A Fractional Variational Approach for Modelling Dissipative Mechanical Systems: Continuous and Discrete Settings

TL;DR

This paper introduces a fractional variational principle using Riemann-Liouville derivatives to model dissipative mechanical systems, providing invariant equations in continuous and discrete settings that account for energy loss and are reversible in time.

Contribution

It develops a novel fractional variational framework for dissipative systems, including a discrete analogue, with invariance properties and specific restrictions on variations.

Findings

Derivation of fractional Euler-Lagrange equations for dissipative systems.

Invariance of equations under linear change of variables and time reversal.

Discrete equations accurately approximate continuous fractional equations.

Abstract

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linear change of variables. This principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-Lagrange equations model a dissipative system in both directions of time, summing up to a set of equations that is invariant under time reversal. Finally, we show that the discrete equations are a meaningful discretisation of the continuous ones.