Functional Lagrange formalism for time-non-local Lagrangians

TL;DR

This paper introduces a variational calculus-based formalism for analyzing time-non-local Lagrangians, deriving generalized equations of motion and Hamiltonian formalism, with applications to second order cases and insights into non-local dynamics.

Contribution

It develops a novel Lagrangian and Hamiltonian formalism for time-non-local systems, extending classical mechanics tools to non-local temporal interactions.

Findings

Reproduces standard results in the time-local limit

Derives generalized Euler-Lagrange equations for TNL Lagrangians

Provides an example illustrating the formalism's application

Abstract

We develop a time-non-local (TNL) formalism based on variational calculus, which allows for the analysis of TNL Lagrangians. We derive the generalized Euler-Lagrange equations starting from the Hamilton's principle and, by defining a generalized momentum, we introduce the corresponding Hamiltonian formalism. We apply the formalism to second order TNL Lagrangians and we show that it reproduces standard results in the time-local limit. An example will show how the formalism works, and will provide an interesting insight on the non-standard features of TNL equations.