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

TL;DR

This paper analyzes the minimality and stability of periodic brake orbits in natural Lagrangian systems on Riemannian manifolds, showing they are generally non-minimizing and often spectrally unstable, with explicit examples provided.

Contribution

It establishes that non-constant periodic brake orbits are not minimizers and explores their Morse index and stability properties under various conditions.

Findings

Non-constant brake orbits are not action minimizers.

Strongly nondegenerate brake orbits are spectrally unstable in dimensions ≥3.

Explicit Morse index calculations for classical problems like Kepler and pendulum.

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.