Brake orbits for Hamiltonian systems of the classical type via geodesics in singular Finsler metrics

Dario Corona; Fabio Giannoni

doi:10.1515/anona-2022-0222

Artikel Open Access

Brake orbits for Hamiltonian systems of the classical type via geodesics in singular Finsler metrics

Dario Corona und Fabio Giannoni

Veröffentlicht/Copyright: 16. März 2022

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Aus der Zeitschrift Advances in Nonlinear Analysis Band 11 Heft 1

Abstract

We consider Hamiltonian functions of the classical type, namely, even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton’s equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of the classical type to the multiplicity problem of orthogonal geodesic chords in a concave Finslerian manifold with boundary. This paper will be used for a generalization of a Seifert’s conjecture about the multiplicity of brake orbits to Hamiltonian functions of the classical type.

Keywords: Hamiltonian systems; brake orbits; Finsler metric; variational methods

MSC 2010: 70H12; 70G75; 70H05; 58E10; 53B40

1 Introduction

Let H : R 2 n → R be an autonomous Hamiltonian function of class C 2 . A curve ( q , p ) : [ 0 , T ] → R 2 n is a solution of Hamilton’s equations if

(1.1) q ˙ = ∂ H ∂ p ( q , p ) , and p ˙ = − ∂ H ∂ q ( q , p ) .

Since the Hamiltonian is autonomous, the conservation law of the energy holds. More formally, if ( q , p ) : [ 0 , T ] → R 2 n is a solution of Hamilton’s equations, then there exists a real number E , called energy, such that

H ( q ( t ) , p ( t ) ) = E , ∀ t ∈ [ 0 , T ] .

Let K : R 2 n → R and V : R n → R be two functions such that the Hamiltonian can be written as follows:

H ( q , p ) = K ( q , p ) + V ( q ) ,

and K ( q , ⋅ ) : R n → R is even and strictly positive unless p = 0 . Then, whenever the set { q ∈ R n : V ( q ) < E } is non-empty, it can be thought as a potential well.

This paper concerns the multiplicity of the brake orbits in a bounded potential well. Roughly speaking, a brake orbit is a periodic solution of Hamilton’s equations with energy E that oscillates back and forth between two points of the boundary of the potential well (Figure 1). When the Hamiltonian is natural, hence, given by

H ( q , p ) = 1 2 ∑ i , j = 1 n a i j ( q ) p i p j + V ( q ) ,

where ( a i j ( q ) ) is a positive definite quadratic form on R n , and the potential well is homeomorphic to the n -dimensional disk in R n , Seifert conjectured the existence of at least n brake orbits (cf. [1]). This conjecture has motivated an extensive literature on the subject (e.g., [2,3,4, 5,6,7, 8,9,10]), and it has been recently proved in [11], exploiting some partial results given by the authors in different previous papers (cf. [12,13,14, 15,16,17]). This work points towards a generalization of Seifert’s conjecture, looking for the multiplicity of brake orbits when the Hamiltonian function is of the classical type (see Definition 1.1). Indeed, the present paper includes some results that will be exploited in the future to generalize the Seifert’s conjecture for Hamiltonian systems of the classical type. In particular, we show that the brake orbits in a bounded potential well for a Hamiltonian function of the classical type have a one–one correspondence with the orthogonal geodesic chords in a strictly concave Finsler manifold with boundary (see Theorem 1.9). Different generalizations of Seifert’s conjecture have been analyzed in the last decades. The papers with the most similar setting to the present one are [18] and [19], where the existence of one brake orbit is proved for Finsler mechanical systems and Hamiltonian systems of the classical type, respectively.

$Figure 1 Projection on the configuration space of a break-orbit. The periodic solution q ( t ) q\left(t) oscillates back and forth between the two points q ( 0 ) , q ( T ) q\left(0),q\left(T) that lies on the boundary of the potential well D = { q ∈ R 2 : V ( q ) < E } D=\left\{q\in {{\mathbb{R}}}^{2}:V\left(q)\lt E\right\} .$

Figure 1

Projection on the configuration space of a break-orbit. The periodic solution q ( t ) oscillates back and forth between the two points q ( 0 ) , q ( T ) that lies on the boundary of the potential well D = { q ∈ R 2 : V ( q ) < E } .

Before formally stating our main result, we need the following definitions (cf. [19]).

Definition 1.1

A Hamiltonian function H ( q , p ) on R 2 n is of the classical type if, for each q 0 ∈ R n , the function p ↦ H ( q 0 , p ) is even and ( ∂ 2 H / ∂ p 2 ) ( q 0 , p ) is strictly positive definite for all p , namely, there exists a continuous function ν : R n → R such that, for all q ∈ R n , ν ( q ) > 0 and

(1.2) ∂ 2 H ∂ p 2 ( q , p ) [ ξ , ξ ] ≥ ν ( q ) ‖ ξ ‖ 2 , ∀ p , ξ ∈ R n .

Remark 1.2

If H is a Hamiltonian of the classical type, by (1.2), the inverse of ( ∂ H / ∂ p ) ( q , ⋅ ) is well defined for all q ∈ D ¯ . Hence, with a slight abuse of notation, we will say that a curve q : [ 0 , T ] → D ¯ is a solution of the Hamilton’s equations if ( q , p ) : [ 0 , T ] → R 2 n is a solution of (1.1), where p is implicitly defined by

q ˙ ( t ) = ∂ H ∂ p ( q ( t ) , p ( t ) ) , ∀ t ∈ [ 0 , T ] .

Definition 1.3

Let H : R 2 n → R be a Hamiltonian function of the classical type. We define the potential energy function V : R n → R as follows:

V ( q ) = H ( q , 0 ) , ∀ q ∈ R n ,

and the kinetic energy function K : R 2 n → R as

K ( q , p ) = H ( q , p ) − V ( q ) .

By Definitions 1.1 and 1.3, a Hamiltonian of the classical type can be written as follows:

H ( q , p ) = K ( q , p ) + V ( q ) ,

where K ( q , p ) is even with respect to p , strictly positive unless p = 0 and, for each q ∈ R n ,

∂ 2 K ∂ p 2 ( q , p ) [ ξ , ξ ] ≥ ν ( q ) ‖ ξ ‖ 2 , ∀ p , ξ ∈ R n .

Definition 1.4

Let H be a Hamiltonian function of the classical type. A potential well for H is an open set D ⊂ R n with boundary ∂ D of class C 2 such that, for some real number E , the followings hold:

V ( q ) < E on D ;
V ( q ) = E on ∂ D ;
∇ V ( q ) ≠ 0 , for all q ∈ ∂ D .

Definition 1.5

Let D ⊂ R n be a potential well for a Hamiltonian H , with V ( q ) = E on ∂ D . A solution ( q ( t ) , p ( t ) ) of Hamilton’s equations for H is called brake orbit if it has energy E , and there exists T > 0 such that q ( t ) ∈ D for 0 < t < T , while q ( 0 ) , q ( T ) ∈ ∂ D .

Following the notation of Remark 1.2, we say that q : [ 0 , T ] → D ¯ is a brake orbit if it is a solution of (1.1) with energy E , q ( ] 0 , T [ ) ⊂ D and q ( 0 ) , q ( T ) ∈ ∂ D .

Remark 1.6

By the conservation law of the energy, if ( q ( t ) , p ( t ) ) is a brake orbit, then p ( 0 ) and p ( T ) must be zero. Since H is of the classical type (hence, even in p ), the solution can be continued so that it will be periodic. In other words, q ( t ) oscillates back and forth along a curve in D with endpoints in ∂ D .

Let us introduce the following notation. Let ( ℳ , F ) be a Finsler manifold of class C 3 and let Ω ⊂ ℳ be an open subset with boundary ∂ Ω ∈ C 2 (we refer to [20,21] for a background material about Finsler geometry).

Definition 1.7

A curve γ : [ a , b ] → Ω ¯ is a Finsler geodesic chord if

It is a geodesic with respect to the Finsler metric F ;
γ ( a ) , γ ( b ) ∈ ∂ Ω and γ ( ] a , b [ ) ⊂ Ω .

If γ ˙ ( a ) and γ ˙ ( b ) are orthogonal, with respect to the Finsler metric F , to T γ ( a ) ∂ Ω and T γ ( b ) ∂ Ω , respectively, namely,

(1.3) ∂ F 2 ∂ v ( γ ( t ) , γ ˙ ( t ) ) [ ξ ] ≔ d d s F 2 ( γ ( t ) , γ ˙ ( t ) + s ξ ) s = 0 = 0 ,

for all ξ ∈ T γ ( t ) ∂ Ω , with t = a , b , then γ is called the orthogonal Finsler geodesic chord.

This paper reduces the multiplicity problem of the brake orbits in a bounded potential well of a Hamiltonian of the classical type to the related problem of orthogonal geodesic chords in a Finslerian manifold with smooth boundary.

The last ingredient to state our main theorem is the notion of strong concavity of a Finsler manifold with boundary. We say that Ω ¯ is strongly concave with respect to the Finsler metric F if every geodesic, which is tangent to ∂ Ω on one point q , lies inside Ω on a neighborhood of q . Thus, differently from the notion of concavity (cf. [22] for the dual notion of convexity), the strong concavity allows the geodesics tangent to the boundary to locally touch the boundary only in one point. We formally define the strong concavity as follows.

Definition 1.8

Let Ω ¯ ⊂ ℳ be a manifold with smooth ( C 2 ) boundary and let ψ : ℳ → R be a function of class C 2 such that ψ ( Ω ) ⊂ ] 0 , ∞ [ , ψ ( ∂ Ω ) = 0 , and d ψ ( q ) ≠ 0 for all q ∈ ∂ Ω , where d ψ denotes the differential of ψ . Then, Ω ¯ is strongly concave if and only if for all q ∈ ∂ Ω we have

(1.4) H ψ ( q , v ) [ v , v ] ≔ d 2 d s 2 ( ψ ∘ γ ) ( 0 ) > 0 , ∀ v ∈ T q ∂ Ω , v ≠ 0 ,

where γ : ( − ε , ε ) → ℳ is the unique geodesic such that γ ( 0 ) = q and γ ˙ ( 0 ) = v .

Now we are ready to state our main theorem.

Theorem 1.9

Let D ⊂ R n be a potential well for a Hamiltonian H of the classical type. If D ¯ is compact, there exists an open set Ω ⊂ D , with a Finsler metric F on Ω ¯ , such that the following statements hold:

Ω ¯ ⊂ D ;
∂ Ω is of class C 2 ;
Ω ¯ is homeomorphic to D ¯ ;
Ω ¯ is strongly concave with respect to the Finsler metric F ;
If γ : [ 0 , 1 ] → Ω ¯ is an orthogonal Finsler geodesic chord, then there exists [ α , β ] ⊃ [ 0 , 1 ] and an unique continuous extension γ ˆ : [ α , β ] → D ¯ of γ such that
1. γ ˆ is a geodesic in ] α , β [ ;
2. Up to a time reparametrization, γ ˆ : [ α , β ] → D ¯ is a brake orbit, namely, there exists a diffeomorphism σ : [ 0 , T ] → [ α , β ] such that q = γ ∘ σ : [ 0 , T ] → D ¯ is a brake orbit.

Theorem 1.9 reduces the study of multiple brake orbits of a Hamiltonian of the classical type to the study of multiple orthogonal geodesic chords in a strongly concave Finsler manifold with boundary. Given a bounded potential well D , we will construct a Finsler manifold ( Ω ¯ , F ) , with Ω ¯ ⊂ D , such that there exists a bijection between the brake orbits in D and the orthogonal geodesic chords in ( Ω ¯ , F ) . This result generalizes the one presented in [17] for natural Hamiltonian functions, where the Finsler metric is actually a Riemannian one. Some results about the multiplicity of orthogonal geodesic chords in the case of convex Finsler manifolds with boundary and some generalizations can be found, for instance, in [23,24,25].

