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Bifurcations for Lagrangian systems and geodesics I

Guangcun Lu

Abstract

This paper is Part I of a two-part series. We investigate bifurcation phenomena in Lagrangian systems with various boundary conditions and constraints, focusing on the interplay between Morse theory and the existence of multiple solutions through three principal configurations: Lagrangian trajectories connecting two submanifolds or with endpoints related by an isometry, and brake orbits in Lagrangian systems. For each configuration, we establish necessary and sufficient conditions for bifurcation using Morse index and nullity techniques, including classification of Rabinowitz-type alternative bifurcation scenarios. For Euler-Lagrange curves emanating perpendicularly from a submanifold, we develop a unified Morse-theoretic framework that rigorously connects geometric focal structure (e.g., conjugate points) and analytic bifurcation behavior (e.g., solution branching patterns).

Bifurcations for Lagrangian systems and geodesics I

Abstract

This paper is Part I of a two-part series. We investigate bifurcation phenomena in Lagrangian systems with various boundary conditions and constraints, focusing on the interplay between Morse theory and the existence of multiple solutions through three principal configurations: Lagrangian trajectories connecting two submanifolds or with endpoints related by an isometry, and brake orbits in Lagrangian systems. For each configuration, we establish necessary and sufficient conditions for bifurcation using Morse index and nullity techniques, including classification of Rabinowitz-type alternative bifurcation scenarios. For Euler-Lagrange curves emanating perpendicularly from a submanifold, we develop a unified Morse-theoretic framework that rigorously connects geometric focal structure (e.g., conjugate points) and analytic bifurcation behavior (e.g., solution branching patterns).
Paper Structure (30 sections, 46 theorems, 326 equations)

This paper contains 30 sections, 46 theorems, 326 equations.

Table of Contents

  1. Introduction and overview of the series
  2. Background and motivation
  3. Main objectives and structure of the series
  4. Relation to existing literature
  5. Basic assumptions, conventions, and statement of the research problems
  6. Bifurcations of Lagrangian system trajectories connecting two submanifolds
  7. Bifurcations of Lagrangian trajectories with endpoints constrained in ${\rm Graph}(\mathbb{I}_{g})$
  8. Bifurcations of generalized periodic solutions in non-autonomous Lagrangian systems
  9. Bifurcations of brake orbits in Lagrangian systems
  10. Organization of the article
  11. Preliminaries and notation
  12. Notation and Conventions
  13. Technical lemmas
  14. Proofs of Theorems \ref{['th:bif-nessLagrGener']}, \ref{['th:bif-existLagrGener']}, \ref{['th:bif-suffLagrGener']} and \ref{['th:MorseBif']}
  15. Proofs of Theorems \ref{['th:bif-nessLagrGener']}, \ref{['th:bif-existLagrGener']}, \ref{['th:bif-suffLagrGener']}
  16. ...and 15 more sections

Key Result

Theorem 1.4

Let Assumptions ass:Lagr6, ass:LagrGenerB be satisfied, and $\mu\in\Lambda$ be such that This assumption is to guarantee the existence of a Riemannian metric $g$ on $M$ such that $S_0$ (resp. $S_1$) is totally geodesic near $\gamma_\mu(0)$ (resp. $\gamma_\mu(\tau)$) when we reduce the problems to Eu

Theorems & Definitions (80)

  • 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
  • Theorem 1.13
  • Theorem 1.14: Existence for bifurcations
  • ...and 70 more