Euler–Lagrange–Herglotz equations on Lie algebroids

Abstract

We introduce Euler–Lagrange–Herglotz equations on Lie algebroids. The methodology is to extend the Jacobi structure from $$TQ\times \mathbb {R}$$ T Q × R and $$T^{*}Q \times \mathbb {R}$$ T ∗ Q × R to $$A\times \mathbb {R}$$ A × R and $$A^{*}\times \mathbb {R}$$ A ∗ × R , respectively, where A is a Lie algebroid and $$A^{*}$$ A ∗ carries the associated Poisson structure. We see that $$A^*\times \mathbb {R}$$ A ∗ × R possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems on Lie algebroids, generalizing previous models on $$TQ\times \mathbb {R}$$ T Q × R and introducing new ones as for instance for reduced systems on Lie algebras, semidirect products (action Lie algebroids) and Atiyah bundles.