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Hamilton-Pontryagin Integrators on Lie Groups: Introduction and Structure-Preserving Properties
Nawaf Bou-Rabee, Jerrold E. Marsden
TL;DR
Mechanical systems on Lie groups require efficient time integrators that preserve geometric structure. This paper derives such methods from a discrete Hamilton-Pontryagin principle, producing variational partitioned Runge-Kutta schemes that generalize symplectic Euler and Störmer-Verlet. The paper’s planned numerical studies examine their behavior on rigid-body and underwater-vehicle dynamics.
Problem
The paper addresses the design of efficient, structure-preserving time integrators for mechanical systems whose configuration space is a Lie group.
Method
The paper discretizes the left-trivialized Hamilton-Pontryagin principle using Runge-Kutta-Munthe-Kaas methods to construct variational partitioned Runge-Kutta integrators on Lie groups.
Results
The resulting methods generalize variational Euler and Störmer-Verlet integrators to Lie groups and inherit properties including efficiency, order of accuracy, symplecticity, and symmetry.
Takeaways & Limitations
The framework provides a practical route for designing discrete Lagrangians and structure-preserving integrators for Lie-group mechanical systems.
Takeaways & Limitations
The paper assumes endpoint conditions on the group variables and uses local Lie-group coordinates defined near the identity.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper uses the Hamilton-Pontryagin principle to design discrete Lagrangians and extend variational partitioned Runge-Kutta methods from vector spaces to Lie groups.
- Hamilton-Pontryagin principle: The Hamilton-Pontryagin action combines a Lagrangian with a momentum-paired kinematic constraint, while varying configuration, velocity, and momentum independently.Varying momentum recovers Hamilton’s principle; varying velocity yields Hamilton’s phase-space principle.
- Discrete construction: An s-stage Runge-Kutta discretization enforces the kinematic constraint inside a discrete action sum with weighted Lagrangian terms and discrete constraint pairings.The weights come from the RK Butcher tableau.
- Lie-group formulation: In Lie-group systems, the paper presents generalized coordinates through a left-trivialized action rather than embedding the group as a holonomically constrained submanifold.The alternative constrained-coordinate approach embeds G in a larger linear space and uses Lagrange multipliers.
- Lie-group formulation: RKMK discretization of the reconstruction equation and quadrature of the left-trivialized Lagrangian yield variational partitioned Runge-Kutta methods on Lie groups.The resulting class includes generalizations of symplectic Euler and Störmer-Verlet methods.
2 Background and Setting
The background contrasts variational integrators with standard, coadjoint-preserving, and energy-preserving approaches, emphasizing long-time structure preservation and the paper’s reduced-Hamilton-Pontryagin perspective.
- Variational integration: Variational integrators are symplectic and are designed to preserve statistical properties in long-time mechanical simulations.The paper discusses Poincaré sections and time-averaged instantaneous temperature as examples.
- Numerical motivation: At h = 0.05, fourth-order RK4 corrupts chaotic invariant sets, whereas second-order variational Euler preserves the benchmark structure.At h = 0.025, both methods agree with the benchmark.
- Lie-group structure preservation: Lie-group systems with symmetry have flow maps that preserve both symplectic structure and a momentum map associated with the symmetry.The background surveys multiple strategies for constructing such integrators.
- Prior approaches: Earlier approaches include Lie-Newmark, coadjoint-preserving, energy-preserving, constrained, and reduced variational integrators on Lie groups.These approaches use mechanisms such as coadjoint actions, gradient estimates, holonomic constraints, or discrete reduction.
- Paper perspective: This paper instead discretizes the reduced Hamilton-Pontryagin principle so reconstruction and discrete Euler-Poincaré equations can be solved independently on a lower-dimensional linear space.The paper presents this decoupling as its distinctive perspective on Lie-group variational integrators.
3 HP Mechanics
The section formulates Hamilton–Pontryagin mechanics on Lie groups and derives its left-trivialized equations and flow. This formulation is equivalent to Hamiltonian mechanics and preserves a symplectic two-form.
- Hamilton–Pontryagin formulation: The Hamilton–Pontryagin principle varies configuration, velocity, and momentum independently, incorporating the Legendre transformation and Euler–Lagrange equations in one variational principle.
- Left trivialization: Left trivialization rewrites the principle using curves in G × g × g∗, with ξ = g−1v and µ = g−1p.
- Left trivialization: For left-invariant Lagrangians, the left-trivialized principle unifies the system’s Euler–Poincaré and Lie–Poisson descriptions.
- HP flow: The left-trivialized HP equations include the reconstruction equation ξ = g−1ġ and the Legendre relation µ = ∂ℓ/∂ξ.
- HP flow: The left-trivialized HP flow is equivalent to the HP and Hamiltonian flows on the admissible space through diffeomorphic restrictions and projection.
- Symplecticity: Left-trivialized HP flows preserve the symplectic two-form ωIhp.
4 Lie Group VPRK Integrators
The paper derives Lie-group VPRK integrators by combining a discrete Hamilton-Pontryagin principle with canonical Lie-group coordinates and RKMK discretization. The resulting schemes include Lie-group Störmer-Verlet and Euler methods, with variational critical-point and symplectic-preservation properties.
- Examples: The framework includes Lie-group Störmer-Verlet, variational Euler, and Euler-Poincaré integrators.The Störmer-Verlet update uses the two-stage implicit trapezoidal-rule tableau and is generally implicit after eliminating the next configuration.
- Canonical coordinates: The method uses a map τ from the Lie algebra to the Lie group as a local chart, including the exponential map as one example.The map is analytic near zero, satisfies τ(0)=e and τ(ξ)·τ(−ξ)=e, and its local coordinates are called canonical coordinates of the first kind.
- RKMK discretization: RKMK discretization applies Runge-Kutta methods to the Lie-algebra equation obtained from the reconstruction equation.The method uses external and internal stage configurations and is determined by the Runge-Kutta a-matrix and b-vector.
- RKMK discretization: If the approximant order satisfies q ≥ p − 2, an underlying order-p Runge-Kutta method yields an order-p RKMK method.The condition controls truncation of dτ^−1 without degrading the underlying method’s order.
- VPRK construction: The discrete Hamilton-Pontryagin principle produces VPRK schemes as critical points of a discrete action and yields a discrete flow preserving the symplectic form.The discrete action uses a quadrature approximation and treats the reconstruction relation as a constraint, with independent external and internal variations.
5 Conclusion
The paper derives a left-trivialized Hamilton-Pontryagin framework for Lie-group mechanics and uses RKMK discretization to construct variational partitioned Runge-Kutta methods. Its conclusion emphasizes Lie-group generalizations of variational Euler and Störmer-Verlet methods, with further analysis planned for accuracy and numerical performance.
- Contributions: The left-trivialized Hamilton-Pontryagin principle unifies Euler-Poincaré and Lie-Poisson descriptions for left-invariant Lagrangians.It also provides a practical route to designing discrete Lagrangians for mechanical systems on Lie groups.
- Contributions: RKMK discretization of the kinematic constraint leads to variational partitioned Runge-Kutta methods generalized from flat spaces to Lie groups.The construction includes Lie-group versions of variational Euler and Störmer-Verlet methods.
- Properties: The methods retain properties associated with their flat-space counterparts, including efficiency, order of accuracy, symplecticity, and symmetry.These properties are stated as inherited features of the Lie-group generalizations.
- Planned analysis: The companion work is intended to prove VPRK order of accuracy and study rigid-body and underwater-vehicle dynamics numerically.Planned experiments include Poincaré sections and comparison with symmetric rigid-body integrators for a nonreversible turntable system.