¯ n(R))|B = (B_i,k ), _∂x^∂

iB_k,l (x)+_∂x^∂_lB_i,k (x)+_∂x^∂_kB_l,i (x)=0, x∈ ¯, i, k, l=1. . . n}andAn(R)denotes the space ofn×nreal antisymmetric matrices.

ByVC²,we denote the supremum of the set{|∂_x^jV (x)| | x ∈ , j = (j1, . . , jn)∈ (N∪{0})ⁿ,_n

i=1j_i ≤2}and byBC¹,we denote the supremum of the set{|∂_x^jB_i,k(x)| |x ∈, i, k=1. . . n,j =(j1, . . , j_n)∈(N∪ {0})ⁿ,_n

i=1j_i ≤1}.

Math Subject Classifications. 34A55, 83A05, 78A46.

Key Words and Phrases. Newton-Einstein equation, inverse kinematic problem in electromagnetic field, inverse problems at fixed energy.

©2007 The Journal of Geometric Analysis ISSN 1050-6926

The Equation (1.1) is an equation forx =x(t )and is the equation of motion inRⁿ of a relativistic particle of massm=1 and chargee=1 in an external electromagnetic field described byV andB(see [4] and, for example, Section 17 of [11]). In this equationxis the position of the particle,pis its impulse,tis the time, andcis the speed of light.

For the Equation (1.1), the energy E=c²

1+|p(t )|²

c² +V (x(t )) (1.2)

is an integral of motion. We denote byB_cthe Euclidean open ball whose radius is c and whose centre is 0.

In this article we consider the Equation (1.1) in two situations. We study Equation (1.1) when =DwhereDis a bounded strictly convex in the strong sense

open domain ofRⁿ, n≥2,withC²boundary. (1.3) And we study Equation (1.1) when

=Rⁿand|∂_x^j1V (x)| ≤β_|j1|(1+ |x|)^−α−|j1|, x∈Rⁿ, n≥2,

|∂_x^j2B_i,k(x)| ≤β_|j2|+1(1+ |x|)^{−α−1−|j2|}, x∈Rⁿ, (1.4) for|j1| ≤2,|j2| ≤1,i, k=1. . . nand someα >1 (herej is the multiindexj ∈(N∪ {0})ⁿ,

|j| =_n

i=1jiandβ_|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 energyE (i.e.,E > E(VC²,D,BC¹,D, D)), the solutionsxat energyEhave the following properties (see Properties (2.1) and (2.2) in Section 2 and see Section 6):

for each solutionx(t )there aret1, t2∈R, t1< t2, such that x ∈C³([t₁, t₂],Rⁿ), x(t₁), x(t₂)∈∂D, x(t )∈Dfor t ∈]t₁, t₂[,

x(s₁)=x(s₂)fors₁, s₂∈ [t₁, t₂], s₁=s₂; (1.5) for any two distinct pointsq0, q ∈ ¯D, there is one and only one solution

x(t )=x(t, E, q0, q)such thatx(0)=q0, x(s)=qfor somes >0. (1.6) Let (q0, q) be two distinct points of ∂D. By sV ,B(E, q0, q) we denote the time at which x(t, E, q0, q)reachesqfromq0. Byk0,V ,B(E, q0, q)we denote the velocity vectorx(0, E, q˙ 0, q).

BykV ,B(E, q0, q)we denote the velocity vectorx(s˙ V ,B(E, q0, q), E, q0, q). We considerk0,V ,B

(E, q₀, q),k_{V ,B}(E, q₀, q),q₀, q∈∂D, q₀=q, as the boundary value data.

Remark 1.1. Forq₀, q ∈ ∂D, q₀ = q, the trajectory ofx(t, E, q₀, q)and the trajectory of x(t, E, q, q₀)are distinct, in general.

Note that in the present article we always assume that the aforementioned real constant E(VC²,D,BC¹,D, D), considered as function ofVC²,DandBC¹,D, satisfies

E(λ1, λ2, D)≤E

λ₁, λ₂, D

if λ1≤λ₁ and λ2≤λ₂, (1.7) forλ₁, λ₂, λ₁, λ₂∈ [0,+∞[.

