# Fractional damping through restricted calculus of variations

## TL;DR

This paper introduces a new variational approach to fractional damping in mechanical systems, establishing a restricted Hamilton's principle and developing numerical integrators called Fractional Variational Integrators (FVIs) that outperform traditional methods in energy tracking.

## Contribution

It proposes a novel restricted Hamilton's principle incorporating fractional derivatives, and develops FVIs for better numerical simulation of fractional damping systems.

## Key findings

- FVIs have local truncation order 1.
- FVIs outperform explicit and implicit Euler schemes in energy tracking.
- The restricted Hamilton's principle accurately captures fractional damping dynamics.

## Abstract

We deliver a novel approach towards the variational description of Lagrangian mechanical systems subject to fractional damping by establishing a restricted Hamilton's principle. Fractional damping is a particular instance of non-local (in time) damping, which is ubiquitous in mechanical engineering applications. The restricted Hamilton's principle relies on including fractional derivatives to the state space, the doubling of curves (which implies an extra mirror system) and the restriction of the class of varied curves. We will obtain the correct dynamics, and will show rigorously that the extra mirror dynamics is nothing but the main one in reversed time; thus, the restricted Hamilton's principle is not adding extra physics to the original system. The price to pay, on the other hand, is that the fractional damped dynamics is only a sufficient condition for the extremals of the action. In addition, we proceed to discretise the new principle. This discretisation provides a set of numerical integrators for the continuous dynamics that we denote Fractional Variational Integrators (FVIs). The discrete dynamics is obtained upon the same ingredients, say doubling of discrete curves and restriction of the discrete variations. We display the performance of the FVIs, which have local truncation order 1, in two examples. As other integrators with variational origin, for instance those generated by the discrete Lagrange-d'Alembert principle, they show a superior performance tracking the dissipative energy, in opposition to direct (order 1) discretisations of the dissipative equations, such as explicit and implicit Euler schemes.