This paper is organized as follows. Some standard notations are presented at the end of this introduction. In Section 2, we present and study a Jacobi-Finsler metric F defined on the potential well such that its geodesics are, up to a time reparametrization, the solution of the Hamiltonian system. However, F cannot be defined on the boundary ∂ D , since it degenerates to the zero function. Therefore, in Section 3, we analyze the behavior of the solutions of Hamilton equations near the boundary. Following a variational approach, we see the geodesics as critical points of the energy functional J of the Jacobi-Finsler metric. Through the energy functional, in Section 4, we define the function ψ : D ¯ → R , which is the infimum of J among all the geodesics that connect a point to the boundary of the potential well. Hence, ψ ( y ) → 0 as y approaches the boundary ∂ D , and we also prove that, if y is sufficiently near to the boundary, there exists a unique geodesic γ y connecting y and ∂ D such that ψ ( y ) = J ( γ y ) . In Section 5, we prove that ψ is of class C 2 near the boundary ∂ D and that there exists a δ ˆ > 0 such that Ω ¯ = ψ − 1 ( [ δ ˆ , ∞ [ ) is a strongly concave set with respect to F . In Section 6, we finally give the proof of Theorem 1.9 exploiting all the previous results. In this proof, the main idea is to connect an orthogonal geodesic Finsler chord in Ω ¯ with the unique geodesic that realizes ψ ( y ) = J ( y ) to obtain a brake-orbit, up to a time reparametrization (Figure 2).

$Figure 2 The setting of Theorem 1.9. The orthogonal Finsler geodesic chord γ \gamma in Ω ¯ \overline{\Omega } can be extended to γ ˆ \hat{\gamma } , which is a brake orbit in the potential well D D , up to a time reparametrization. The extension is obtained through the unique geodesics that realize ψ ( γ ( 0 ) ) \psi \left(\gamma \left(0)) and ψ ( γ ( 1 ) ) \psi \left(\gamma \left(1)) .$

Figure 2

The setting of Theorem 1.9. The orthogonal Finsler geodesic chord γ in Ω ¯ can be extended to γ ˆ , which is a brake orbit in the potential well D , up to a time reparametrization. The extension is obtained through the unique geodesics that realize ψ ( γ ( 0 ) ) and ψ ( γ ( 1 ) ) .

1.1 Notation

If f is a real-valued function defined on R 2 n , then ∂ f / ∂ q and ∂ f / ∂ p will denote the differentials of f with respect to q and p , respectively. We denote by f ′ the differential of f , hence, f ′ ( q , p ) = ( ∂ f / ∂ q , ∂ f / ∂ p ) . We will denote by v the conjugate variable of p via Legendre transform of a function. Hence, ∂ f / ∂ v will denote the partial derivative with respect to v . We denote by ⟨ ⋅ , ⋅ ⟩ : R 2 n → R the Euclidean scalar product and ‖ ⋅ ‖ : R 2 n → R the Euclidean norm. We denote by J ∈ M 2 n × 2 n ( R ) the symplectic matrix

J = 0 I n × n − I n × n 0 .

Let z : [ 0 , T ] → R 2 n be a curve with z ( t ) = ( q ( t ) , p ( t ) ) . Using this notation, (1.1) can be written as follows:

z ˙ ( t ) = J H ′ ( z ( t ) ) .

For every compact interval I ⊂ R and every A ⊂ R n , we denote by W 1 , 2 ( I , A ) the Sobolev space:

W 1 , 2 ( I , A ) = { γ : I → A : γ is absolutely continuous and γ ˙ ∈ L 2 ( I , R n ) } .

2 The Jacobi-Finsler metric

Let H be a Hamiltonian of the classical type and D ⊂ R n be the (open) potential well such that V ( q ) ≡ E on ∂ D and D ¯ is compact. In this section, following the same construction of [19], we endow the potential well with a Finsler metric whose geodesics are linked to the solution of the Hamiltonian system via time reparametrization. Let us define

Σ = { ( q , p ) ∈ R 2 n : q ∈ D , H ( q , p ) = E } .

If ( q 0 , p 0 ) ∈ Σ , then p 0 ≠ 0 , and this implies that H ′ ( q 0 , p 0 ) is different from zero. As a consequence, Σ is a regular level surface for H .

Lemma 2.1

There exists a function U : D × R n → R such that

U is of class C 1 ;
U is of class C 2 on D × ( R n \ { 0 } ) ;
U ( q , p ) is even and homogeneous of degree 2 in p ;
Σ = U − 1 ( 1 ) and Σ is a regular level surface for U .

Proof

Since H is convex with respect to p , for every q 0 ∈ D , the set

{ p : H ( q 0 , p ) = E }

is a nonempty, convex, compact hypersurface in R n , symmetric about the origin. As a consequence, there exists a unique function U : R 2 n → R , which is homogeneous of degree 2 in p and which is identically 1 on Σ . Since H is of class C 2 , so it is U on D × ( R n \ { 0 } ) . Moreover, the homogeneity of degree 2 in p implies both the C 1 -regularity of U and that U ′ ( q , p ) ≠ 0 for all ( q , p ) ∈ Σ .□

Since Σ is a regular level surface for the Hamiltonian functions H and U , we have the following result.

Lemma 2.2

A curve ( q , p ) : [ 0 : T ] → D × R n is a solution of Hamilton’s equations for H if and only if it is a solution of Hamilton’s equations for U, up to time reparametrization.

Proof

See [19, Lemma 2.1].□

Remark 2.3

Let x = ( q , p ) : [ 0 , S ] → R 2 n be a solution of Hamilton’s equations with Hamiltonian U ; hence,

d x d s = J U ′ ( x ( s ) ) .

By Lemma 2.2, there exists a function λ : [ 0 , T ] → [ 0 , S ] such that z = x ∘ λ : [ 0 , T ] → R 2 n is a solution of Hamilton’s equations with Hamiltonian H ; hence,

d z d t = J H ′ ( z ( t ) ) .

As a consequence, we obtain

J H ′ ( x ( λ ( t ) ) ) = J H ′ ( z ( t ) ) = d z d t ( t ) = d λ d t ( t ) d x d s ( λ ( t ) ) = d λ d t ( t ) J U ′ ( x ( λ ( t ) ) ) .

Imposing that λ is an orientation preserving reparametrization, we obtain

d λ d t ( t ) = ⟨ H ′ ( z ( t ) ) , U ′ ( z ( t ) ) ⟩ ‖ U ′ ( z ( t ) ) ‖ 2 = ‖ H ′ ( z ( t ) ) ‖ ‖ U ′ ( z ( t ) ) ‖ .

Hence, we have

(2.1) d x d s ( s ) = ‖ U ′ ( z ( t ) ) ‖ ‖ H ′ ( z ( t ) ) ‖ d z d t ( t ) , with λ ( t ) = s .

The inverse function of λ satisfies

d ( λ − 1 ) d s ( s ) = ‖ U ′ ( x ( s ) ) ‖ ‖ H ′ ( x ( s ) ) ‖ .

Hence, we can obtain the time reparametrization solving the following integral:

(2.2) t ( s ) = ∫ 0 s ‖ U ′ ( x ( σ ) ) ‖ ‖ H ′ ( x ( σ ) ) ‖ d σ .

The following result provides the Finsler metric that we will employ in our study.

Lemma 2.4

Let G : D × R n → R be the Legendre transform of U with respect to p ; hence,

G ( q , v ) = sup p ∈ R n ( ⟨ v , p ⟩ − U ( q , p ) ) ,

and define ℒ : D × R n → D × R n as follows:

(2.3) ℒ ( q , p ) = q , ∂ U ∂ p ( q , p ) = ( q , v ) .

Then, the function F : D × R n → R , defined as F = G ( q , v ) , is a Finsler metric on D . Moreover, a curve q : [ 0 , S ] → D is a Finsler geodesic parametrized by arc length if and only if ( q ( s ) , p ( s ) ) = ℒ − 1 ( q ( s ) , q ˙ ( s ) ) is a solution of Hamilton’s equations with Hamiltonian U and U ( q ( s ) , p ( s ) ) ≡ 1 .

Proof

Since U is convex and homogeneous of degree 2 in p , for every ( q , v ) ∈ D × R n , the function

p ↦ ⟨ v , p ⟩ − U ( q , p )

has a unique maximum. Therefore, the function G is well defined, convex, and homogeneous of degree 2 in v . Moreover, G is of class C 2 on D × R n \ { 0 } , while it is of class C 1 on D × R n . Thus, the function F : D × R n → R defined as F ( q , v ) = G ( q , v ) is a Finsler metric on D . Since

∂ 2 U ∂ p 2 ( q , p ) > 0 , ∀ q ∈ D , ∀ p ∈ R n \ { 0 } ,

the map ( ∂ U / ∂ p ) ( q , ⋅ ) is invertible; thus, ℒ is a diffeomorphism, and it is homogeneous of degree 1 with respect to p . The equivalence between the Finsler geodesics parametrized by arc length and the solutions of (1.1) with energy E is a direct consequence of the Legendre transform (see, for instance, [20, Chapter I, p. 22]).□

By Lemmas 2.2 and 2.4, if q is a geodesic in D , then ℒ − 1 ( q , q ˙ ) is a solution of Hamilton’s equations with the original Hamiltonian H , up to a time reparametrization. As a consequence, finding a Finsler geodesic in D is equivalent to finding a solution of (1.1) in D with energy E . The following result provides the reparametrization that links the geodesics to the solutions of (1.1), combining the time reparametrization (2.2) with the Legendre transform defined in (2.3). For the sake of presentation, we use the following notation:

(2.4) ϕ ( q , v ) ≔ ‖ U ′ ( ℒ − 1 ( q , v ) ) ‖ ‖ H ′ ( ℒ − 1 ( q , v ) ) ‖ , ∀ ( q , v ) ∈ ℒ ( Σ ) .

Remark 2.5

The function ϕ given by (2.4) is well defined. Indeed, since Σ ¯ is compact, ∂ V ∂ q ( q ) ≠ 0 for all q ∈ ∂ D and H is strictly convex with respect to p , there exist two constants h 1 and h 2 such that

(2.5) 0 < h 1 ≤ ‖ H ′ ( q , p ) ‖ ≤ h 2 , ∀ ( q , p ) ∈ Σ ¯ .

Lemma 2.6

Let γ : [ 0 , 1 ] → D be a Finsler geodesic such that

G ( γ ( s ) , γ ˙ ( s ) ) = c γ , ∀ s ∈ [ 0 , 1 ] ,

and let λ : [ 0 , T ] → [ 0 , 1 ] be the reparametrization such that γ ∘ λ is a solution of (1.1) with energy E . Then, the inverse of λ is given by

t ( s ) = c γ ∫ 0 s ϕ γ ( σ ) , γ ˙ ( σ ) c γ d σ .