1.3. Scattering data

For the Equation (1.1) under condition (1.4), the following is valid (see [20]): For any (v₋, x₋)∈B_c×Rⁿ, v₋=0, the Equation (1.1) has a unique solutionx ∈C²(R,Rⁿ)such that x(t )=v₋t+x₋+y₋(t ) , (1.8) wherey˙₋(t ) → 0, y₋(t ) → 0, as t → −∞; in addition for almost any(v₋, x₋) ∈ B_c× Rⁿ, v₋=0,

x(t )=v₊t+x₊+y₊(t ) , (1.9) wherev₊ = 0,|v₊|< c,v₊ =a(v₋, x₋),x₊ =b(v₋, x₋),y˙₊(t )→ 0,y₊(t )→ 0, ast → +∞.

For an energyE > c², the mapS_E :SE×Rⁿ→ SE×Rⁿ

whereSE =

v∈B_c| |v| = c

1−_c2

E

2

given by the formulas

v₊=a(v₋, x₋), x₊=b(v₋, x₋) , (1.10) is called the scattering map at fixed energyEfor the Equation (1.1) under condition (1.4). By D(S_E)we denote the domain of definition ofS_E. The dataa(v₋, x₋),b(v₋, x₋)for(v₋, x₋)∈ D(S_E)are called the scattering data at fixed energyEfor the Equation (1.1) under condition (1.4).

1.4. Inverse scattering and boundary value problems

In the present article, we consider the following inverse boundary value problem at fixed energy for the Equation (1.1) under condition (1.3).

Problem 1: GivenkV ,B(E, q0, q), k0,V ,B(E, q0, q)for allq0, q∈∂D , q0=q, at fixed sufficiently large energyE, findV andB .

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

Theorem 1.2.

At fixed E > E(VC²,D,BC¹,D, D), the boundary datak0,V ,B(E, q0, q), (q0, q) ∈

∂D×∂D,q0=q, uniquely determineV , B.

Theorem 1.2 follows from Theorem 3.5 given in Section 3.

In the present article, we also consider the following inverse scattering problem at fixed energy for the Equation (1.1) under condition (1.4).

Problem 2: GivenS_Eat fixed energyE, findV andB .

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

Theorem 1.3.

max(V1C²,D,V2C²,D,B1C¹,D,B2C¹,D)≤λ, andsupp(V1)∪supp(V2)∪supp(B1)∪ supp(B2) ⊆ D. LetS_E^µ be the scattering map at fixed energyE subordinate to(V_µ, B_µ)for µ=1,2. Then there exists a nonnegative real constantE(λ, D)such that for anyE > E(λ, D), (V₁, B₁)≡(V₂, B₂)if and only ifS_E¹ ≡S_E².

Theorem 1.3 follows from Theorem 1.2, (1.7), and Proposition 2.4 of Section 2.

Remark 1.4. Note that forV ∈C₀²(Rⁿ,R), ifE < c²+sup{V (x)|x ∈Rⁿ}thenS_Edoes not determine uniquelyV.

Remark 1.5. Theorems 1.2 and 1.3 give uniqueness results. In this article 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 [6]

for the multidimensional nonrelativistic Newton equation (without magnetic field) in a bounded open strictly convex domain. In [6] 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 [13], [2], and [3]).

Novikov [17] studied inverse scattering for nonrelativistic multidimensional Newton equa- tion (without magnetic field). Novikov [17] 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 [6] and [17], the author [10] studied an inverse bound- ary problem and inverse scattering problem for the multidimensional relativistic Newton equation (without magnetic field) at fixed energy. In [10] results of uniqueness and stability are obtained.

Inverse scattering at high energies for the relativistic multidimensional Newton equation was studied by the author (see [8, 9]).

As regards analogs of Theorems 1.2, 1.3, and Proposition 2.4 for the case B ≡ 0 for nonrelativistic quantum mechanics see [16, 14, 18] and further references therein. As regards an analog of Theorem 1.3 for the caseB ≡ 0 for relativistic quantum mechanics see [7]. As regards analogs of Theorems 1.2, 1.3 for the caseB≡0 for nonrelativistic quantum mechanics, see [5, 15] 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 [9].

1.6. Structure of the article

The article 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 1.2 (based on Theorem 3.5 formulated in Section 3). Section 4 is devoted to the proof of Lemma 3.4 and Theorem 3.5 formulated in Section 3. Section 5 is devoted to the proof of Lemma 2.2 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 1.2, 1.3 for the nonrelativistic case.

