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Periodic orbits of reversible Lagrangian systems without self-intersections and Mañé genericity

Hans-Bert Rademacher

Abstract

Bernard [3] showed that a Mañé generic convex Hamiltonian has only non-degenerate periodic orbits on a given energy level. We show that one can use this result to prove that for a generic potential the prime periodic orbits of fixed energy of a Lagrangian system of classical type on a compact manifold of dimension $n\ge 3$ do not have self-intersections and do not intersect each other.

Periodic orbits of reversible Lagrangian systems without self-intersections and Mañé genericity

Abstract

Bernard [3] showed that a Mañé generic convex Hamiltonian has only non-degenerate periodic orbits on a given energy level. We show that one can use this result to prove that for a generic potential the prime periodic orbits of fixed energy of a Lagrangian system of classical type on a compact manifold of dimension do not have self-intersections and do not intersect each other.
Paper Structure (6 sections, 5 theorems, 58 equations)

This paper contains 6 sections, 5 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Finsler metrics, Lagrangian and Hamiltonian systems of classical type
  3. Perturbing a single segment of an orbit
  4. Intersection of orbits of Lagrangian systems
  5. Proof of Theorem \ref{['thm:main']}
  6. Proof of Corollary \ref{['cor:periodic']}

Key Result

Theorem 1

Let $F$ be a reversible Finsler metric on the compact smooth manifold $M$ of dimension $n=\dim M\ge 3$ defining the kinetic energy $T=F^2/2$ and let $E \in \mathbb{R}.$ Then for a generic function $U \in C^{\infty}(M)$ the number $E$ is a regular value of the function $U$ as well as of the total ene

Theorems & Definitions (12)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1: Jacobi-Finsler metric
  • Remark 4
  • Example 1
  • Lemma 1: Perturbation Lemma
  • Remark 5
  • ...and 2 more