Proof

If G ( γ ( s ) , γ ˙ ( s ) ) = c γ for all s , then the reparametrization

λ 1 : [ 0 , c γ ] → [ 0 , 1 ] , λ 1 ( τ ) = τ c γ

is such that the curve γ ˆ = γ ∘ λ 1 : [ 0 , c γ ] → D is a geodesic parametrized by arc length. By Lemma 2.4, the curve x : [ 0 , c γ ] → D , which is defined as follows:

x ( τ ) = ( q ( τ ) , p ( τ ) ) = ℒ − 1 ( γ ˆ ( τ ) , γ ˆ ˙ ( τ ) ) ,

is a solution of Hamilton’s equations with respect to U . Let λ 2 : [ 0 , T ] → [ 0 , c γ ] be the inverse of

τ ↦ ∫ 0 τ ‖ U ′ ( x ( u ) ) ‖ ‖ H ′ ( x ( u ) ) ‖ d u = ∫ 0 τ ϕ ( γ ˆ ( u ) , γ ˆ ˙ ( u ) ) d u .

With the change of variable σ = u / c γ , σ ∈ [ 0 , 1 ] , we have

λ 2 − 1 ( τ ) = c γ ∫ 0 τ / c γ ϕ ( γ ˆ ( c γ σ ) , γ ˆ ˙ ( c γ σ ) ) d σ = c γ ∫ 0 τ / c γ ϕ γ ( σ ) , γ ˙ ( σ ) c γ d σ .

By Remark 2.3, in particular by (2.2), x ∘ λ 2 is a solution of (1.1). As a consequence, since ℒ − 1 is the identity map with respect to the first variable, the curve γ ∘ λ 1 ∘ λ 2 : [ 0 , T ] → D is the reparametrization of γ such that it is a solution of (1.1) with energy E . Hence, the desired reparametrization λ : [ 0 , T ] → [ 0 , 1 ] is given by λ = λ 1 ∘ λ 2 and its inverse t : [ 0 , 1 ] → [ 0 , T ] is given by

t ( s ) = λ 2 − 1 ( λ 1 − 1 ( s ) ) = λ 2 − 1 ( c γ s ) = c γ ∫ 0 s ϕ γ ( σ ) , γ ˙ ( σ ) c γ d σ ,

and we are done.□

Remark 2.7

When H is a Hamiltonian of natural type, the previous construction leads to the well-known Maupertuis principle (cf. [17]). Indeed, set

H ( q , p ) = 1 2 ∑ i , j = 1 n a i j ( q ) p i p j + V ( q ) ,

where ( a i j ( q ) ) is a positive definite quadratic form on R n . Then, using the aforementioned construction,

U ( q , p ) = 1 2 ( E − V ( q ) ) ∑ i , j = 1 n a i j ( q ) p i p j ,

and its Legendre transform is expressed as follows:

(2.6) G ( q , v ) = 1 2 ( E − V ( q ) ) ∑ i , j = 1 n a i j ( q ) v i v j ,

where ( a i j ( q ) ) is the inverse of ( a i j ( q ) ) . We observe that G ( q , v ) degenerates on the boundary ∂ D , where, by continuity, it can be extended to 0. Since

U ′ ( q , p ) = 1 E − V ( q ) H ′ ( q , p ) , ∀ ( q , p ) ∈ Σ ,

then

‖ U ′ ( q , p ) ‖ ‖ H ′ ( q , p ) ‖ = 1 E − V ( q ) , ∀ ( q , p ) ∈ Σ .

Using Lemma 2.6, if γ : [ 0 , 1 ] → D is a geodesic of constant speed with respect to the Riemannian metric G , then we can obtain the reparametrization λ : [ 0 , T ] → [ 0 , 1 ] such that q = γ ∘ λ : [ 0 , T ] → D is a solution of (1.1) for H . Using (2.2), the inverse of λ is given by

t ( s ) = c γ ∫ 0 s 1 E − V ( γ ( σ ) ) d σ ,

where G ( γ y ( s ) , γ ˙ ( s ) ) ≡ c γ .

3 Jacobi-Finsler metric near the boundary

Since U and G are not defined on ∂ D , the aforementioned construction does not allow to see the brake orbits in D ¯ as Finsler geodesics. In this section, we estimate the behavior of U near the boundary ∂ D , and we will show that G degenerates on ∂ D to the zero function, as it can be seen in (2.6) for the case of natural Hamiltonian systems. Differently from [19], we are interested in the multiplicity of the brake orbits, not only in their existence. Hence, in addition to the construction given in [19], we give an upper and a lower bound for the Finsler metric that depend only on H and the potential well D , and these bounds will be exploited to obtain the one-one correspondence between the brake orbits and the orthogonal geodesic chords.

As a preliminary step, we give the following result, which is available up to a modification of H ( q , p ) = K ( q , p ) + V ( q ) far away from Σ .

Lemma 3.1

There exist two constants ν 1 , ν 2 > 0 such that the followings hold for every q ∈ D ¯ and p ∈ R n :

(3.1) ν 1 ‖ ξ ‖ 2 ≤ ∂ 2 K ∂ p 2 ( q , p ) [ ξ , ξ ] ≤ ν 2 ‖ ξ ‖ 2 , ∀ ξ ∈ R n ;

(3.2) ν 1 ‖ p ‖ ≤ ∂ K ∂ p ( q , p ) ≤ ν 2 ‖ p ‖ ;

(3.3) 1 2 ν 1 ‖ p ‖ 2 ≤ K ( q , p ) ≤ 1 2 ν 2 ‖ p ‖ 2 .

Proof

Since we are interested in the solutions of Hamilton’s equations for H in Σ , which is a bounded set, we can modify H far away from Σ . Hence, we may assume that H is fiber-wise quadratic for ‖ p ‖ sufficiently large. By (1.2) and the compactness of D ¯ , there exist ν 1 , ν 2 > 0 such that (3.1) holds. Since ( ∂ K / ∂ p ) ( q , 0 ) = 0 and K ( q , 0 ) = 0 for all q , from (3.1), we infer (3.2) and (3.3) by integration.□

Lemma 3.2

Let ν 1 and ν 2 be the constants defined by Lemma 3.1. Then, the followings hold:

(3.4) ν 1 2 ( E − V ( q ) ) ‖ p ‖ 2 ≤ U ( q , p ) ≤ ν 2 2 ( E − V ( q ) ) ‖ p ‖ 2 , ∀ ( q , p ) ∈ D × R n ,

(3.5) ( E − V ( q ) ) 2 ν 2 ‖ v ‖ 2 ≤ G ( q , v ) ≤ ( E − V ( q ) ) 2 ν 1 ‖ v ‖ 2 , ∀ ( q , v ) ∈ D × R n .

Moreover, there exists a constant ν 3 > 0 such that

(3.6) ‖ U ′ ( q , p ) ‖ ≥ ν 3 E − V ( q ) , ∀ ( q , p ) ∈ Σ .

Proof

Set S n − 1 = { θ ∈ R n : ‖ θ ‖ = 1 } . We define H ˜ : D × S n − 1 × R + → R as follows:

H ˜ ( q , θ , ω ) = H ( q , ω θ ) − E .

Since H ˜ ( q , θ , 0 ) < 0 for all ( q , θ ) ∈ D × S n − 1 , exploiting also the convexity of H , we obtain that for all ( q , θ ) ∈ D × S n − 1 , there exists an unique ω > 0 such that H ˜ ( q , θ , ω ) = 0 . As a consequence, the function ω : D × S n − 1 → R + such that

H ˜ ( q , θ , ω ( q , θ ) ) = 0 , ∀ ( q , θ ) ∈ D × S n − 1

is well defined. Moreover, by (3.2), we have

∂ H ˜ ∂ ω ( q , θ , ω ) = ∂ H ∂ p ( q , ω θ ) , θ = 1 ω ∂ K ∂ p ( q , ω θ ) , ω θ ≥ ω ν 1 > 0 .

So we can apply the implicit function theorem to obtain that the function ω is of class C 2 and it satisfies

(3.7) ∂ ω ∂ q ( q , θ ) = − ∂ H ∂ p ( q , ω ( q , θ ) θ ) , θ − 1 ∂ H ∂ q ( q , ω ( q , θ ) ) = − ∂ K ∂ p ( q , ω ( q , θ ) θ ) , θ − 1 ∂ H ∂ q ( q , ω ( q , θ ) ) .

By definition of ω ( q , θ ) , K ( q , ω ( q , θ ) θ ) = E − V ( q ) for all ( q , θ ) ∈ D × S n − 1 . By (3.3), we have

1 2 ν 1 ω 2 ( q , θ ) ≤ K ( q , ω ( q , θ ) θ ) = E − V ( q ) ≤ 1 2 ν 2 ω 2 ( q , θ ) ,

and hence,

(3.8) 0 < 2 ( E − V ( q ) ) ν 2 ≤ ω 2 ( q , θ ) ≤ 2 ( E − V ( q ) ) ν 1 , ∀ q ∈ D , ∀ θ ∈ S n − 1 .

By definition of ω ( q , θ ) , we have also

U ( q , ω ( q , θ ) θ ) = 1 , ∀ q ∈ D , ∀ θ ∈ S n − 1 .

Since U is homogeneous of degree 2 in p , for all q ∈ D and p ≠ 0 , we obtain

(3.9) U ( q , p ) = ‖ p ‖ 2 ω 2 ( q , p / ‖ p ‖ ) U q , ω q , p ‖ p ‖ p ‖ p ‖ = ‖ p ‖ 2 ω 2 ( q , p / ‖ p ‖ ) .

Using (3.8) and (3.9), we obtain (3.4). Since the Legendre transform inverts the order relation, (3.4) implies (3.5).

It remains to prove (3.6). Let us fix δ > 0 and set

D δ = { q ∈ D : V ( q ) ≤ E − δ } ,

so every point in D δ is far away from the boundary ∂ D . By the bounds on the function U given by (3.4) and recalling that U is homogeneous of degree 2 in p , there exists a constant c δ such that

(3.10) ‖ U ′ ( q , p ) ‖ ≥ ∂ U ∂ p ( q , p ) ≥ c δ > 0 , ∀ ( q , p ) ∈ Σ , q ∈ D δ .

By the arbitrariness of δ , we can obtain (3.6) by proving it for all ( q , p ) ∈ Σ with q sufficiently near the boundary. More precisely, we prove the existence of a constant c 1 such that

(3.11) ‖ U ′ ( q , p ) ‖ ≥ ∂ U ∂ q ( q , p ) ≥ c 1 E − V ( q ) ,

for all ( q , p ) ∈ Σ with q sufficiently near the boundary. For every ( q , p ) ∈ Σ , U ( q , p ) = 1 , so by (3.9), we obtain

∂ U ∂ q ( q , p ) = − 2 ‖ p ‖ 2 ω 3 ( q , p / ‖ p ‖ ) ∂ ω ∂ q q , p ‖ p ‖ = − 2 ω ( q , p / ‖ p ‖ ) ∂ ω ∂ q q , p ‖ p ‖ , ∀ ( q , p ) ∈ Σ .

As a consequence, using also (3.7) and denoting p / ‖ p ‖ by θ , we have

(3.12) ∂ U ∂ q ( q , p ) = 2 ω ( q , θ ) ∂ K ∂ p ( q , ω ( q , θ ) θ ) , θ − 1 ∂ H ∂ q ( q , ω ( q , θ ) ) ,

for all ( q , p ) ∈ Σ . By (3.2), we have

∂ K ∂ p ( q , ω ( q , θ ) θ ) , θ − 1 ≥ 1 ν 2 ω ( q , θ ) .

Hence, by (3.12) and using again (3.8), we obtain

(3.13) ∂ U ∂ q ( q , p ) ≥ 2 ν 2 ω 2 ( q , θ ) ∂ H ∂ q ( q , ω ( q , θ ) θ ) ≥ ν 1 ν 2 ( E − V ( q ) ) ∂ H ∂ q ( q , ω ( q , θ ) θ ) , ∀ ( q , p ) ∈ Σ .

The existence of a strictly positive constant c 1 such that (3.11) holds for all ( q , p ) ∈ Σ with q sufficiently near the boundary ∂ D can be obtained by (3.13), recalling that D ¯ is compact and ( ∂ V / ∂ q ) ( q ) ≠ 0 in ∂ D . Finally, we obtain (3.6) by (3.10) and (3.11), recalling the arbitrariness of δ .□

Remark 3.3

By (3.5), we can extend G on the boundary ∂ D by continuity. Denoting this extension again with G , we have

G ( q , v ) = 0 , ∀ q ∈ ∂ D , ∀ v ∈ R n .

3.1 Behaviour of the solutions near the boundary

In this section, we present some preliminary results about the behavior of the solutions of Hamilton’s equations near the boundary of the potential well D . These results are required to analyze the time reparametrization of the Finsler geodesics that correspond to the brake orbits and to study the strong concavity of the set Ω ¯ described in Theorem 1.9.

Lemma 3.4

There exists ε ¯ > 0 such that, if ( q ( t ) , p ( t ) ) is a solution of (1.1) with Hamiltonian H and energy E such that V ( q ( t ) ) ≥ E − ε ¯ for t ∈ [ a , b ] , then

d 2 d t 2 V ( q ( t ) ) ≤ − ε ¯ , ∀ t ∈ [ a , b ] .

Proof

See [19, Lemma 5.2].□

The following result provides an upper bound for the length of a time interval in which a solution of Hamilton’s equations with energy E can be uniformly near the boundary.

Lemma 3.5

Let ε ¯ given by Lemma 3.4. If ( q ( t ) , p ( t ) ) is a solution of Hamilton’s equations with total energy E and V ( q ( t ) ) ≥ E − ε ¯ / 2 for t ∈ [ a , b ] , then b − a ≤ 2 .

Proof

See [19, Corollary 5.3].□

For every Q ∈ ∂ D , we denote by z ( t , Q ) = ( q ( t , Q ) , p ( t , Q ) ) the solution of Hamilton’s equations for H with total energy E and such that q ( 0 ) = Q . Since z ( t , Q ) is the solution of the Cauchy problem

z ˙ ( t , Q ) = J H ′ ( z ( t , Q ) ) , z ( 0 , Q ) = ( Q , 0 ) ,

it is well defined and of class C 1 .

Remark 3.6

Since J H ′ is a function of class C 1 , also z ˙ ( t , Q ) is of class C 1 with respect to the variables t and Q .

Lemma 3.7

For every Q 0 ∈ ∂ D , there exists a function ρ : [ 0 , + ∞ [ × ∂ D → R n of class C 1 such that d ρ ( 0 , Q 0 ) = 0 and

(3.14) q ˙ ( t , Q ) = − t ∂ 2 H ∂ p 2 ( Q 0 , 0 ) ∇ V ( Q 0 ) + ρ ( t , Q ) , ∀ t ∈ [ 0 , + ∞ [ , ∀ Q ∈ ∂ D .

Proof

We define ρ 0 : [ 0 , + ∞ [ × ∂ D → R n as follows:

ρ 0 ( t , Q ) = q ¨ ( t , Q ) − q ¨ ( 0 , Q 0 ) .

Recalling that z ( t , Q ) is of class C 1 both respect to t and Q and taking the derivative with respect to t of z ˙ ( t , Q ) = J H ′ ( z ( t , Q ) ) , we obtain that z ¨ ( t , Q ) is a continuous function, so ρ 0 ( t , Q ) is a continuous function. Since p ( 0 , Q ) = 0 for all Q ∈ ∂ D , then q ˙ ( 0 , Q ) = 0 and

q ¨ ( 0 , Q ) = d d t q ˙ ( t , Q ) t = 0 = ∂ 2 H ∂ q ∂ p ( q ( t , Q ) , p ( t , Q ) ) q ˙ ( t , Q ) + ∂ 2 H ∂ p 2 ( q ( t , Q ) , p ( t , Q ) ) p ˙ ( t , Q ) t = 0 = − ∂ 2 H ∂ p 2 ( Q , 0 ) ∇ V ( Q ) , ∀ Q ∈ ∂ D .

By definition of ρ 0 , we have

q ¨ ( t , Q ) = q ¨ ( 0 , Q 0 ) + ρ 0 ( t , Q ) = − ∂ 2 H ∂ p 2 ( Q 0 , 0 ) ∇ V ( Q 0 ) + ρ 0 ( t , Q ) .

Integrating the previous equation, recalling that q ˙ ( 0 , Q ) = 0 and setting ρ ( t , Q ) = ∫ 0 t ρ 0 ( τ , Q ) d τ , we obtain (3.14). Since ρ 0 ( 0 , Q 0 ) = 0 , we have d ρ ( 0 , Q 0 ) = 0 .□

4 The Jacobi-Finsler energy function

In this section, we introduce the function ψ : D ¯ → R , which will be exploited to define the strongly concave set Ω ¯ with the properties required by Theorem 1.9. Given a point y ∈ D ¯ , the function ψ ( y ) is the infimum of the energy of the curves connecting y with ∂ D . We prove that, if y is sufficiently near the boundary, ψ ( y ) is attained on exactly one curve that is a solution of (1.1) with energy E , up to time reparametrization.

Let us define the functional J : W 1 , 2 ( [ 0 , 1 ] , D ¯ ) → R as follows:

J ( γ ) = ∫ 0 1 G ( γ ( s ) , γ ˙ ( s ) ) d s .

If γ ( [ 0 , 1 ] ) ⊂ D , then J is differentiable at γ and its differential

d J ( γ ) : W 1 , 2 ( [ 0 , 1 ] , R n ) → R

is given by

d J ( γ ) [ ξ ] = ∫ 0 1 ∂ G ∂ q ( γ ( s ) , γ ˙ ( s ) ) [ ξ ( s ) ] + ∂ G ∂ v ( γ ( s ) , γ ˙ ( s ) ) [ ξ ˙ ( s ) ] d s .

For every y ∈ D , we define X y as follows:

X y ≔ { γ ∈ W 1 , 2 ( [ 0 , 1 ] , D ¯ ) : γ ( 0 ) = y , γ ( [ 0 , 1 [ ) ⊂ D and γ ( 1 ) ∈ ∂ D } .

Definition 4.1

We define the function ψ : D ¯ → R as follows:

(4.1) ψ ( y ) ≔ inf γ ∈ X y J ( γ ) .

The function ψ will be the main focus of our analysis. Indeed, from now on, we will state and prove some results that will lead to define the set Ω described in Theorem 1.9 as ψ − 1 ( ] δ , ∞ [ ) , for some δ sufficiently small.

Lemma 4.2

There exists a constant d ¯ > 0 such that

ψ ( y ) ≤ d ¯ , ∀ y ∈ D ¯ .

Proof

The thesis directly follows from the upper bound given in (3.5) and the compactness of D ¯ .□

Proposition 4.3

For every y ∈ D , ψ ( y ) is attained on at least one curve γ y ∈ X y . Moreover, γ y satisfies

(4.2) ∫ 0 1 ∂ G ∂ q ( γ y ( s ) , γ ˙ y ( s ) ) [ ξ ( s ) ] + ∂ G ∂ v ( γ y ( s ) , γ ˙ y ( s ) ) [ ξ ˙ ( s ) ] d s = 0 , ∀ ξ ∈ W 0 1 , 2 ( [ 0 , 1 ] , R n ) ,

and there exist a T > 0 and a diffeomorphism σ : [ 0 , T ] → [ 0 , 1 ] such that, setting γ ˆ y = γ y ∘ σ : [ 0 , T ] → D ¯ , the pair ( q , p ) : [ 0 , T ] → D ¯ × R n given by

( q ( t ) , p ( t ) ) = ℒ − 1 ( γ ˆ y ( t ) , γ ˆ ˙ y ( t ) )

is a solution of (1.1) with energy E , q ( 0 ) = y and q ( T ) ∈ ∂ D .

To prove Proposition 4.3, we obtain γ y as the weak limit of a sequence of Finsler geodesics ( γ k ) ⊂ W 1 , 2 ( [ 0 , 1 ] , D ) . We exploit the fact that ( T k ) is uniformly bounded, where T k are given by

T k = t k ( 1 ) = J ( γ k ) ∫ 0 1 ϕ γ ( σ ) , γ ˙ ( σ ) J ( γ k ) d σ .

Recalling the reparametrization given by Lemma 2.6, T k is the final time of the reparametrization of γ k , which is a solution of (1.1) with energy E . More formally, we require the following lemma.

Lemma 4.4

Let ( γ k ) ⊂ W 1 , 2 ( [ 0 , 1 ] , D ) be a sequence of Finsler geodesics. If there exist two constants c 1 , c 2 such that

(4.3) 0 < c 1 ≤ J ( γ k ) ≤ c 2 , ∀ k ∈ N ,

then there exist two constants c 3 , c 4 such that

(4.4) 0 < c 3 ≤ T k ≤ c 4 , ∀ k ∈ N .

Proof

Since ( E − V ( q ) ) → 0 when q → ∂ D and D ¯ is compact, there exists a strictly positive constant c 5 such that

1 E − V ( q ) ≥ c 5 , ∀ q ∈ D .

By the definition of ϕ given by (2.4), using (2.5) and (3.6), we have

∫ 0 1 ϕ γ k ( s ) , γ ˙ k ( s ) J ( γ k ) d s ≥ ∫ 0 1 ν 3 h 2 ( E − V ( γ k ( s ) ) ) d s ≥ c 5 ν 3 h 2 > 0 .

By using also (4.3), we obtain

T k = J ( γ k ) ∫ 0 1 ϕ γ ( σ ) , γ ˙ ( σ ) J ( γ k ) d σ ≥ c 1 c 5 ν 3 h 2 ≕ c 3 > 0 .

To prove the existence of a constant c 4 such that (4.4) holds, we work directly on the reparametrizations of γ k . Following the construction given in [19, Lemma 5.1], let ε ¯ > 0 be given by Lemma 3.4 and let us divide D ¯ into (Figure 3):

The rim { q ∈ D ¯ : V ( q ) ≥ E − ε ¯ / 2 } ;
The band { q ∈ D ¯ : E − ε ¯ ≤ V ( q ) ≤ E − ε ¯ / 2 } ;
The core { q ∈ D ¯ : V ( q ) ≤ E − ε ¯ } .

Let us set

Σ ¯ ε ¯ / 2 = { ( q , p ) ∈ Σ : V ( q ) ≤ E − ε ¯ / 2 } .

Since Σ ¯ ε ¯ / 2 is compact, there exists a constant ϕ ¯ > 0 such that ϕ ( q , p ) ≤ ϕ ¯ for all ( q , p ) ∈ Σ ¯ ε ¯ / 2 . For every k , we set I k = { s ∈ [ 0 , 1 ] : V ( γ k ( s ) ) < E − ε ¯ / 2 } and C k = [ 0 , 1 ] ⧹ I k . Hence,

(4.5) ∫ 0 1 ϕ γ k ( s ) , γ ˙ k ( s ) J ( γ k ) d s = ∫ I k ϕ γ k ( s ) , γ ˙ k ( s ) J ( γ k ) d s + ∫ C k ϕ γ k ( s ) , γ ˙ k ( s ) J ( γ k ) d s ≤ ϕ ¯ + ∫ C k ϕ γ k ( s ) , γ ˙ k ( s ) J ( γ k ) d s .