2. Scattering data and boundary value data 2.1. Properties of the boundary value data

LetDbe a bounded strictly convex open subset ofRⁿ,n≥2, withC²boundary.

At fixed sufficiently large energyE(i.e.,E > E(VC²,D,BC¹,D, D)≥c²+sup_{x∈ ¯D} V (x)) solutionsx(t )of the Equation (1.1) under condition (1.3) have the following properties (see Section 6):

For each solutionx(t )there aret₁, t₂∈R, t₁< t₂, such that x ∈C³([t1, t2],Rⁿ), x(t1), x(t2)∈∂D, x(t )∈Dfort ∈]t1, t2[, x(s₁)=x(s₂)fors₁, s₂∈ [t₁, t₂], s₁=s₂,x(t˙ ₁)◦N (x(t₁)) <0 andx(t˙ ₂)◦N (x(t₂)) >0, whereN (x(t_i))is the unit outward normal vector of∂Datx(t_i)fori=1,2;

(2.1)

for any two pointsq₀, q∈ ¯D, q =q₀, there is one and only one solution x(t )=x(t, E, q₀, q)such thatx(0)=q₀, x(s)=q for somes >0;

˙

x(0, E, q0, q)∈C¹((D¯ × ¯D)\ ¯G,Rⁿ), whereG¯ is the diagonal inD¯ × ¯D;

(2.2) where◦ denotes the usual scalar product on Rⁿ (and where by “x(0, E, q˙ 0, q)∈ C¹((D¯ × D)¯ \ ¯G,Rⁿ)” we mean thatx(0, E, q˙ 0, q)is the restriction to(D¯ × ¯D)\ ¯Gof a function which belongs toC¹((Rⁿ×Rⁿ)\)whereis the diagonal ofRⁿ×Rⁿ).

Remark 2.1. IfB ∈ C¹(D, A¯ _n(R))andB ∈ Fmag(D)¯ (whereA_n(R)denotes the space of n×nreal antisymmetric matrices), then at fixed energyE > E(VC²,D,BC¹,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 constantE(VC²,D,BC¹,D, D), considered as function ofVC²,D andBC¹,D, satisfies (1.7). In addition, real constantE(VC²,D, BC¹,D, D)has the following property: For anyC²continuationV˜ ofV onRⁿ, and for any B˜ ∈C¹(Rⁿ, A_n(R))such thatB˜ ≡BonD, one has¯

E V˜

C²,Dx0,ε, B˜

C¹,Dx0,ε, D_x0,ε

→E(VC²,D,BC¹,D, D), asε→0, (2.3) whereDx₀,ε= {x0+(1+ε)(x−x0)|x ∈D}for anyx0∈Dandε >0 (note thatDx₀,εis a bounded, open, strictly convex (in the strong sense) domain ofRⁿwithC²boundary).

LetE > E(V_C²_,D,B_C¹_,D, D). Consider the solutionx(t, E, q₀, q)from (2.2) for q0, q∈ ¯D,q0=q. We define vectorskV ,B(E,q0, q)andk0,V ,B(E, q0, q)by

k_{V ,B}(E, q₀, q) = ˙x(s_{V ,B}(E, q₀, q), E, q₀, q) , k_{0,V ,B}(E, q₀, q) = ˙x(0, E, q0, q) ,

where we defines=s_{V ,B}(E, q0, q)as the root of the equation x(s, E, q₀, q)=q, s >0. Forq0=q ∈ ¯D, we puts_{V ,B}(E, q0, q)=0.

Note that

|k0,V ,B(E, q0, q)| =c

1−

E−V (q₀) c²

₋2

,

|kV ,B(E, q0, q)| =c

1−

E−V (q) c²

₋2

,

(2.4)

for(q, q0)∈(D¯ × ¯D)\ ¯G.

Using Properties (2.1) and (2.2), we obtain the following.

Lemma 2.2.

We consider k_{V ,B}(E, q₀, q), k_{0,V ,B}(E, q₀, q), q₀, q ∈ ∂D, q₀ = q as the boundary val- ue data.

