A Canonical Formalism For Lagrangians With Nonlocality Of Finite Extent

TL;DR

This paper develops a canonical formalism for finite-extent nonlocal Lagrangians by relating them to higher derivative theories, revealing their inherent instabilities through a specific example.

Contribution

It introduces a new canonical formalism for nonlocal Lagrangians with finite extent, connecting them to higher derivative theories and analyzing their stability.

Findings

Formalism derived for nonlocal Lagrangians with finite extent

Explicit formula for conjugate momenta provided

Demonstrates the inherent instability of such nonlocal systems

Abstract

I consider Lagrangians which depend nonlocally in time but in such a way that there is no mixing between times differing by more than some finite value $\Delta t$. By considering these systems as the limits of ever higher derivative theories I obtain a canonical formalism in which the coordinates are the dynamical variable from $t$ to $t + {\Delta t}$. A simple formula for the conjugate momenta is derived in the same way. This formalism makes apparent the virulent instability of this entire class of nonlocal Lagrangians. As an example, the formalism is applied to a nonlocal analog of the harmonic oscillator.