For every k , the set C k is the union of closed and disjoint intervals in which the orbit γ k is in the rim. The orbit can enter the rim many times, but, as a consequence of Lemma 2.6, each pair of passages into the rim must be separated by a dip into the core, and this requires the solution to cross the band twice. This bounds the number of closed disjoint intervals that constitute C k , independently of k . Indeed, let us set

d ¯ = min { J ( γ ) : γ ∈ W 1 , 2 ( [ 0 , 1 ] , D ) , V ( γ ( 0 ) ) = E − ε ¯ , V ( γ ( 1 ) ) = E − ε ¯ / 2 } .

Since (4.3) holds, we have that γ k can cross the band at most N times, where N is a positive integer strictly greater than c 2 / ( 2 d ¯ ) , independent of k . As a consequence, by Lemma 3.5, we have

J ( γ k ) ∫ C k ϕ γ k ( s ) , γ ˙ k ( s ) J ( γ k ) d s ≤ 2 N ,

and, by (4.3) and (4.5), we have

T k ≤ c 2 ϕ ¯ + 2 N ≕ c 4 ,

so (4.4) holds.□

Figure 3

The division of the potential well D into the rim, the band and the core.

The next lemma provides the sequence of geodesics that will be exploited in the proof of Proposition 4.3. For every k ∈ N , let D k = V ( ] − ∞ , E − 1 / k [ ) ∩ D and

X y k = { γ ∈ W 1 , 2 ( [ 0 , 1 ] , D ¯ k ) : γ ( 0 ) = y , γ ( 1 ) ∈ ∂ D k } .

Lemma 4.5

For k sufficiently large, the functional J restricted to X y k has a minimum γ k that is a Finsler geodesic and such that γ k ( [ 0 , 1 [ ) ⊂ D k .

Proof

By the continuity of the function V , for k sufficiently large, we have y ∈ D k , so X y k ≠ ∅ , and D k is homeomorphic to D ; hence, D ¯ k is a closed and bounded set; thus it is compact. In the following, let us denote by J the restriction of it to X y k . Since J is bounded from below and D ¯ k is compact, a minimizing sequence ( γ m ) m ⊂ X y k is equibounded with respect to norm of W 1 , 2 ( [ 0 , 1 ] , D ¯ k ) . By the Ascola-Arzelà theorem, unless to consider a subsequence, ( γ m ) m converges uniformly to a curve γ k and, since D ¯ k is closed, γ k ∈ X y k . Moreover, ( γ m ) m converges weakly to γ k in W 1 , 2 ( [ 0 , 1 ] , D ¯ k ) . By Lemma 2.4, the function G ( q , v ) is strongly convex in D ¯ k with respect to the variable v . As a consequence, J is lower weakly semi-continuous; hence,

J ( γ k ) ≤ liminf m → ∞ J ( γ m ) = inf γ ∈ X y k J ( γ ) ,

so γ k is a minimum for J in X y k . Let us prove that γ k ( [ 0 , 1 [ ) ∈ D k . Assume, by contradiction, that there exists s ¯ ∈ ] 0 , 1 [ such that γ k ( s ¯ ) ∈ ∂ D k . Then, defining the curve γ ˜ k ∈ X y k as γ ˜ k ( s ) = γ k ( s / s ¯ ) , we obtain

J ( γ k ) = ∫ 0 1 G ( γ k , γ ˙ k ) d s ≥ ∫ 0 s ¯ G ( γ k , γ ˙ k ) d s = 1 s ¯ ∫ 0 1 G ( γ ˜ k , γ ˜ ˙ k ) d s > ∫ 0 1 G ( γ ˜ k , γ ˜ ˙ k ) d s = J ( γ ˜ k ) ,

which contradicts the minimality of γ k . Since γ k is a curve, which minimizes the energy functional J and γ k ( [ 0 , 1 [ ) ⊂ D k , we obtain that γ k is a geodesic.□

Proof of Proposition 4.3

For k sufficiently large, let ( γ k ) k be a sequence of geodesics obtained with Lemma 4.5. Setting ℓ k = J ( γ k ) , by definition of ψ in (4.1), we have

liminf k → ∞ ℓ k ≥ ψ ( y ) .

We claim that

(4.6) liminf k → ∞ ℓ k = ψ ( y ) .

By absurd, if it was liminf k → ∞ ℓ k > ψ ( y ) , then we could find a curve x ∈ X y such that J ( x ) < liminf k → ∞ ℓ k and a suitable reparametrization of x would yield a curve x k ∈ X y k such that J ( x k ) < ℓ k , which contradicts the minimality of ℓ k . Hence, (4.6) holds. Since γ k minimizes J on X y k , it is a geodesic with constant speed; hence,

G ( γ k ( s ) , γ ˙ k ( s ) ) = ℓ k , ∀ s ∈ [ 0 , 1 ] .

Using (3.5) and Lemma 4.2, there are two constants c 1 , c 2 such that

(4.7) 0 < c 1 ≤ ℓ k ≤ c 2 ,

for all k sufficiently large. As a consequence, we can apply Lemma 4.4, so there exist two constants c 3 , c 4 such that (4.4) holds for every k sufficiently large. Using also (2.5), (3.6), and (4.7), we have

c 4 ≥ T k ≥ c 3 ∫ 0 1 ν 3 h 2 ( E − V ( γ k ( s ) ) ) d s .

Then, the sequence

∫ 0 1 1 E − V ( γ k ( s ) ) d s

is bounded. By (3.5) and (4.7), we have

∫ 0 1 ‖ γ ˙ k ( s ) ‖ 2 d s ≤ ∫ 0 1 2 ν 2 E − V ( γ k ( s ) ) G ( γ k ( s ) , γ ˙ k ( s ) ) d s = ℓ k ∫ 0 1 2 ν 2 E − V ( γ k ( s ) ) d s ≤ 2 ν 2 c 2 ∫ 0 1 1 E − V ( γ k ( s ) ) d s ,

so ( γ k ) is bounded in W 1 , 2 ( [ 0 , 1 ] , D ¯ ) . By the Ascoli Arzelà theorem, γ k uniformly converges to a curve γ y , up to a subsequence. We claim that γ y is a minimizer for J in X y . Let us show that γ y ∈ X y . Since γ k converges uniformly to γ y , γ y ( 0 ) = y and γ ( 1 ) ∈ ∂ D . We show that γ ( [ 0 , 1 [ ) ⊂ D arguing by contradiction. Let s ¯ ∈ ( 0 , 1 ) be the first instant, where γ y ( s ¯ ) ∈ ∂ D . By the minimality of γ k , we have that γ y ( [ s ¯ , 1 ] ) ⊂ ∂ D . Thus, we obtain

lim k → ∞ ( 1 − s ¯ ) ℓ k = lim k → ∞ ∫ s ¯ 1 G ( γ k ( s ) , γ ˙ k ( s ) ) d s = ∫ s ¯ 1 G ( γ y ( s ) , γ ˙ y ( s ) ) d s = 0 ,

in contradiction with ℓ k ≥ c 1 > 0 , given by (4.7). Hence, γ y belongs to X y and, since J ( γ y ) ≤ liminf k → ∞ ℓ k , by (4.6), we obtain J ( γ y ) = ψ ( y ) . Being a minimizer, γ y satisfies (4.2). By Lemma 2.6, the diffeomorphism σ : [ 0 , T ] → [ 0 , 1 ] such that γ y ∘ σ is a solution of (1.1) with energy E has inverse

t ( s ) = J ( γ y ) ∫ 0 s ϕ γ y ( σ ) , γ y ˙ ( σ ) J ( γ y ) d σ .

By (4.4),

T = t ( 1 ) = J ( γ y ) ∫ 0 1 ϕ γ y ( σ ) , γ ˙ y ( σ ) J ( γ y ) d σ

is bounded and strictly greater than 0.□

The next main step is proving that if y is sufficiently near ∂ D , then the minimizer of J in X y is unique. To this aim, we require the following lemma, which provides a coordinate system of a neighborhood of ∂ D through the solutions of (1.1) that start from the boundary.

Lemma 4.6

There exists a constant δ ¯ > 0 such that the following property holds:

(4.8) ∀ y ∈ D with ψ ( y ) ≤ δ ¯ there exists a unique solution ( q y , p y ) of ( 1.1 ) with energy E and a unique t y > 0 such that q y ( 0 ) ∈ ∂ D , q y ( t y ) = y and ψ ( q y ( t ) ) ≤ δ ¯ , for all t ∈ [ 0 , t y ] .

A representation of Property (4.8) is given in Figure 4.

$Figure 4 A representation of Property (4.8). Each point y i {y}_{i} , i = 1 , 2 , 3 i=1,2,3 , is such that ψ ( y i ) ≤ δ ¯ \psi ({y}_{i})\le \bar{\delta } . Hence, there exists a unique q y i ∈ ∂ D {q}_{{y}_{i}}\in \partial D and a unique t y i {t}_{{y}_{i}} such that q ( 0 ) = q y i q\left(0)={q}_{{y}_{i}} , q ( t y i ) = y i q\left({t}_{{y}_{i}})={y}_{i} and ψ ( q ( t ) ) ≤ δ ¯ \psi \left(q\left(t))\le \bar{\delta } for every t ∈ [ 0 , t y i ] t\in \left[0,{t}_{{y}_{i}}] , where q q is the unique solution of Hamilton’s equations with energy E E starting from q y i {q}_{{y}_{i}} .$

Figure 4

A representation of Property (4.8). Each point y i , i = 1 , 2 , 3 , is such that ψ ( y i ) ≤ δ ¯ . Hence, there exists a unique q y i ∈ ∂ D and a unique t y i such that q ( 0 ) = q y i , q ( t y i ) = y i and ψ ( q ( t ) ) ≤ δ ¯ for every t ∈ [ 0 , t y i ] , where q is the unique solution of Hamilton’s equations with energy E starting from q y i .

Proof

By Lemma 3.7, for every Q 0 ∈ ∂ D , there exists a function ρ : ∂ D × R → R n such that

q ( t , Q ) = Q − t 2 2 ∂ 2 H ∂ p 2 ( Q 0 , 0 ) ∂ V ∂ q ( Q 0 ) + ∫ 0 t ρ ( τ , Q ) d τ , ∀ Q ∈ ∂ D ,

where the vector ∂ 2 H ∂ p 2 ( Q 0 , 0 ) ∂ V ∂ q ( Q 0 ) is not tangent to ∂ D for every Q 0 ∈ ∂ D . Indeed, ∂ V ∂ q ( Q 0 ) is orthogonal to ∂ D by definition of D and by (3.1), we have

∂ 2 H ∂ p 2 ( Q 0 , 0 ) ∂ V ∂ q ( Q 0 ) , ∂ V ∂ q ( Q 0 ) ≥ ν 1 ∂ V ∂ q ( Q 0 ) 2 > 0 , ∀ Q 0 ∈ ∂ D .

