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On tensor invariants for integrable cases of Euler, Lagrange and Kovalevskaya rigid body motion

A. V. