Nonlocal Mechanics

TL;DR

This paper develops a Hamiltonian framework for nonlocal Lagrangian systems that avoids infinite-derivative expansions, using a variational principle and symplectic geometry to define phase space and energy.

Contribution

It introduces a novel Hamiltonian formalism for nonlocal mechanics based on a trajectory variational principle and symplectic structures, without relying on infinite derivatives.

Findings

Defined canonical momenta and energy for nonlocal systems

Constructed a (pre)symplectic form on the kinematic space

Demonstrated the approach with three example models

Abstract

We introduce a Hamiltonian framework for nonlocal Lagrangian systems without relying on infinite-derivative expansions. Starting from a (trajectory-based) variational principle and a generalized Noether theorem, we define the canonical momenta and energy. Moreover, we construct a (pre)symplectic form on the kinematic space, and show that its restriction to the phase space (by implementing the constraints) yields a true (pre)symplectic structure encoding the dynamics. Three examples -- a finite nonlocal oscillator, the fully nonlocal Pais-Uhlenbeck model, and a delayed harmonic oscillator -- demonstrate how phase space and the Hamiltonian emerge without explicitly solving the Euler-Lagrange equations.