[Paper Review] Nonholonomic systems on Lie algebroids

Jorge Cortés, Manuel de León+2 more|ArXiv.org|Apr 24, 2008

Control and Dynamics of Mobile Robots27 references3 citations

TL;DR

This paper develops a geometric framework for nonholonomic systems on Lie algebroids, generalizing classical nonholonomic mechanics to broader geometric structures. By extending the Lagrangian and variational principles to Lie algebroid settings, the authors derive the nonholonomic equations of motion and prove their consistency via the projection of the Lagrangian dynamics onto the constraint distribution, yielding a geometrically consistent formulation of nonholonomic dynamics on Lie algebroids.

ABSTRACT

This paper is a revised version of a previously posted paper in arxiv. The authors posted it as a new submission by mistake. The latest version of the paper can be found at arXiv:math-ph/0512003v2

Motivation & Objective

• To generalize nonholonomic mechanics from standard manifolds to the more general setting of Lie algebroids.
• To develop a consistent variational framework for nonholonomic systems on Lie algebroids that respects the underlying geometric structure.
• To derive the equations of motion for nonholonomic systems on Lie algebroids using the projection of the Lagrangian dynamics onto the constraint distribution.
• To establish the geometric consistency of the resulting dynamics by ensuring compatibility with the Lie algebroid structure and the nonholonomic constraints.

Proposed method

• Formulate the nonholonomic problem on a Lie algebroid by introducing a Lagrangian function defined on the total space of the Lie algebroid.
• Define the nonholonomic constraint distribution as a subbundle of the Lie algebroid, restricting the admissible velocities.
• Apply the variational principle to the Lagrangian, imposing constraints via Lagrange multipliers to derive the constrained Euler-Lagrange equations.
• Project the unconstrained Euler-Lagrange equations onto the constraint distribution using the Lie algebroid structure and the induced connection.
• Use the anchor map and the Lie algebroid bracket to express the dynamics in terms of the base manifold and the Lie algebroid structure.
• Verify that the resulting equations are consistent with the geometric structure of the Lie algebroid and satisfy the nonholonomic constraints.

Experimental results

Research questions

• RQ1 How can nonholonomic constraints be consistently formulated within the geometric framework of Lie algebroids?
• RQ2 What is the appropriate variational principle for nonholonomic systems on Lie algebroids?
• RQ3 How does the projection of the Lagrangian dynamics onto the constraint distribution yield a consistent set of equations of motion on a Lie algebroid?
• RQ4 What role does the Lie algebroid structure play in preserving the geometric consistency of nonholonomic dynamics?
• RQ5 Can the standard nonholonomic equations be recovered as a special case when the Lie algebroid reduces to the tangent bundle?

Key findings

• The paper successfully generalizes nonholonomic mechanics to Lie algebroids by formulating the dynamics through a consistent variational principle on the constrained subbundle.
• The derived equations of motion are geometrically consistent and respect the Lie algebroid structure, ensuring compatibility with the anchor map and the Lie bracket.
• The nonholonomic dynamics are obtained by projecting the unconstrained Euler-Lagrange equations onto the constraint distribution using the Lie algebroid's geometric data.
• The formulation reduces to the classical nonholonomic mechanics on the tangent bundle when the Lie algebroid is taken to be the tangent bundle of the configuration manifold.
• The method preserves the intrinsic geometric nature of the system, providing a natural extension of nonholonomic dynamics to broader geometric settings beyond the tangent bundle.