Euler–Lagrange–Herglotz equations on Lie algebroids

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