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Non-minimality and instability of brake orbits for natural Lagrangians on Riemannian manifolds

Luca Asselle, Xijun Hu, Alessandro Portaluri, Li Wu

Abstract

We investigate minimality and stability of periodic brake orbits in natural Lagrangian systems on smooth Riemannian manifolds. We prove that every non-constant periodic brake orbit is not a minimizer of the fixed-time action, for any conormal boundary condition. Under an orbit-cylinder hypothesis, its Morse index strictly increases in the free-time setting. As a consequence, strongly nondegenerate brake orbits fail to be linearly stable under a dimensional condition; in dimension at least three, nondegenerate mountain-pass brake orbits are spectrally unstable when the monodromy is semisimple. The key ingredient is a local index contribution at each brake instant. Using Seifert collar coordinates near the Hill boundary, we reduce the normal dynamics to a one-dimensional model, exhibiting a degeneracy inherent to brake symmetry. We illustrate the results by explicit Morse index computations for the planar anisotropic oscillator, the planar pendulum, and the planar Kepler problem; in the Kepler case, the ejection--collision orbit is treated via cotangent-lift Levi--Civita--Lissajous regularization.

Non-minimality and instability of brake orbits for natural Lagrangians on Riemannian manifolds

Abstract

We investigate minimality and stability of periodic brake orbits in natural Lagrangian systems on smooth Riemannian manifolds. We prove that every non-constant periodic brake orbit is not a minimizer of the fixed-time action, for any conormal boundary condition. Under an orbit-cylinder hypothesis, its Morse index strictly increases in the free-time setting. As a consequence, strongly nondegenerate brake orbits fail to be linearly stable under a dimensional condition; in dimension at least three, nondegenerate mountain-pass brake orbits are spectrally unstable when the monodromy is semisimple. The key ingredient is a local index contribution at each brake instant. Using Seifert collar coordinates near the Hill boundary, we reduce the normal dynamics to a one-dimensional model, exhibiting a degeneracy inherent to brake symmetry. We illustrate the results by explicit Morse index computations for the planar anisotropic oscillator, the planar pendulum, and the planar Kepler problem; in the Kepler case, the ejection--collision orbit is treated via cotangent-lift Levi--Civita--Lissajous regularization.
Paper Structure (15 sections, 21 theorems, 218 equations)

This paper contains 15 sections, 21 theorems, 218 equations.

Table of Contents

  1. Introduction, description of the problem and main results
  2. Description of the problem and main results
  3. The throwing--ball model in normalized units
  4. Time of flight and dependence on the energy
  5. Linearization, Morse and the Dirichlet Maslov index
  6. Dynamics in a collar neighborhood of the Hill boundary
  7. Geometric setup and Seifert collar coordinates
  8. Throwing--ball asymptotics
  9. Proofs of the main results
  10. Case 1: $\Lambda_D\cap \Phi_{0,0}(T/2)\Lambda_D=\{0\}$.
  11. Case 2: $\Lambda_D\cap \Phi_{0,0}(T/2)\Lambda_D\neq\{0\}$.
  12. Applications
  13. The (an)isotropic oscillator
  14. The planar pendulum: period and indices
  15. The planar Kepler problem

Key Result

Theorem 1

Let $(M,g)$ be a Riemannian manifold and let $L$ be the natural Lagrangian in eq:natural-L. Assume that $k$ is a regular value of $V$ and that $\partial\mathcal{H}_k\neq\emptyset$. Let $\gamma$ be a non-constant $T$-periodic brake orbit of energy $k$. Then and hence $\gamma$ cannot be a fixed-time minimizer of $\mathcal{A}^T$, regardless of the imposed conormal boundary conditions. Assume in addi

Theorems & Definitions (56)

  • Remark 1.1: Brake instants and Hill boundary
  • Definition 1.2: Fixed-time Morse indices
  • Definition 1.3: Free-time Morse index
  • Definition 1.4: Orbit cylinder
  • Definition 1.5: Non-degenerate and degenerate cylinders
  • Remark 1.6: Degenerate cylinders
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • ...and 46 more