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

Nawaf Bou‐Rabee, Jerrold E. Marsden|ArXiv.org|Jan 7, 2008

Numerical methods for differential equations39 references9 citations

TL;DR

This paper introduces a novel class of variational partitioned Runge-Kutta (VPRK) integrators on Lie groups by discretizing the Hamilton-Pontryagin variational principle using Runge-Kutta-Milstein-Kaas (RKMK) methods. The method generalizes symplectic Euler and Störmer-Verlet integrators to Lie groups, preserving momentum maps and symplectic structure, with superior long-term stability and accuracy compared to energy- or momentum-preserving methods.

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 Störmer-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

• Develop structure-preserving time integrators for mechanical systems on Lie groups with guaranteed geometric properties.
• Address the challenge of discretizing kinematic constraints on non-flat configuration spaces using variational principles.
• Generalize symplectic and variational integrators (e.g., Störmer-Verlet, symplectic Euler) from flat spaces to Lie groups.
• Ensure preservation of momentum map under symmetry and symplectic structure via a variational design.
• Provide a foundation for numerical analysis and experiments in Part II on rigid bodies and underwater vehicles.

Proposed method

• Formulate a left-trivialized Hamilton-Pontryagin action principle on Lie groups, unifying Euler-Poincaré and Lie-Poisson dynamics for left-invariant Lagrangians.
• Discretize the kinematic constraint using an s-stage Runge-Kutta-Munthe-Kaas (RKMK) method to handle group-valued trajectories.
• Construct a discrete action sum combining weighted Lagrangians and momentum-paired kinematic constraints via Butcher tableau coefficients.
• Derive the resulting variational partitioned Runge-Kutta (VPRK) integrators as solutions to the discrete Euler-Lagrange equations.
• Use the adjoint representation and coadjoint action to express momentum map evolution and verify conservation properties.
• Implement Padé approximants and projectors (e.g., skew-symmetrization for SO(n)) to maintain group membership in discrete updates.

Experimental results

Research questions

• RQ1 Can the Hamilton-Pontryagin variational principle be generalized to construct structure-preserving integrators on Lie groups?
• RQ2 How can the kinematic constraint on a Lie group be discretized using Runge-Kutta methods while preserving geometric structure?
• RQ3 Do the resulting VPRK integrators preserve symplecticity and momentum maps, and do they outperform standard energy- or momentum-preserving methods?
• RQ4 What is the order of accuracy of these integrators, and how does it relate to the underlying variational structure?
• RQ5 How do these integrators perform on nonreversible systems such as a rigid body on a turntable compared to symmetric integrators?

Key findings

• The proposed VPRK integrators preserve both the symplectic form and the momentum map under symmetry, ensuring long-term stability.
• The methods generalize symplectic Euler and Störmer-Verlet integrators to Lie groups via a variational formulation on the tangent bundle of the group.
• The discrete action sum is constructed using Butcher tableau coefficients and RKMK discretization of the reconstruction equation.
• The momentum map evolution is derived as a discrete version of the Lie-Poisson bracket, with conservation verified via the adjoint action.
• The integrators achieve second-order accuracy, as proven via the variational proof of order of accuracy (Marsden & West, 2001).
• Numerical experiments in Part II show superior performance over energy- or momentum-preserving methods in chaotic dynamics and nonreversible systems.