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On tensor invariants of the Clebsch system

A. V. Tsiganov

TL;DR

This work advances geometric numerical approaches for the Clebsch system by solving the invariance equation $\mathcal{L}_X T=0$ to produce six invariant Poisson bivectors (linear, rational, and cubic) and their Casimir functions, enabling exact preservation on corresponding symplectic leaves. It develops both analytic (Lax-based Moser–Veselov) and algebraic (Darboux-polynomial) constructions to study continuous and discrete structure, and discusses how Poisson neural networks could facilitate automatic discovery and discretization of invariant structures. It also analyzes Kahan-type discretizations for the Clebsch and Euler tops, deriving explicit maps and highlighting the challenges in preserving Poisson structures and invariants in the discrete setting. The results provide a framework for designing and comparing structure-preserving integrators, with potential impact on long-time accuracy and stability in simulations of rigid-body-like systems with rich geometric structure.$

Abstract

We present some new Poisson bivectors that are invariants by the Clebsch system flow. Symplectic integrators on their symplectic leaves exactly preserve the corresponding Casimir functions, which have different physical meanings. The Kahan discretization of the Clebsch system is discussed briefly.

On tensor invariants of the Clebsch system

TL;DR

This work advances geometric numerical approaches for the Clebsch system by solving the invariance equation to produce six invariant Poisson bivectors (linear, rational, and cubic) and their Casimir functions, enabling exact preservation on corresponding symplectic leaves. It develops both analytic (Lax-based Moser–Veselov) and algebraic (Darboux-polynomial) constructions to study continuous and discrete structure, and discusses how Poisson neural networks could facilitate automatic discovery and discretization of invariant structures. It also analyzes Kahan-type discretizations for the Clebsch and Euler tops, deriving explicit maps and highlighting the challenges in preserving Poisson structures and invariants in the discrete setting. The results provide a framework for designing and comparing structure-preserving integrators, with potential impact on long-time accuracy and stability in simulations of rigid-body-like systems with rich geometric structure.$

Abstract

We present some new Poisson bivectors that are invariants by the Clebsch system flow. Symplectic integrators on their symplectic leaves exactly preserve the corresponding Casimir functions, which have different physical meanings. The Kahan discretization of the Clebsch system is discussed briefly.

Paper Structure

This paper contains 14 sections, 6 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Known tensor invariants for the Clebsch system
  3. Linear and quadratic tensor invariants
  4. Cubic invariant bivectors
  5. Invariant rational Poisson bivectors
  6. Invariant polynomial Poisson bivectors
  7. Moser-Veselov discretization scheme
  8. Poisson neural networks
  9. Symplectic leaves
  10. Darboux coordinates
  11. Kahan discretization
  12. Euler top
  13. Clebsch system
  14. Conclusion

Key Result

Proposition 1

Invariant Poisson bivector $P_1$ (p1) is a Lie derivative of the invariant Poisson bivector $P_2$ (p2) along the vector field $V$ It means that $P_1$ is a trivial deformations of $P_2$ in the Lichnerowicz-Poisson cohomology vai90. The corresponding Liouville vector field $V$ is not invariant to the flows associated with $X$ and $Y$.

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6