Morse index theorem for heteroclinic, homoclinic and halfclinic orbits of Lagrangian systems

TL;DR

This paper extends the Morse index theorem to a broader class of solutions in Lagrangian systems and applies it to classical mechanical models to compute their Morse indices.

Contribution

It introduces a more general Morse index theorem for heteroclinic, homoclinic, and half-clinic orbits in Lagrangian systems, with explicit index computations for specific models.

Findings

Morse index theorem is generalized for complex solutions.

Explicit Morse index calculations for classical models.

Applications to pendulum, Nagumo equation, and diffusion systems.

Abstract

The purpose of this paper is to prove a new, more general version of the Morse index theorem for heteroclinic, homoclinic, and half-clinic solutions in general Lagrangian systems. In the final section, we compute the Morse index for specific heteroclinic and half-clinic solutions in classical mechanical models such as the mathematical pendulum, the Nagumo equation, and a four-dimensional competition-diffusion system.