As a consequence, if { y 1 , … , y n − 1 } is a coordinate system of ∂ D in a neighborhood of Q 0 , then { y 1 , … , y n − 1 , t } is a local coordinate system on the manifold with boundary ∂ D and ( t , Q ) ↦ q ( t , Q ) defines a local chart. By the compactness of ∂ D , we can construct a neighborhood N ⊂ D ¯ of ∂ D as union of a finite number of such local charts. By the upper bound on G given by (3.5), ψ ( y ) → 0 as y → ∂ D , and there exists a δ ¯ > 0 such that ψ − 1 ( [ 0 , δ ¯ ] ) ⊂ N . As a consequence, if y ∈ D ¯ satisfies ψ ( y ) ≤ δ ¯ , then y ∈ N , and it is uniquely represented by a coordinate of the constructed local chart, so there exists a unique solution ( q y , p y ) of (1.1) with energy E and a unique t y > 0 such that q y ( 0 ) ∈ ∂ D , q y ( t y ) = y and q y ( t ) ∈ N for each t ∈ [ 0 , t y ] . It remains to prove that ψ ( q y ( t ) ) ≤ δ ¯ for every t ∈ [ 0 , t y ] . By Proposition 4.3, there exists γ y such that J ( γ y ) = ψ ( y ) and a reparametrization of it is a solution of (1.1) with energy E and with an endpoint in ∂ D . We recall that, since H is even with respect to p , a backward parametrization of a solution is still a solution. Thus, there exists a solution q γ y : [ 0 , T ] → D ¯ of (1.1) and energy E such that q γ y ( 0 ) ∈ ∂ D , q γ y ( T ) = y , and for each t ∈ [ 0 , T ] , there exists an s ∈ [ 0 , 1 ] such that q γ y ( t ) = γ y ( s ) . Using the reparametrizations of γ y , it can be proved that for each s ∈ [ 0 , 1 ] , we have ψ ( γ y ( s ) ) ≤ δ ¯ , so ψ ( q γ y ( t ) ) ≤ δ ¯ for every t ∈ [ 0 , T ] . By definition of δ ¯ , this implies that q γ y ( t ) ∈ N for every t ∈ [ 0 , T ] . By the uniqueness of q y , we obtain that T = t y and q γ y ( t ) = q y ( t ) for every t ∈ [ 0 , t y ] , so we are done.□

Notation: If y ∈ D is such that ψ ( y ) ≤ δ ¯ , we denote by ( t y , Q y ) the unique element in R + × ∂ D such that q ( t y , Q y ) = y and ψ ( q ( t , Q y ) ) ≤ δ ¯ , for all t ∈ [ 0 , t y ] .

Remark 4.7

Both t y and Q y are functions of class C 1 with respect to y , since they are implicitly defined by the coordinate system given by the proof of Lemma 4.6.

Proposition 4.8

For every y ∈ D such that ψ ( y ) ≤ δ ¯ , the minimizer of J in the space X y is unique.

Proof

By contradiction argument, let us assume the existence of two different curves, γ 1 , γ 2 ∈ X y , such that ψ ( y ) = J ( γ 1 ) = J ( γ 2 ) . Since γ 1 and γ 2 are two minimizers, by Proposition 4.3, they are reparametrizations of two solutions of (1.1) with energy E and final points on ∂ D . Moreover, ψ ( γ 1 ( s ) ) ≤ δ ¯ and ψ ( γ 2 ( s ) ) ≤ δ ¯ for every s ∈ [ 0 , 1 ] . Hence, if γ 1 ( 1 ) ≠ γ 2 ( 1 ) , then setting Q 1 = γ 1 ( 1 ) and Q 2 = γ 2 ( 1 ) , we have

y = q ( t 1 , Q 1 ) and y = q ( t 2 , Q 2 ) ,

for some t 1 , t 2 > 0 ,

ψ ( q ( t , Q 1 ) ) ≤ δ ¯ ∀ t ∈ [ 0 , t 1 ] and ψ ( q ( t , Q 2 ) ) ≤ δ ¯ ∀ t ∈ [ 0 , t 2 ] .

As a consequence, y is given by two different coordinates of the local chart constructed in Lemma 4.6, which is a contradiction. If γ 1 ( 1 ) = γ 2 ( 1 ) , by the uniqueness of the solution q ( t , Q y ) of (1.1) with energy E , we infer that γ 1 ( s ) = γ 2 ( s ) for all s ∈ [ 0 , 1 ] , which is a contradiction.□

Remark 4.9

By Propositions 4.3 and 4.8, for every y ∈ D such that ψ ( y ) ≤ δ ¯ , the minimizer γ y and the curve q ( t , Q y ) are linked by a reparametrizations, which invert the orientation.

5 Differentiability and concavity

In this section, we prove that ψ is of class C 2 near the boundary and that, if δ ˆ is sufficiently small, the set ψ − 1 ( [ δ ˆ , + ∞ [ ) is strongly concave with respect to the Finsler metric F .

Let δ ¯ satisfies property (4.8), and set

D δ ¯ = { y ∈ D : ψ ( y ) ≤ δ ¯ } .

Proposition 5.1

For every y ∈ D δ ¯ , ψ is differentiable at y and

(5.1) d ψ ( y ) [ ξ ] = − ∂ G ∂ v ( y , γ ˙ y ( 0 ) ) [ ξ ] , ∀ ξ ∈ R n .

Proof

Let us fix ξ ∈ R n , ‖ ξ ‖ ≤ 1 and set

v ξ ( s ) = max { 0 , 1 − 2 s } ξ ,

and hence, v ξ ( s ) = 0 for all s ∈ [ 1 / 2 , 1 ] . Let us define J ˜ : W 1 , 2 ( [ 0 , 1 ] , D ) → R as follows:

J ˜ ( γ ) = ∫ 0 1 2 G ( γ ( s ) , γ ˙ ( s ) ) d s .

Since the curve γ y ∣ [ 0 , 1 / 2 ] is uniformly far from ∂ D , so are the curves ( γ y + ε v ξ ) ∣ [ 0 , 1 / 2 ] for ε sufficiently small. Moreover, by Proposition 4.3, there exists a constant c γ > 0 such that G ( γ y ( s ) , γ ˙ y ( s ) ) = c γ , for all s ∈ [ 0 , 1 [ . As a consequence, for ε sufficiently small, we can assume that

(5.2) γ ˙ y ( s ) + σ ε v ˙ ξ ( s ) ≠ 0 , ∀ s ∈ 0 , 1 2 , ∀ σ ∈ [ 0 , 1 ] ,

and thus, we shall work in a region where G is of class C 2 . By definition of ψ , we have

ψ ( y + ε ξ ) ≤ J ( γ y + ε v ξ ) .

Since ψ ( y ) = J ( γ y ) , we obtain

ψ ( y + ε ξ ) − ψ ( y ) ≤ J ( γ y + ε v ξ ) − J ( γ y ) = J ˜ ( γ y + ε v ξ ) − J ˜ ( γ y ) ,

and hence,

limsup ε → 0 1 ε ( ψ ( y + ε ξ ) − ψ ( y ) ) ≤ limsup ε → 0 1 ε ( J ˜ ( γ y + ε v ξ ) − J ˜ ( γ y ) ) .

Then, using the dominated convergence theorem, an integration by parts and recalling that γ y satisfies (4.2), we have

lim ε → 0 1 ε ( J ˜ ( γ y + ε v ξ ) − J ˜ ( γ y ) ) = lim ε → 0 1 ε ∫ 0 1 2 ( G ( γ y + ε v ξ , γ ˙ y + ε v ˙ ξ ) − G ( γ y , γ ˙ y ) ) d s = ∫ 0 1 2 ∂ G ∂ q ( γ y , γ ˙ y ) [ v ξ ] + ∂ G ∂ v ( γ y , γ ˙ y ) [ v ˙ ξ ] d s = ∂ G ∂ v ( γ y , γ ˙ y ) [ v ˙ ξ ] 0 1 / 2 = − ∂ G ∂ v ( y , γ ˙ y ( 0 ) ) [ ξ ] ,

and hence,

limsup ε → 0 1 ε ( ψ ( y + ε ξ ) − ψ ( y ) ) ≤ − ∂ G ∂ v ( y , γ ˙ y ( 0 ) ) [ ξ ] .

It remains to prove that

(5.3) liminf ε → 0 1 ε ( ψ ( y + ε ξ ) − ψ ( y ) ) ≥ − ∂ G ∂ v ( y , γ ˙ y ( 0 ) ) [ ξ ] .

Since ψ ( y + ε ξ ) = J ( γ y + ε ξ ) and ψ ( y ) ≤ J ( γ y + ε ξ − ε v ξ ) , we have

(5.4) ψ ( y + ε ξ ) − ψ ( y ) ≥ J ( γ y + ε ξ ) − J ( γ y + ε ξ − ε v ξ ) = J ˜ ( γ y + ε ξ ) − J ˜ ( γ y + ε ξ − ε v ξ ) .

By (5.2), J ˜ is of class C 2 in a neighborhood of γ y . Hence, there exists some σ ε ∈ ] 0 , 1 [ such that

(5.5) J ˜ ( γ y + ε ξ ) − J ˜ ( γ y + ε ξ − ε v ξ ) = ε d J ˜ ( γ y + ε ξ ) [ v ξ ] − ε 2 2 d 2 J ˜ ( γ y + ε ξ − σ ε ε v ξ ) [ v ξ , v ξ ] .

Now, we are going to prove that

(5.6) lim ε → 0 ε d 2 J ˜ ( γ y + ε ξ − σ ε ε v ξ ) [ v ξ , v ξ ] = 0 .

Since γ y is uniformly far from ∂ D on the interval [ 0 , 1 / 2 ] , the same holds for γ y + ε ξ whenever ε is sufficiently small. As a consequence, there exists a constant c 1 > 0 such that

1 E − V ( γ y + ε ξ ( s ) ) ≤ c 1 , ∀ s ∈ 0 , 1 2 .

Since γ y + ε ξ is a minimal geodesic, we also have

G ( γ y + ε ξ ( s ) , γ ˙ y + ε ξ ( s ) ) = ψ ( y + ε ξ ) , ∀ s ∈ 0 , 1 2 .

Moreover, using (3.5), there exists a constant c 2 > 0 such that

(5.7) ∫ 0 1 2 ‖ γ ˙ y + ε ξ ( s ) ‖ 2 d s ≤ ∫ 0 1 2 2 ν 2 E − V ( γ y + ε ξ ( s ) ) G ( γ y + ε ξ ( s ) , γ ˙ y + ε ξ ( s ) ) d s ≤ c 1 ν 2 ψ ( y + ε ξ ) ≤ c 2 .

Hence, γ y + ε ξ is uniformly bounded in W 1 , 2 0 , 1 2 , D . Since v ξ ( s ) = 0 on 0 , 1 2 , we have that d 2 J ˜ ( γ y + ε ξ − σ ε ε v ξ ) [ v ξ , v ξ ] is uniformly bounded with respect to ε sufficiently small; hence, (5.6) holds. By (5.5) and (5.6), we have

(5.8) lim ε → 0 1 ε ( J ˜ ( γ y + ε ξ ) − J ˜ ( γ y + ε ξ − ε v ξ ) ) = lim ε → 0 d J ˜ ( γ y + ε ξ ) [ v ξ ] .

Since γ y + ε ξ satisfies (4.2), integration by parts leads to

(5.9) d J ˜ ( γ y + ε ξ ) [ v ξ ] = − ∂ G ∂ v ( y + ε ξ , γ ˙ y + ε ξ ( 0 ) ) [ ξ ] .

To obtain (5.3) and conclude the proof, it suffices to show that

(5.10) lim ε → 0 γ ˙ y + ε ξ ( 0 ) = γ ˙ y ( 0 ) .

To this aim, we exploit the uniqueness of γ y ensured by Proposition 4.8. Arguing by contradiction, let ( ε n ) be a sequence such that ε n → 0 and

lim n → ∞ γ ˙ y + ε n ξ ( 0 ) ≠ γ ˙ y ( 0 ) .

By (5.7), γ y + ε ξ are uniformly bounded in W 1 , 2 ( [ 0 , 1 ] , D ¯ ) ; hence, there exists v ∈ R n such that lim n → ∞ γ ˙ y + ε n ξ ( 0 ) = v ≠ γ ˙ y ( 0 ) . Since ( γ y + ε n ξ ) is a sequence of geodesics, it converges with respect to the C 1 norm to a minimum. Since the minimum is unique by Proposition 4.8, then γ y + ε n ξ → γ y in C 1 , so γ ˙ y + ε n ξ ( 0 ) → γ ˙ y ( 0 ) , which is a contradiction.

