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Bifurcations for Lagrangian systems and geodesics

Guangcun Lu

TL;DR

This work generalizes abstract bifurcation theory to Lagrangian systems on manifolds, linking boundary-value and periodic problems to Morse-theoretic invariants. By reducing to Euclidean models and employing $S^1$-equivariant analyses, it derives comprehensive criteria for when families of Lagrangian trajectories bifurcate from trivial branches, including geodesic and brake-orbit settings on Finsler and Riemannian manifolds. The results yield Rabinowitz-type alternatives and explicit multiplicity statements, applicable to connecting submanifolds, generalized periodic solutions, autonomous regimes, and brake orbits. Overall, the paper provides a robust variational framework for detecting and characterizing bifurcations of geodesics and related Lagrangian trajectories under parameter variation, with clear geometric implications for geodesic structure on curved spaces. $m^0$-positive focal points and Morse index changes emerge as central drivers of bifurcation, with the methodology extendable to higher-order Lagrangians in future work.$

Abstract

In this paper we shall use the abstract bifurcation theorems developed by the author in previous papers to study bifurcations of solutions for Lagrangian systems on manifolds linearly or nonlinearly dependent on parameters under various boundary value conditions. As applications, many bifurcation results for geodesics on Finsler and Riemannian manifolds are derived.

Bifurcations for Lagrangian systems and geodesics

TL;DR

This work generalizes abstract bifurcation theory to Lagrangian systems on manifolds, linking boundary-value and periodic problems to Morse-theoretic invariants. By reducing to Euclidean models and employing -equivariant analyses, it derives comprehensive criteria for when families of Lagrangian trajectories bifurcate from trivial branches, including geodesic and brake-orbit settings on Finsler and Riemannian manifolds. The results yield Rabinowitz-type alternatives and explicit multiplicity statements, applicable to connecting submanifolds, generalized periodic solutions, autonomous regimes, and brake orbits. Overall, the paper provides a robust variational framework for detecting and characterizing bifurcations of geodesics and related Lagrangian trajectories under parameter variation, with clear geometric implications for geodesic structure on curved spaces. -positive focal points and Morse index changes emerge as central drivers of bifurcation, with the methodology extendable to higher-order Lagrangians in future work.$

Abstract

In this paper we shall use the abstract bifurcation theorems developed by the author in previous papers to study bifurcations of solutions for Lagrangian systems on manifolds linearly or nonlinearly dependent on parameters under various boundary value conditions. As applications, many bifurcation results for geodesics on Finsler and Riemannian manifolds are derived.
Paper Structure (39 sections, 82 theorems, 582 equations, 1 figure)

This paper contains 39 sections, 82 theorems, 582 equations, 1 figure.

Table of Contents

  1. Bifurcations of Lagrangian systems
  2. Statement of main results
  3. Bifurcations for Lagrangian trajectories connecting submanifolds
  4. Bifurcations for generalized periodic solutions of time dependent Lagrangian systems
  5. Bifurcations for generalized periodic solutions of autonomous Lagrangian systems
  6. Bifurcations of (\ref{['e:Lagr10*']}) starting at constant solutions
  7. Bifurcations of (\ref{['e:Lagr10*']}) starting at nonconstant solutions
  8. Bifurcations for brake orbits of Lagrangian systems
  9. Preparations and some technical lemmas
  10. Technical lemmas
  11. Notations and conventions
  12. Proofs of Theorems \ref{['th:bif-nessLagrGener']}, \ref{['th:bif-existLagrGener']}, \ref{['th:bif-suffLagrGener']} and \ref{['th:MorseBif']}
  13. Proofs of Theorems \ref{['th:bif-nessLagrGener']}, \ref{['th:bif-existLagrGener']}, \ref{['th:bif-suffLagrGener']}
  14. Reduction to Euclidean spaces
  15. Proofs of Theorems \ref{['th:bif-nessLagrGener']}, \ref{['th:bif-suffLagrGener']}
  16. ...and 24 more sections

Key Result

Theorem 1

The exponential map $\exp^F$ of a $C^\infty$ Finsler space $(M, F)$ is not locally injective near any critical point.

Figures (1)

  • Figure :

Theorems & Definitions (162)

  • Theorem 1
  • Theorem 2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5: Existence for bifurcations
  • Theorem 1.6: Alternative bifurcations of Rabinowitz's type
  • Definition 1.8
  • Theorem 1.9
  • Remark 1.10
  • Definition 1.12
  • ...and 152 more