Remark 2.3. Note that ifx(t )is solution of (1.1) under condition (1.3), thenx(−t )is solution of (1.1) withBreplaced by−B∈Fmag(D)¯ under condition (1.3). Hence, the following equalities are valid: At fixedE > E(VC²,D,BC¹,D, D),

k0,V ,B(E, q0, q) = −kV ,−B(E, q, q0) , (2.5) k_{V ,B}(E, q₀, q) = −k_{0,V ,−B}(E, q, q₀), (2.6) sV ,B(E, q0, q) = sV ,−B(E, q, q0) , (2.7) forq₀, q∈ ¯D,q₀=q.

2.2. Properties of the scattering operator

For Equation (1.1) under condition (1.4), the following is valid (see [20]): For any(v₋, x₋)∈ B_c×Rⁿ, v₋=0, the Equation (1.1) under condition (1.4) has a unique solutionx ∈C²(R,Rⁿ) such that

x(t )=v₋t+x₋+y₋(t ) , (2.8) wherey˙₋(t ) → 0, y₋(t ) → 0, as t → −∞; in addition for almost any(v₋, x₋) ∈ Bc× Rⁿ, v₋=0,

x(t )=v₊t+x₊+y₊(t ) , (2.9) wherev₊=0,|v₊|< c, v₊=a(v₋, x₋), x₊=b(v₋, x₋),y˙₊(t )→0, y₊(t )→0, ast→ +∞.

The mapS:Bc×Rⁿ→Bc×Rⁿgiven by the formulas

v₊=a(v₋, x₋), x₊=b(v₋, x₋) (2.10) is called the scattering map for the Equation (1.1) under condition (1.4). The functionsa(v₋, x₋), b(v₋, x₋)are called the scattering data for the Equation (1.1) under condition (1.4).

ByD(S)we denote the domain of definition ofS; byR(S)we denote the range ofS(by definition, if(v₋, x₋)∈D(S), thenv₋=0 anda(v₋, x₋)=0).

The mapShas the following simple properties (see [20]):D(S)is an open set ofBc×Rⁿ and Mes((Bc×Rⁿ)\D(S))=0 for the Lebesgue measure onBc×Rⁿinduced by the Lebesgue measure onRⁿ×Rⁿ; the mapS : D(S) → R(S)is continuous and preserves the element of volume; for any(v, x)∈D(S),a(v, x)²=v².

ForE > c², the mapSrestricted to E=



(v₋, x₋)∈Bc×Rⁿ|v₋| =c

1− c²

E 2





is the scattering operator at fixed energyEand is denoted bySE.

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

M=

(v₋, x₋)∈Bc×Rⁿ|v₋=0, v₋x₋=0 .

This observation is based on the fact that ifx(t )satisfies (1.1), thenx(t+t0)also satisfies (1.1) for anyt0∈R. In particular, the mapSat fixed energyEis uniquely determined by its restriction toME(S)=D(S)∩ME, whereME=_E∩M.

2.3. Relation between scattering data and boundary value data

V ∈C₀²D,¯ R

, B ∈C₀¹D, A¯ n(R)

, B∈Fmag

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 datakV ,B(E, q0, q), k0,V ,B(E, q0, q)for E > E(VC²,D, BC¹,D, D)and(q, q0)∈(∂D×∂D)\∂G, to the scattering dataa, b.

Proposition 2.4.

Proof of Proposition 2.4. First of all we introduce functionsχ,τ₋, andτ₊dependent onD.

For(v, x)∈Rⁿ\{0} ×Rⁿ,χ (v, x)denotes the nonnegative number of points contained in the intersection of∂Dwith the straight line parametrized byR→Rⁿ, t →t v+x. AsDis a strictly convex open subset ofRⁿ,χ (v, x)≤2 for allv, x∈Rⁿ,v=0.

Let(v, x)∈Rⁿ\{0} ×Rⁿ. Assume thatχ (v, x)≥1. The realτ₋(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 numbert such thatτ₊(v, x)v+x ∈∂D(ifχ (v, x)=1 thenτ₋(v, x)=τ₊(v, x)).

Direct statement. Let(q0, q)∈(∂D×∂D)\∂G. Under conditions (2.11) and from (2.1) and (2.2), it follows that there exists an unique(v₋, x₋)∈ME(S)such that

χ (v₋, x₋)=2,

q₀=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, q₀, q) = τ₊(a(v₋, x₋), b(v₋, x₋))−τ₋(v₋, x₋)andk_{V ,B}(E,q₀, q) = a(v₋, x₋)andk_{0,V ,B}(E, q₀, q)=v₋.

