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From retraction maps to symplectic-momentum numerical integrators

María Barbero-Liñán, Juan Carlos Marrero, David Martín de Diego

TL;DR

The paper addresses the challenge of designing numerical integrators for Hamiltonian systems that preserve both the symplectic structure and symmetry-induced momentum maps. It introduces discretization maps that generalize retraction maps and uses them to construct symplectic-momentum integrators via Lagrangian submanifolds and an antisymplectomorphism, yielding a discrete flow that preserves $\omega$ and, under symmetry-preserving maps, the momentum map $J_\xi$. Higher-order schemes are obtained by composing Lagrangian submanifolds and employing adjoint discretizations, with explicit midpoint-based examples on Euclidean spaces and Lie groups. The framework lays groundwork for reduction to Lie-Poisson structures and future extensions to Lie groupoids/algebroids, offering robust, structure-preserving tools for long-time simulations in geometric mechanics.

Abstract

Retraction maps have been generalized to discretization maps in (Barbero Liñán and and Martín de Diego, 2022). Discretization maps are used to systematically derive numerical integrators that preserve the symplectic structure, as well as the discrete momemtum map under the assumption of symmetric preservation for the discretization map. The procedure described here gives a geometrical construction that can be easily adapted to discretize dynamics on more general structures and open the door to reduction processes.

From retraction maps to symplectic-momentum numerical integrators

TL;DR

The paper addresses the challenge of designing numerical integrators for Hamiltonian systems that preserve both the symplectic structure and symmetry-induced momentum maps. It introduces discretization maps that generalize retraction maps and uses them to construct symplectic-momentum integrators via Lagrangian submanifolds and an antisymplectomorphism, yielding a discrete flow that preserves and, under symmetry-preserving maps, the momentum map . Higher-order schemes are obtained by composing Lagrangian submanifolds and employing adjoint discretizations, with explicit midpoint-based examples on Euclidean spaces and Lie groups. The framework lays groundwork for reduction to Lie-Poisson structures and future extensions to Lie groupoids/algebroids, offering robust, structure-preserving tools for long-time simulations in geometric mechanics.

Abstract

Retraction maps have been generalized to discretization maps in (Barbero Liñán and and Martín de Diego, 2022). Discretization maps are used to systematically derive numerical integrators that preserve the symplectic structure, as well as the discrete momemtum map under the assumption of symmetric preservation for the discretization map. The procedure described here gives a geometrical construction that can be easily adapted to discretize dynamics on more general structures and open the door to reduction processes.
Paper Structure (7 sections, 1 theorem, 28 equations)

This paper contains 7 sections, 1 theorem, 28 equations.

Table of Contents

  1. Introduction
  2. Retraction maps and discretization maps
  3. Symplectic-momentum integrators
  4. Symplectic integrators based on discretization maps
  5. Symmetry preservation
  6. Higher-order symplectic-momentum integrators
  7. Conclusions

Key Result

Theorem 2

Let $H\colon T^*Q\rightarrow \mathbb{R}$ be a Hamiltonian function and $\xi\in {\mathfrak g}$. If $J_{\xi}$ is a constant of motion of the Hamiltonian vector field $X_{H}$, then the method in Equation eq:DH_method derived from a symmetry-preserving discretization map $R_d$ satisfies that for all $(-\mu_{q_k}, \mu_{q_{k+1}})\in D^h_{H}$.

Theorems & Definitions (8)

  • Example 1
  • Example 2
  • Definition 1
  • Remark 1
  • Example 3
  • Remark 2
  • Theorem 2
  • Example 4