Therefore, (5.10) holds, and using also (5.9), we have

(5.11) lim ε → 0 d J ( γ y + ε ξ ) [ v ξ ] = − ∂ G ∂ v ( y , γ ˙ y ( 0 ) ) [ ξ ] .

Finally, combining (5.4), (5.8), and (5.11), we obtain (5.3), and we are done.□

Lemma 5.2 will play a central role because it links the initial velocity of the curve γ y with q ˙ ( t y , Q y ) through a function of class C 1 .

Lemma 5.2

There exists a function φ : D δ ¯ × R n → R of class C 1 such that

(5.12) ψ ( y ) φ ( y , q ˙ ( t y , Q y ) ) q ˙ ( t y , Q y ) = − γ ˙ y ( 0 ) .

Proof

Let ζ y : [ 0 , ψ ( y ) ] → D ¯ the backward arc-length reparametrization of γ y , namely,

ζ y ( s ) = γ y 1 − s ψ ( y ) .

As a consequence,

(5.13) ζ ˙ y ( ψ ( y ) ) = − 1 ψ ( y ) γ ˙ y ( 0 ) .

By Lemma 2.4, the curve x : [ 0 , ψ ( y ) ] → R 2 n given by

x ( s ) = ℒ − 1 ( ζ y ( s ) , ζ ˙ y ( s ) ) ,

is a solution of Hamilton’s equations with respect to U and U ( x ( s ) ) ≡ 1 . Since y ∈ D δ ¯ , By Lemma 4.6 and Remark 4.9, x ( s ) is actually a reparametrization of z ( t , Q y ) = ( q ( t , Q y ) , p ( t , Q y ) ) , with x ( ψ ( y ) ) = z ( t y , Q y ) . Hence, using (2.1) and recalling that ℒ is the identity map with respect to the first variable, we have

(5.14) ζ ˙ y ( ψ ( y ) ) = ‖ U ′ ( z ( t y , Q y ) ) ‖ ‖ H ′ ( z ( t y , Q y ) ) ‖ q ˙ ( t y , Q y ) = ‖ U ′ ( y , p ( t y , Q y ) ) ‖ ‖ H ′ ( y , p ( t y , Q y ) ) ‖ q ˙ ( t y , Q y ) .

Since the map

p ( t y , Q y ) ↦ q ˙ ( t y , Q y ) = ∂ H ∂ p ( y , p ( t y , Q y ) )

is invertible, there exists φ : D δ ¯ × R n → R such that

(5.15) φ ( y , q ˙ ( t y , Q y ) ) = ‖ U ′ ( y , p ( t y , Q y ) ) ‖ ‖ H ′ ( y , p ( t y , Q y ) ) ‖ .

Combining (5.13)–(5.15), we obtain (5.12). Recalling that both t y and Q y are of class C 1 by Remark 4.7, and that q ˙ ( t y , Q y ) ≠ 0 for every y ∈ D δ ¯ , the function φ is of class C 1 as a composition of the derivatives of H and U ,□

Lemma 5.3

The function ψ is of class C 2 in D δ ¯ .

Proof

By (5.12), we deduce that γ ˙ y ( 0 ) is continuous as a function of y ∈ D δ ¯ . Hence, by (5.1), ψ is of class C 1 in D δ ¯ . Using again (5.12) and the C 1 -regularity of φ and q ˙ ( t y , Q y ) , we deduce that γ ˙ y ( 0 ) is of class C 1 . By using (5.1), we obtain the thesis.□

Recalling the notion of strong concavity given in Definition 1.8 and the definition of H ψ ( y , v ) [ v , v ] in (1.4), the next proposition shows that the set ψ ( [ δ ˆ , ∞ [ ) is strongly concave, provided δ ˆ > 0 sufficiently small.

Proposition 5.4

There exists δ ˆ ∈ ] 0 , δ ¯ ] such that for every y ∈ D with 0 < ψ ( y ) ≤ δ ˆ , we have

H ψ ( y , ξ ) [ ξ , ξ ] > 0 , ∀ ξ ∈ R n \ { 0 } : d ψ ( y ) [ ξ ] = 0 .

Proof

For every ξ ∈ R n \ { 0 } such that d ψ ( y ) [ ξ ] = 0 , we denote by η the unique Finsler geodesic such that η ( 0 ) = y and η ˙ ( 0 ) = ξ . We have to prove that, for y sufficiently near the boundary ∂ D ,

d 2 d s 2 ( ψ ∘ η ) ( 0 ) > 0 .

Let ζ be the reparametrization of η , which is a solution of (1.1) with energy E . By Remark 2.3, there exists a function λ of class C 2 such that ζ ( t ) = η ( λ ( t ) ) , λ ( 0 ) = 0 , and λ ˙ ( t ) > 0 . Hence,

d 2 d s 2 ( ψ ∘ ζ ) ( 0 ) = λ ˙ ( 0 ) d 2 d s 2 ( ψ ∘ η ) ( 0 ) + λ ¨ ( 0 ) d ψ ( y ) [ ξ ] = λ ˙ ( 0 ) d 2 d s 2 ( ψ ∘ η ) ( 0 ) .

As a consequence, it suffices to prove that

d 2 d s 2 ( ψ ∘ ζ ) ( 0 ) > 0 .

By using (5.1) and (5.12), we obtain

d d s ψ ( ζ ( s ) ) = d ψ ( ζ ( s ) ) [ ζ ˙ ( s ) ] = − ∂ G ∂ v ( ζ ( s ) , γ ˙ ζ ( s ) ( 0 ) ) [ ζ ˙ ( s ) ] = ψ ( ζ ( s ) ) φ ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) ∂ G ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] .

Let us set w = ζ ˙ ( 0 ) . Since q ˙ ( t y , Q y ) is parallel to γ ˙ y ( 0 ) , we have

∂ G ∂ v ( y , q ˙ ( t y , Q y ) ) [ w ] = 0 ,

and thus, we obtain

d 2 d s 2 ( ψ ∘ ζ ) ( 0 ) = d d s ( ψ ( ζ ( s ) ) φ ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) ) s = 0 ∂ G ∂ v ( y , q ˙ ( t y , Q y ) ) [ w ] + ψ ( y ) φ ( y , q ˙ ( t y , Q y ) ) d d s ∂ G ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] s = 0 = ψ ( y ) φ ( y , q ˙ ( t y , Q y ) ) d d s ∂ G ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] s = 0 .

Since ψ ( y ) and φ ( y , q ˙ ( t y , Q y ) ) are two strictly positive functions with respect to y ∈ D , it remains to prove that, for y sufficiently near the boundary ∂ D , we have

(5.16) d d s ∂ G ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] s = 0 > 0 .

Let us define the function Γ : D × R n → R as follows:

Γ ( q , v ) = G ( q , v ) E − V ( q ) .

Since d ψ ( y ) [ w ] = 0 , we have

∂ Γ ∂ v ( y , q ˙ ( t y , Q y ) ) [ w ] = 1 E − V ( y ) ∂ G ∂ v ( y , q ˙ ( t y , Q y ) ) [ w ] = 0 .

As a consequence,

d d s ∂ G ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] s = 0 = ( E − V ( y ) ) d d s ∂ Γ ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] s = 0 .

Hence, to obtain (5.16), it suffices to prove that

(5.17) d d s ∂ Γ ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] s = 0 > 0 ,

for y sufficiently near the boundary ∂ D . Setting

I 1 ( y ) = d d s ∂ Γ ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) s = 0 [ w ] , I 2 ( y ) = ∂ Γ ∂ v ( y , q ˙ ( t y , Q y ) ) [ ζ ¨ ( 0 ) ] ,

we have

d d s ∂ Γ ∂ v ( ζ ( s ) , q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) [ ζ ˙ ( s ) ] s = 0 = I 1 ( y ) + I 2 ( y ) .

Let us study I 1 ( y ) and I 2 ( y ) separately. We have

I 1 ( y ) = ∂ 2 Γ ∂ q ∂ v ( y , q ˙ ( t y , Q y ) ) [ w , w ] + ∂ 2 Γ ∂ v 2 ( y , q ˙ ( t y , Q y ) ) d d s ( q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) s = 0 , w .

Since q ( t , Q y ) is a solution of (1.1) with energy E , by (3.2), we have

ν 1 ‖ p ( t y , Q y ) ‖ ≤ ∂ K ∂ p ( y , p ( t y , Q y ) ) = ‖ q ˙ ( t y , Q y ) ‖ ≤ ν 2 ‖ p ( t y , Q y ) ‖ ,

and by (3.3), we obtain that there exist two constants c 1 , c 2 > 0 such that

(5.18) c 1 ( E − V ( y ) ) ≤ ‖ q ˙ ( t y , Q y ) ‖ 2 ≤ c 2 ( E − V ( y ) ) .

Similarly, since w = ζ ˙ ( 0 ) , we have

(5.19) c 1 ( E − V ( y ) ) ≤ ‖ w ‖ 2 ≤ c 2 ( E − V ( y ) ) .

Since Γ is homogeneous of degree two with respect to v and recalling the bounds for G given by (3.5), there exists a constant c 3 such that

∂ 2 Γ ∂ q ∂ v ( q , v ) [ ω , ω ] ≤ c 3 ‖ v ‖ ‖ ω ‖ 2 , ∀ ( q , v ) ∈ D × R n .

As a consequence, using (5.18) and (5.19), we have

(5.20) ∂ 2 Γ ∂ q ∂ v ( y , q ˙ ( t y , Q y ) ) [ w , w ] ≤ c 3 ( E − V ( y ) ) 3 2 .

Let us set

v y ≔ ∂ 2 H ∂ p 2 ( y , 0 ) ∂ V ∂ q ( y ) .

We remark that, since ( ∂ V / ∂ q ) ( q ) ≠ 0 for every q ∈ ∂ D , and by the strictly convexity of H given by (3.1), ‖ v y ‖ is uniformly greater than zero if y is sufficiently near the boundary. Since lim s → 0 Q ζ ( s ) = Q y , using (3.14), we have

q ˙ ( t , Q ζ ( s ) ) = − t v y + ρ ( t , Q ζ ( s ) ) , ∀ t ∈ [ 0 , + ∞ [ ,

with d ρ ( 0 , Q y ) = 0 . By Remark 4.7, t y and Q y are functions of class C 1 with respect to y . Moreover, by Remark 3.6, q ˙ ( t , Q ) is of class C 1 . Hence, we obtain

(5.21) d d s ( q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) ∣ s = 0 = − d t y [ w ] v y + ∂ ρ ∂ t ( t y , Q y ) d t y [ w ] + ∂ ρ ∂ Q ( t y , Q y ) ∂ Q ∂ y [ w ] .

Since q ( t y , Q y ) = y , we get

(5.22) d t y [ v ] q ˙ ( t y , Q y ) + ∂ q ∂ Q ( t y , Q y ) ∂ Q y ∂ y [ v ] = v , ∀ v ∈ R n .

We recall that ( t , Q ) is a coordinate system in a neighborhood of ∂ D , where y = q ( t y , Q y ) . Hence, if y tends to ∂ D , then t y → 0 and ( ∂ q / ∂ Q ) ( t y , Q y ) goes to the identity map. Similarly, when y → ∂ D , ( ∂ Q y / ∂ y ) [ v ] tends to v uniformly as ‖ v ‖ ≤ 1 . Then, by (5.22), d t y [ v ] q ˙ ( t y , Q y ) → 0 uniformly in v , as y → ∂ D . Therefore, since by (5.18) and (5.19) w and q ˙ ( t y , Q y ) , we have

0 < c 1 c 2 ≤ ‖ q ˙ ( t y , Q y ) ‖ ‖ w ‖ c 2 c 1 ,

and we obtain

(5.23) lim y → ∂ D d t y [ w ] = 0 .

