Non-local Lagrangian Mechanics: Noether theorem and Hamiltonian formalism

TL;DR

This paper extends classical mechanics frameworks to non-local in time Lagrangian systems, deriving a Noether theorem and Hamiltonian formalism, and applies these to examples including non-local oscillators and the p-adic particle.

Contribution

It introduces a novel extension of Noether's theorem and Hamiltonian formalism for non-local time-dependent Lagrangian systems, unifying local and non-local cases.

Findings

Extended Noether theorem for non-local Lagrangians

Established Hamiltonian formalism for non-local systems

Applied framework to non-local oscillators and p-adic particle

Abstract

We study Lagrangian systems with a finite number of degrees of freedom that are non-local in time. We obtain an extension of Noether theorem and Noether identities to this kind of Lagrangians. A Hamiltonian formalism is then set up for this systems. $n$-order local Lagrangians can be treated as a particular case and the standard results for them are recovered. The method is then applied to several other cases, namely two examples of non-local oscillators and the p-adic particle.