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Variational Dissipative Mechanics on Lie Algebroids

Alexandre Anahory Simoes, Leonardo Colombo

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--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 coordinate-free Euler--Lagrange--Poincaré--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.

Variational Dissipative Mechanics on Lie Algebroids

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 . 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 coordinate-free Euler--Lagrange--Poincaré--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.

Paper Structure

This paper contains 11 sections, 7 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Lie Algebroids
  3. Herglotz Variational Principle on a Lie Algebroid
  4. Energy Balance and Noether--Herglotz Theorems
  5. Energy balance law
  6. Noether--Herglotz theorem on a Lie algebroid
  7. Coordinate-free formulation of the Euler--Lagrange--Herglotz equations
  8. Euler--Lagrange--Herglotz equations
  9. Hamilton--Pontryagin--Herglotz principle on a Lie algebroid
  10. Conclusions and Future Work
  11. Appendix: Local expression of the intrinsic Herglotz equation

Key Result

Theorem 3.2

Let $(E\to M,[\cdot,\cdot],\rho)$ be a Lie algebroid whose anchor components are given by $\rho^i_{\ \alpha}(x)$ and strcuture functions are $C^\gamma_{\ \alpha\beta}(x)$ with respect to local coordinates denoted by $(x^{i},y^{\alpha},z)$. Let $L\colon E\times\mathbb{R}\to\mathbb{R}$ be a Herglotz-t Then $(x^{i}(t),y^{\alpha}(t),z(t))$ is a Herglotz extremal, i.e. satisfies eq:stationary for all a

Theorems & Definitions (37)

  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.2: Euler--Lagrange--Herglotz equations on a Lie algebroid
  • proof
  • Remark 3.3
  • Example 3.4: Euler--Lagrange--Herglotz equations
  • Example 3.5: Hamel-Herglotz equations on Lie algebroids
  • Example 3.6: Euler--Poincaré--Herglotz equation on a Lie algebra
  • Example 3.7: Euler--Poincaré--Herglotz equations on an action Lie algebroid
  • ...and 27 more