Canonical formalism for Lagrangians with nonlocality of finite extent

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 $\Delta t.\mathrm{}$ 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 $t+\Delta t.\mathrm{}$ 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.