Converse statement. Let(v₋, x₋)∈ ME(S). Under conditions (2.11), ifχ (v₋,x₋)≤1 then (a(v₋, x₋), b(v₋, x₋))=(v₋, x₋).

Assume thatχ (v₋, x₋)=2. Let

q₀=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, q₀, q)=v₋, q∈∂D, q=q₀. (2.12) We denote byq(v₋, x₋)the unique solution of (2.12). Hence, we obtain

a(v₋, x₋)= k_{V ,B}(E, q0, q(v₋, x₋)) ,

b(v₋, x₋)= q(v₋, x₋)−k_{V ,B}(E, q₀, q(v₋, x₋))(s_{V ,B}(E, q₀, q(v₋, x₋))+τ₋(v₋, x₋)) . Proposition 2.4 is proved.

For a more complete discussion about connection between scattering data and boundary value data, see [17] considering the nonrelativistic Newton equation (without magnetic field).

3. Inverse boundary value problem

3.1. Notations

Forx ∈ ¯D, and forE > V (x)+c², we define

rV ,E(x)=c

E−V (x) c²

2

−1.

AtE > E(V_C²_,D,B_C¹_,D, D)for q₀, q ∈ ¯D× ¯D, q₀ = q, we define the vectors k¯_{V ,B}(E, q₀, q)andk¯_{0,V ,B}(E, q₀, q)by

k¯V ,B(E, q0, q)= ^{kV ,B(E,q0,q)}

1−|kV ,B (E,q0,q)|2 c2

, k¯_{0,V ,B}(E, q₀, q)= ^{k0,V ,B(E,q0,q)}

1−^|k0,V ,B (E,q0,q)|2 c2

,

(3.1)

and k¯V ,B(E, q0, q) = (k¯_{V ,B}¹ (E, q0, q), . . . ,k¯ⁿ_{V ,B}(E, q0, q)),k¯0,V ,B(E, q0, q) = ( k¯_{0,V ,B}¹ (E, q0, q), . . . ,k¯ⁿ_{0,V ,B}(E, q0, q)). Note that from (2.4), it follows that

k¯_{V ,B}(E, q₀, q)=r_{V ,E}(q) ,

k¯_{0,V ,B}(E, q₀, q)=r_{V ,E}(q₀) . (3.2) ForB∈Fmag(D), let¯ Fpot(D, B)be the set ofC¹magnetic potentials for the magnetic field B, i.e.,

Fpot(D, B):=

A∈C¹D,¯ Rⁿ B_i,k(x)= ∂

∂xi

Ak(x)− ∂

∂xk

Ai(x), x∈ ¯D, i, k=1. . . n

.(3.3) (The setFpot(D, B)is not empty: Take, for example, A(x)= −₁

0 sB(x₀+s(x−x₀))(x−x₀) ds, forx ∈ ¯Dand some fixed pointx₀ofD.)¯

3.2. Hamiltonian mechanics

Let A ∈ Fpot(D, B). The Equation (1.1) in D is the Euler-Lagrange equation for the LagrangianLdefined byL(x, x)˙ = −c²

1−^x_c^˙2²+c⁻¹A(x)◦ ˙x−V (x),x˙∈B_candx ∈D, where

◦denotes the usual scalar product onRⁿ. The HamiltonianHassociated to the LagrangianLby Legendre’s transform (with respect tox˙) isH (P , x)=c²

1+c⁻²|P− c⁻¹A(x)|²1/2

+V (x) whereP ∈Rⁿandx ∈D. Then Equation (1.1) inDis equivalent to the Hamilton’s equation

˙

x= ^∂H_∂P(P , x) ,

P˙ = −^∂H_∂x(P , x) , (3.4)

forP ∈Rⁿ,x∈D.

For a solutionx(t )of Equation (1.1) inD, we define the impulse vector P (t )= x(t )˙

1−^{| ˙x(t )}_c2^|2

+c⁻¹A(x(t )) .

