Finite reduction and Morse index estimates for mechanical systems

TL;DR

This paper introduces a straightforward finite-dimensional reduction method for mechanical systems using a global implicit function theorem, ensuring the reduced Hessian retains key information and extending results to Dirichlet problems.

Contribution

It provides a novel, simple reduction technique for variational mechanical systems with a thorough theoretical foundation and extends to boundary value problems.

Findings

Finite reduction preserves the Hessian's essential information.

Method applies to Dirichlet problems on bounded domains.

The approach is based on a global implicit function theorem.

Abstract

A simple version of exact finite dimensional reduction for the variational setting of mechanical systems is presented. It is worked out by means of a thorough global version of the implicit function theorem for monotone operators. Moreover, the Hessian of the reduced function preserves all the relevant information of the original one, by Schur's complement, which spontaneously appears in this context. Finally, the results are straightforwardly extended to the case of a Dirichlet problem on a bounded domain.