State dependent Hamiltonian delay equations and Neumann one-forms

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.