Further forq₀, q∈ ¯D,q₀=q, andt ∈ [0, s(E, q0, q)], we consider P (t, E, q₀, q)= x(t, E, q˙ 0, q)

1−x(t,E,q_{˙ 0},q)²

c²

+c⁻¹A(x(t, E, q0, q)) , (3.5)

wherex(., E, q₀, q)is the solution given by (2.2). From Maupertuis’s principle (see [1]), it follows that ifx(t ),t ∈ [t₁, t₂], is a solution of (1.1) inD with energyE, thenx(t )is a critical point of the functionalA(y)=_t2

t1

r_{V ,E}(y(t ))| ˙y(t )| +c⁻¹A(y(t ))◦ ˙y(t )

dtdefined on the set of the functionsy ∈C¹([t1, t2], D), with boundary conditionsy(t1)=x(t1)andy(t2)=x(t2). Note that forq0, q∈D,q0=q, functionalAtaken along the trajectory of the solutionx(., E, q0, q) given by (2.2) is equal to the reduced action S0_{V ,A,E}(q0, q) from q0 to q at fixed energy E for (3.4), where

S0_{V ,A,E}(q0, q)=

0, if q₀=q , _s(E,q0,q)

0 P (s, E, q₀, q)◦ ˙x(s, E, q₀, q) ds, if q₀=q , (3.6) forq₀, q∈ ¯D.

3.3. Properties of

at fixed and sufficiently large energy

The following Propositions 3.1, 3.2 give properties ofS0_{V ,A,E} at fixed and sufficiently large energyE.

Proposition 3.1.

S0_{V ,A,E} ∈CD¯ × ¯D,R

, (3.7)

S0_{V ,A,E} ∈C²D¯ × ¯D

\ ¯G,R

, (3.8)

∂S0_{V ,A,E}

∂x_i (ζ, x)= ¯kⁱ_{V ,B}(E, ζ, x)+c⁻¹Ai(x) , (3.9)

∂S0_{V ,A,E}

∂ζ_i (ζ, x)= −¯k_{0,V ,B}ⁱ (E, ζ, x)−c⁻¹Ai(ζ ) , (3.10)

∂²S0_{V ,A,E}

∂ζ_i∂x_j (ζ, x)= −∂k¯ⁱ_{0,V ,B}

∂x_j (E, ζ, x)=∂k¯_{V ,B}^j

∂ζ_i (E, ζ, x) , (3.11)

for(ζ, x)∈(D¯ × ¯D)\ ¯G,ζ =(ζ1, . . , ζn),x=(x1, . . , xn), andi, j=1. . . n. In addition, max∂S0V ,A,E

∂xi

(ζ, x),∂S0V ,A,E

∂ζi

(ζ, x)

≤M1, (3.12)

∂²S0_{V ,A,E}

∂ζi∂xj

(ζ, x)≤ M2

|ζ−x| , (3.13)

for(ζ, x)∈ (D¯ × ¯D)\ ¯G,ζ =(ζ1, . . , ζn),x =(x1, . . , xn), andi, j = 1. . . n, and whereM1

andM2depend onV,BandD.

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 [1]).

Proposition 3.2.

νV ,B,E(ζ, x)= − k_{V ,B}(E, ζ, x)

|k_{V ,B}(E, ζ, x)|, for (ζ, x)∈∂D×D , (3.14) has the following properties:

ν_{V ,B,E} ∈C¹(∂D×D,Sⁿ⁻¹),

the mapν_{V ,B,E,x} :∂D→Sⁿ⁻¹, ζ →ν_{V ,B,E}(ζ, x), is a C¹orientation preserving diffeomorphism from∂DontoSⁿ⁻¹

(3.15)

forx ∈D(where we choose the canonical orientation ofSⁿ⁻¹and the orientation of∂Dgiven by the canonical orientation of Rⁿand the unit outward normal vector).

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

Remark 3.3. Taking account of (3.9) and (3.10), we obtain the following formulas: AtE >

E(VC²,D,BC¹,D, D), for anyx, ζ ∈ ¯D,x=ζ,

Bi,j(x) = −c ∂k¯_{V ,B}^j

∂x_i (E, ζ, x)−∂k¯ⁱ_{V ,B}

∂x_j (E, ζ, x)

, (3.16)

B_i,j(x) = −c

∂k¯_{0,V ,B}^j

∂x_i (E, x, ζ )−∂k¯_{0,V ,B}ⁱ

∂x_j (E, x, ζ )

. (3.17)

3.4. Results of uniqueness

We denote byω0,V ,Bthen−1 differential form on∂D×Dobtained in the following manner:

