Variational Dissipative Mechanics on Lie Algebroids

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–Poincaré–Herglotz equations on a Lie algebra, and the Lagrange–Poincaré–Herglotz equations on Atiyah algebroids of principal bundles. Starting from the local formulation, we then use Lie algebroid connections to obtain a global connection-based Euler–Lagrange–Poincaré-Herglotz and Hamilton–Pontryagin–Herglotz theory, where the connection serves as an auxiliary device for the horizontal-vertical splitting of the dynamics. Finally, we establish energy balance laws and Noether–Herglotz-type results, in which classical conserved quantities are replaced by dissipated invariants.