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A new perspective on symplectic integration of constrained mechanical systems via discretization maps

María Barbero Liñán, David Martín de Diego, Rodrigo T. Sato Martín de Almagro

TL;DR

The paper tackles the challenge of designing structure-preserving integrators for constrained mechanical systems, particularly holonomic systems on general manifolds. It introduces discretization maps derived from retraction maps to tailor discretization to the continuous problem, ensuring the constraint submanifold is preserved exactly by the discrete flow via $R^h_d: TQ \rightarrow Q\times Q$ and its cotangent lift. The authors develop a geometric framework based on Lagrangian submanifolds, cotangent lifts, and Lie-group trivializations to produce symplectic integrators for holonomic systems in Euclidean spaces and on Lie groups, including explicit Symplectic Euler A/B, Mid-point, and RATTLE-type schemes, with a discrete-null-space interpretation. This approach provides a unified view of constrained symplectic integration, clarifies how to extend to nonlinear configuration spaces and multibody systems, and offers practical, exact-constraint-preserving methods with potential applications in robotics and computational mechanics.

Abstract

A new geometric procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the discretization rule rather than viceversa. As a result, the constraint submanifold is exactly preserved by the symplectic discrete flow and the extension of these methods to the case of non-linear configuration spaces is doable.

A new perspective on symplectic integration of constrained mechanical systems via discretization maps

TL;DR

The paper tackles the challenge of designing structure-preserving integrators for constrained mechanical systems, particularly holonomic systems on general manifolds. It introduces discretization maps derived from retraction maps to tailor discretization to the continuous problem, ensuring the constraint submanifold is preserved exactly by the discrete flow via and its cotangent lift. The authors develop a geometric framework based on Lagrangian submanifolds, cotangent lifts, and Lie-group trivializations to produce symplectic integrators for holonomic systems in Euclidean spaces and on Lie groups, including explicit Symplectic Euler A/B, Mid-point, and RATTLE-type schemes, with a discrete-null-space interpretation. This approach provides a unified view of constrained symplectic integration, clarifies how to extend to nonlinear configuration spaces and multibody systems, and offers practical, exact-constraint-preserving methods with potential applications in robotics and computational mechanics.

Abstract

A new geometric procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the discretization rule rather than viceversa. As a result, the constraint submanifold is exactly preserved by the symplectic discrete flow and the extension of these methods to the case of non-linear configuration spaces is doable.
Paper Structure (20 sections, 7 theorems, 125 equations)

This paper contains 20 sections, 7 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Constrained mechanical systems
  3. Intrinsic viewpoint
  4. Extrinsic viewpoint
  5. Geometric relation between both approaches
  6. Discretization maps
  7. Symplectic integration of holonomic mechanical systems
  8. Construction of constrained symplectic integrators
  9. A geometrical interpretation of the discrete null space method
  10. Application to mechanical Lagrangian systems with holonomic constraints
  11. On Euclidean vector spaces
  12. Symplectic Euler method-A
  13. Symplectic Euler method-B
  14. Mid-point discretization
  15. Composition of symplectic Euler methods: RATTLE discretization
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $N$ be a submanifold of $Q$ and $L\colon TN \rightarrow \mathbb{R}$ be a regular Lagrangian function. Then,

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • Corollary 3.4
  • Example 3.5
  • Definition 4.1
  • ...and 11 more