– Forx ∈D, letωV ,B,xbe the pull-back ofω0byνV ,B,E,x whereω0denotes the canonical orientation form onSⁿ⁻¹(i.e., ω0(ζ )(v1, . . , vn−1) = det(ζ, v1, . . , vn−1), for ζ ∈ Sⁿ⁻¹ and v1, . . , vn−1∈TζSⁿ⁻¹),

– for(ζ, x)∈∂D×Dand for anyv1, . . , vn−1∈T(ζ,x)(∂D×D), ω_{0,V ,B}(ζ, x)(v₁, . . , v_n−1)=ω_{V ,B,x}(ζ )

σ_(ζ,x) (v₁), . . , σ_(ζ,x) (v_n−1) ,

whereσ : ∂D×D → ∂D,(ζ, x)→ ζ, andσ_(ζ,x) denotes the derivative (linear part) ofσ at(ζ, x).

From smoothness ofν_{V ,B,E},σ andω₀, it follows thatω_{0,V ,B} is a continuousn−1 form on∂D×D.

Now letλ ∈ R⁺ andV₁, V₂ ∈ C²(D,¯ R),B₁, B₂ ∈ Fmag(D), such that max(¯ V_1C²_,D, V_2C²_,D,B_1C¹_,D,B_2C¹_,D)≤λ. Forµ=1,2, let Aµ∈Fpot(D, B_µ).

LetE > E(λ, λ, D)whereE(λ, λ, D)is defined in (1.7). Considerβ¹,β²the differential one forms defined on(∂D× ¯D)\ ¯Gby

β^µ(ζ, x)= n j=1

k¯_V^j

µ,Bµ(E, ζ, x) dxj , (3.18) for(ζ, x)∈(∂D× ¯D)\ ¯G,x=(x1, . . . , xn)andµ=1,2.

Consider the differential forms₀on(∂D× ¯D)\ ¯Gand₁on(∂D× ¯D)\ ¯Gdefined by 0(ζ, x) = −(−1)^n(n2+1)

β²−β¹

(ζ, x)∧dζ(S0_V

2,A2,E −S0_V

1,A1,E)(ζ, x)

∧

p+q=n−2

(dd_ζS0_V1,A1,E(ζ, x))^p∧(dd_ζS0_V2,A2,E(ζ, x))^q, (3.19)

for(ζ, x)∈(∂D× ¯D)\ ¯G, whered =d_ζ+d_x, 1(ζ, x) = −(−1)^n(n2−1)

β¹(ζ, x)∧(ddζS0_V

1,A1,E(ζ, x))ⁿ⁻¹

+β²(ζ, x)∧(dd_ζS0_V2,A2,E(ζ, x))ⁿ⁻¹−β¹(ζ, x) (3.20)

∧(dd_ζS0_V2,A2,E(ζ, x))ⁿ⁻¹−β²(ζ, x)∧(dd_ζS0_V1,A1,E(ζ, x))ⁿ⁻¹

, for(ζ, x)∈(∂D× ¯D)\ ¯G, whered =dζ+dx.

Consider theC²map incl:(∂D×∂D)\∂G→(∂D× ¯D)\ ¯G,(ζ, x)→(ζ, x).

From (3.8), (3.12), and (3.13), it follows that₀is continuous on(∂D× ¯D)\ ¯Gand incl^∗(₀) is integrable on∂D×∂Dand₁is continuous on(∂D× ¯D)\ ¯Gand integrable on∂D× ¯D(where incl^∗(₀)is the pull-back of the differential form₀by the inclusion map incl).

Lemma 3.4.

∂D×∂D

incl^∗(₀) =

∂D× ¯D

₁; (3.21)

1

(n−1)!₁(ζ, x) =

r_V1,E(x)ⁿω_0,V1,B1(ζ, x)+r_V2,E(x)ⁿω_0,V2,B2(ζ, x)

− ¯kV₁,B₁(E, ζ, x)◦ ¯kV₂,B₂(E, ζ, x) (3.22)

×

r_V1,E(x)ⁿ⁻²ω_0,V1,B1(ζ, x)+r_V2,E(x)ⁿ⁻²

×ω0,V₂,B2(ζ, x)

∧ dx1∧. .∧ dx_n,

for(ζ, x)∈∂D×D.