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<p>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 <i>z</i>. 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.</p>

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Variational Dissipative Mechanics on Lie Algebroids

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月25日

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  • Alexandre Anahory Simoes,
  • Leonardo Colombo

摘要

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.