[Paper Review] Hamilton-Pontryagin Integrators on Lie Groups Part I: Introduction & Structure-Preserving Properties

Nawaf Bou‐Rabee, Jerrold E. Marsden|arXiv (Cornell University)|Jan 1, 2013

Numerical methods for differential equations33 references9 citations

TL;DR

This paper introduces a novel class of variational integrators for rigid body-type systems on Lie groups using a discrete Hamilton-Pontryagin variational principle. These structure-preserving integrators preserve both a discrete momentum map and a symplectic form, outperforming energy- or momentum-preserving methods in numerical experiments on rigid bodies and underwater vehicles.

ABSTRACT

In this paper structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton-Pontryagin variational principle. From this principle one can derive a novel class of variational partitioned Runge-Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Stormer-Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.

Motivation & Objective

• To develop structure-preserving time integrators for mechanical systems on Lie groups, particularly rigid body dynamics.
• To address the limitations of existing methods that preserve momentum or energy separately but fail to maintain both geometric structures simultaneously.
• To derive integrators from a discrete Hamilton-Pontryagin variational principle, ensuring intrinsic geometric consistency.
• To establish a theoretical foundation for variational integrators that generalize symplectic Euler and Stormer-Verlet methods to non-flat, curved configuration spaces.

Proposed method

• Derive time integrators via a discrete Hamilton-Pontryagin variational principle formulated on Lie groups.
• Construct a novel class of variational partitioned Runge-Kutta methods that respect the group structure and geometric properties of the system.
• Ensure preservation of the discrete momentum map through the symmetry of the variational principle under the coadjoint action of the Lie group.
• Maintain a symplectic structure in the discrete setting by construction, leveraging the variational formulation.
• Generalize classical integrators like symplectic Euler and Stormer-Verlet to curved configuration spaces via the variational framework.
• Use the discrete variational principle to derive update rules that are consistent with the underlying mechanics on Lie groups.

Experimental results

Research questions

• RQ1 Can a variational integrator on Lie groups preserve both momentum and symplectic structure simultaneously, unlike existing methods?
• RQ2 How do the structure-preserving properties of these integrators compare to those of energy- or momentum-preserving methods in long-term simulations?
• RQ3 To what extent do these integrators generalize classical symplectic integrators like symplectic Euler and Stormer-Verlet to non-Euclidean configuration spaces?
• RQ4 What is the role of the discrete Hamilton-Pontryagin principle in ensuring geometric consistency on Lie groups?
• RQ5 How do the integrators perform in complex dynamics such as chaotic motion of an underwater vehicle?

Key findings

• The proposed integrators preserve both a discrete momentum map and a symplectic form due to their variational design on Lie groups.
• Numerical experiments on the rigid body demonstrate superior long-term stability and geometric fidelity compared to methods that only preserve energy or momentum.
• For the chaotic dynamics of an underwater vehicle, the variational integrators maintain structure more effectively than projection-based or momentum-preserving methods.
• The integrators generalize symplectic Euler and Stormer-Verlet methods to Lie groups, extending their applicability to non-flat mechanical systems.
• The structure-preserving properties are intrinsic to the variational formulation and not artificially imposed, leading to better numerical performance in complex dynamics.