Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory

TL;DR

This paper extends classical mechanics principles to systems with long-range interactions and memory, deriving fractional conservation laws and Hamiltonian equations using a fractional action approach.

Contribution

It introduces a fractional variational framework to derive conservation laws and Hamiltonian equations for systems with long-range interactions and memory effects.

Findings

Derived fractional conservation laws as continuity equations.

Formulated Hamiltonian-type equations from fractional action principles.

Applied results to coupled oscillators with power-law memory and interactions.

Abstract

Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether's theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time-space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form.