Since d ρ ( 0 , Q y ) = 0 , by (5.21) and (5.23), we infer

(5.24) d d s ( q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) s = 0 = o ( 1 ) ,

as y → ∂ D . Since Γ is homogeneous of degree 2 with respect to v and using (3.5), there exist a constant c 3 > 0 such that

∂ 2 Γ ∂ v 2 ( q , ξ ) [ v 1 , v 2 ] ≤ c 4 ‖ v 1 ‖ ‖ v 2 ‖ , ∀ q ∈ D , ∀ ξ ∈ R n \ { 0 } , ∀ v 1 , v 2 ∈ R n .

As a consequence, by (5.19) and (5.24), we obtain

(5.25) lim y → ∂ D ∂ 2 Γ ∂ v 2 ( y , q ˙ ( t y , Q y ) ) d d s ( q ˙ ( t ζ ( s ) , Q ζ ( s ) ) ) ∣ s = 0 , w = o ( E − V ( y ) ) .

By (5.20) and (5.25), we obtain

(5.26) I 1 ( y ) = o ( E − V ( y ) ) ,

as y → ∂ D .

Let us analyze I 2 ( y ) . Since ( ζ , p ) is a solution of Hamilton’s equations, with p implicitly defined by

ζ ˙ ( s ) = ∂ H ∂ p ( ζ ( s ) , p ( s ) ) ,

we have

ζ ¨ ( 0 ) = ∂ 2 H ∂ q ∂ p ( y , p ( 0 ) ) w − ∂ 2 H ∂ p 2 ( y , p ( 0 ) ) ∂ K ∂ q ( y , p ( 0 ) ) + ∂ V ∂ q ( y ) .

If y → ∂ D , then ‖ p ( 0 ) ‖ → 0 . Therefore,

∂ 2 H ∂ q ∂ p ( y , p ( 0 ) ) w → 0 , ∂ K ∂ q ( y , p ( 0 ) ) → 0 ,

and we obtain lim y → ∂ D ζ ¨ ( 0 ) = − v y . Hence, using also (3.5) and (5.21), we obtain

(5.27) lim y → ∂ D ∂ Γ ∂ v y , q ˙ ( t y , Q y ) ‖ q ˙ ( t y , Q y ) ‖ [ ζ ¨ ( 0 ) ] = ∂ Γ ∂ v ( y , − v y ) [ − v y ] = Γ ( y , v y ) ≥ 1 2 ν 2 ‖ v y ‖ 2 .

As a consequence, by (5.18) and (5.27), we obtain

(5.28) lim y → ∂ D I 2 ( y ) E − V ( y ) > 0 .

Finally, by (5.26) and (5.28), we obtain (5.17), and we are done.□

6 Proof of the main theorem

Finally, we are ready to prove Theorem 1.9.

Proof of Theorem 1.9

Let δ ˆ be as in Proposition 5.4 and set

Ω = ψ − 1 ( ] δ ˆ , + ∞ [ ) .

By continuity, Ω is an open subset of D and ∂ Ω = ψ − 1 ( δ ˆ ) . By Lemma 5.3, ψ is of class C 2 in D δ ¯ , and since d ψ ≠ 0 on ∂ Ω , we have that ∂ Ω is of class C 2 . Since δ ¯ satisfies property (4.8) and δ ˆ ≤ δ ¯ , Ω ¯ is homeomorphic to D ¯ . Since ∂ Ω is a level hyper-surface of ψ , for every y ∈ ∂ Ω , v ∈ T y ∂ Ω if and only if d ψ ( y ) [ v ] = 0 . Recalling Definition 1.8, Proposition 5.4 implies that Ω ¯ is strongly concave with respect to the Finsler metric F .

Let γ : [ 0 , 1 ] → Ω ¯ be an orthogonal Finsler geodesic chord. We will prove the desired properties of the extension γ ˆ : [ α , β ] → D ¯ only in the interval [ 1 , β ] . The case [ α , 0 ] is analog. Set y = γ ( 1 ) . Since γ is an orthogonal Finsler geodesic chord, it satisfies (1.3); hence,

∂ G ∂ v ( y , γ ˙ ( 1 ) ) [ v ] = 0 , ∀ v ∈ T γ ( 1 ) ∂ Ω .

The minimizer curve γ y satisfies

∂ G ∂ v ( y , γ ˙ y ( 0 ) ) [ v ] = 0 , ∀ v ∈ T γ ( 1 ) ∂ Ω ,

and thus, γ ˙ ( 1 ) and γ ˙ y ( 0 ) are parallel. As a consequence, the curve γ ¯ : [ 0 , 2 ] → D ¯ defined as

γ ¯ ( s ) = γ ( s ) , if s ∈ [ 0 , 1 ] , γ y ( s − 1 ) , if s ∈ ] 1 , 2 ] ,

is of class C 1 and it is a geodesic with respect to F , up to a suitable time reparametrization. With the analog extension in [ α , 0 ] , we obtain a geodesic γ ˆ : [ α , β ] → D ¯ such that γ ˆ ( α ) , γ ˆ ( β ) ∈ ∂ D and γ ˆ ( ] α , β [ ) ⊂ D . By Lemmas 2.2 and 2.4, we have that

( q ( t ) , p ( t ) ) = ℒ − 1 ( γ ˆ ( t ) , γ ˆ ˙ ( t ) ) ∀ t ∈ ] α , β [

is a solution of (1.1) with energy E for H , up to time reparametrization. By using also Lemma 4.4 to ensure that the time reparametrization is finite, we obtain the existence of a diffeomorphism σ : [ 0 , T ] → [ α , β ] , with σ ( 0 ) = α and σ ( T ) = β , such that

( q , p ) ∘ σ : [ 0 , T ] → Σ ¯

is a brake orbit.□

Conflict of interest: The authors state no conflict of interest.

References

[1] H. Seifert, Periodische bewegungen mechanischer systeme, Math. Z. 51 (1948), 197–216. 10.1007/BF01291002Suche in Google Scholar

[2] A. Ambrosetti, V. Benci, and Y. Long, A note on the existence of multiple brake orbits, Nonlinear Anal. 21 (1993), no. 9, 643–649. 10.1016/0362-546X(93)90061-VSuche in Google Scholar

[3] X. Hu, L. Wu, and R. Yang, Morse index theorem of Lagrangian systems and stability of brake orbit, J. Dynam. Differ. Equ. 32 (2020), no. 1, 61–84. 10.1007/s10884-018-9711-xSuche in Google Scholar

[4] C. Liu, Y. Long, and D. Zhang, Index iteration theory for brake orbit type solutions and applications, Anal. Theory Appl. 37 (2021), no. 2, 129–156. 10.4208/ata.2021.pr80.05Suche in Google Scholar

[5] C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math. 67 (2014), 1563–1604. 10.1002/cpa.21525Suche in Google Scholar

[6] Y. Long, D. Zhang, and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv. Math. 203 (2006), no. 2, 568–635. 10.1016/j.aim.2005.05.005Suche in Google Scholar

[7] P. H. Rabinowitz, Critical point theory and applications to differential equations: a survey, In: M. Matzeu and A. Vignoli, editors, Topological Nonlinear Analysis, volume 15, chapter Critical Point Theory and Applications to Differential Equations: A Survey, Birkhäuser Boston, Boston, 1995, pp. 464–513. 10.1007/978-1-4612-2570-6_6Suche in Google Scholar

[8] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann. 283 (1989), no. 2, 241–255. 10.1007/BF01446433Suche in Google Scholar

[9] F. Wang and D. Zhang, Multiple brake orbits of even Hamiltonian systems on torus, Sci. China Math. 63 (2020), no. 7, 1429–1440. 10.1007/s11425-018-9456-9Suche in Google Scholar

[10] D. Zhang and C. Liu, Multiple brake orbits on compact convex symmetric reversible hypersurfaces in R2n, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, 531–554. 10.1016/j.anihpc.2013.03.010Suche in Google Scholar

[11] R. Giambò, F. Giannoni, P. Piccione, Multiple orthogonal geodesic chords and a proof of Seifert’s conjecture on brake orbits, 2021, http://arxiv.org/abs/2002.09687v2. Suche in Google Scholar

[12] R. Giambò, F. Giannoni, and P. Piccione, On the multiplicity of orthogonal geodesies in riemannian manifold with concave boundary. Applications to brake orbits and homoclinics, Adv. Nonlinear Stud. 9 (2009), no. 4, 763–782. 10.1515/ans-2009-0409Suche in Google Scholar

[13] R. Giambò, F. Giannoni, and P. Piccione, Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk, Nonlinear Anal. 72 (2010), no. 2, 290–337. 10.1016/j.na.2010.03.019Suche in Google Scholar

[14] R. Giambò, F. Giannoni, and P. Piccione, Multiple brake orbits in m-dimensional disks, Calc. Var. Partial Differ. Equ. 54 (2015), 2553–2580. 10.1007/s00526-015-0875-5Suche in Google Scholar

[15] R. Giambò, F. Giannoni, and P. Piccione, On the normal exponential map in singular conformal metrics, Nonlinear Anal. 127 (2015), 35–44. 10.1016/j.na.2015.06.016Suche in Google Scholar

[16] R. Giambò, F. Giannoni, and P. Piccione, Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles, Calc. Var. Partial Differ. Equ. 57 (2018), no. 117. 10.1007/s00526-018-1394-ySuche in Google Scholar

[17] R. Giambò, F. Giannoni, and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in riemannian manifolds, Adv. Differ. Equ. 10 (2004), 1–24. 10.1080/1023619031000110912Suche in Google Scholar

[18] O. R. Ruiz, Existence of brake orbits in Finsler mechanical systems, In: J. Palis and M. doCarmo, editors, Geometry and Topology, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1977, pp. 542–567. 10.1007/BFb0085375Suche in Google Scholar

[19] A. Weinstein, Periodic orbits for convex hamiltonian systems, Ann. of Math. 108 (1978), no. 3, 507–518. 10.2307/1971185Suche in Google Scholar

[20] H. Rund, The Differential Geometry of Finsler Spaces, Springer-Verlag, Berlin, 1959. 10.1007/978-3-642-51610-8Suche in Google Scholar

[21] Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing, Singapore, 2001. 10.1142/4619Suche in Google Scholar

[22] R. Bartolo, E. Caponio, A. V. Germinario, and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calculus Variat. 40 (2011), 335–356. 10.1007/s00526-010-0343-1Suche in Google Scholar

[23] D. Corona, A multiplicity result for Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Fixed Point Theory Appl. 22 (2020), no. 60. 10.1007/s11784-020-00795-4Suche in Google Scholar

[24] D. Corona, A multiplicity result for orthogonal geodesic chords in Finsler disks, Discrete Contin. Dyn. Syst. 41 (2021), no. 11, 5329–5357. 10.3934/dcds.2021079Suche in Google Scholar

[25] D. Corona and F. Giannoni, A new approach for Euler-Lagrange orbits on compact manifolds with boundary, Symmetry 12 (2020), no. 11, 1917. 10.3390/sym12111917Suche in Google Scholar

Received: 2021-06-11

Revised: 2021-10-13

Accepted: 2022-01-04

Published Online: 2022-03-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2022-0222

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Hamiltonian systems; brake orbits; Finsler metric; variational methods

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