Fractional Noether's Theorem with Classical and Caputo Derivatives: constants of motion for non-conservative systems

TL;DR

This paper generalizes Noether's theorem to include systems with fractional and Caputo derivatives, enabling the derivation of constants of motion for non-conservative, dissipative systems using fractional calculus of variation.

Contribution

It introduces a fractional Noether's theorem for Lagrangians with classical and Caputo derivatives, extending conservation laws to non-conservative systems.

Findings

Derived a generalized Noether's theorem for mixed derivatives.

Established conditions for fractional optimal control problems.

Provided a framework for constants of motion in dissipative systems.

Abstract

Since the seminal work of Emmy Noether it is well know that all conservations laws in physics, \textrm{e.g.}, conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether's theorem can not yields informations about constants of motion for non-conservative systems since it is not possible to formulate physically meaningful Lagragians for this kind of systems in classical calculus of variation. On the other hand, in recent years the fractional calculus of variation within Lagrangians depending on fractional derivatives has emerged as an elegant alternative to study non-conservative systems. In the present work, we obtained a generalization of the Noether's theorem for Lagrangians depending on mixed classical and Caputo derivatives that can be used to obtain constants of motion for dissipative systems. In addition, we also obtained Noether's conditions for the fractional optimal control problem.