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Action-minimizing periodic orbits of the Lorentz force equation with dominant vector potential

Manuel Garzón, Salvador López-Martínez

Abstract

We establish the existence of non-constant periodic solutions to the Lorentz force equation, where no scalar potential is needed to induce the electromagnetic field. Our results extend to cases where a possibly singular scalar potential is present, although the vector potential assumes a leading role. The approach is based on minimizing the action functional associated with the relativistic Lagrangian. The compactness of the minimizing sequences requires the existence of negative values for the functional, which is proven using novel ideas that exploit the sign-indefinite nature of the term involving the vector potential.

Action-minimizing periodic orbits of the Lorentz force equation with dominant vector potential

Abstract

We establish the existence of non-constant periodic solutions to the Lorentz force equation, where no scalar potential is needed to induce the electromagnetic field. Our results extend to cases where a possibly singular scalar potential is present, although the vector potential assumes a leading role. The approach is based on minimizing the action functional associated with the relativistic Lagrangian. The compactness of the minimizing sequences requires the existence of negative values for the functional, which is proven using novel ideas that exploit the sign-indefinite nature of the term involving the vector potential.

Paper Structure

This paper contains 10 sections, 14 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Hypotheses and main results
  3. Minimization of the action functional
  4. Functional framework
  5. Semicontinuity property of the action functional
  6. Compactness of minimizing sequences
  7. Existence of minimizer when Phi = 0
  8. Extensions to with dominant vector potential
  9. Conclusive remarks
  10. Appendix

Key Result

Theorem 2.1

Let $A\in\mathcal{C}^1\left(\mathbb{R}^4;\mathbb{R}^3\right)$ be non-autonomous, $T$-periodic in its first variable, and with uniformly asymptotic decay at infinity, i.e. and Then the Lorentz force equation eq:LFE0 possesses at least one non-constant periodic solution ${\bf q}$. Moreover, for some $\rho\in(0,1)$ independent of ${\bf q}$.

Theorems & Definitions (33)

  • Theorem 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Definition 3.1
  • ...and 23 more