State dependent Hamiltonian delay equations and Neumann one-forms

TL;DR

This paper investigates critical points of a modified action functional in classical mechanics with retarded Hamiltonian terms, using Taylor approximations and perturbation theory to prove Arnold-type conjectures at first order in the fine structure constant.

Contribution

It introduces a novel analysis of Hamiltonian delay equations with retarded terms and establishes Arnold-type conjectures using first-order perturbations in the fine structure constant.

Findings

Relation of the functional to time-dependent symplectic perturbations

Proof of Arnold-type conjectures at first order in the fine structure constant

Development of Taylor approximations for delay Hamiltonian systems

Abstract

In this note we study critical points of a variation of the action functional of classical mechanics, where the Hamiltonian term is retarded. Following a more than hundert and fifty year old paper by Carl Neumann we as well introduce Taylor approximations to this functional in terms of the fine structure constant. We see how in first order in the fine structure constant this functionals are related to time dependent perturbations of the symplectic form. This observation allows us to prove Arnold type conjectures in first order in the fine structure constant.