Variational Dissipative Mechanics on Lie Algebroids

TL;DR

This paper develops a geometric variational framework for dissipative systems on Lie algebroids, generalizing classical equations and establishing energy and invariance laws in a unified setting.

Contribution

It introduces a Herglotz-type variational principle on Lie algebroids, extending classical dissipative mechanics to a broad geometric context.

Findings

Derivation of Euler--Lagrange--Herglotz equations on Lie algebroids

Recovery of classical equations as special cases

Establishment of energy balance and Noether--Herglotz results

Abstract

We formulate a Herglotz-type variational principle on a Lie algebroid and derive the corresponding Euler--Lagrange--Herglotz equations for a Lagrangian depending on an additional scalar variable $z$. This provides a geometric framework for dissipative systems on Lie algebroids and recovers, as special cases, the classical Euler--Lagrange--Herglotz equations on tangent bundles, the Euler--Poincar\'e--Herglotz equations on a Lie algebra, and the Lagrange--Poincar\'e--Herglotz equations on Atiyah algebroids of principal bundles. Starting from the local formulation, we then use Lie algebroid connections to obtain a coordinate-free Euler--Lagrange--Poincar\'e--Herglotz and Hamilton--Pontryagin--Herglotz theory. Finally, we establish energy balance laws and Noether--Herglotz-type results, in which classical conserved quantities are replaced by dissipated invariants.