Bifurcations for Lagrangian systems and geodesics II

TL;DR

This paper advances the understanding of bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on manifolds, providing new necessary and sufficient conditions and refining classical theorems with rigorous examples.

Contribution

It extends bifurcation analysis to generalized periodic solutions and geodesics, refining Morse theory results and developing novel bifurcation theorems for these systems.

Findings

Derived necessary and sufficient bifurcation conditions for Lagrangian systems.

Refined the Morse-Littauer theorem for geodesic behavior.

Confirmed sharpness of results with explicit counterexamples.

Abstract

This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in earlier work and the results of Part I, this study makes contributions in two main directions. In Part A, we focus on bifurcations of generalized periodic solutions in autonomous Lagrangian systems. By employing Morse index and nullity techniques within the normal space to the $\mathbb{R}$-orbits of solutions, we derive necessary and sufficient conditions for bifurcation, encompassing scenarios of both Fadell--Rabinowitz and Rabinowitz type. In Part B, we extend these results to the geometric setting of geodesic bifurcations in Finsler and Riemannian manifolds. A principal achievement is the significant refinement of the classical Morse-Littauer theorem, providing a precise description of geodesic behavior near critical points of the exponential map. The sharpness of these theoretical results is rigorously tested and confirmed through explicit counterexamples, such as the round sphere. The work is technically rigorous, leveraging a specialized technique developed by the author to establish novel bifurcation theorems. These findings have profound theoretical implications and potential applications in related fields such as the Zermelo navigation problem and the study of stationary spacetimes.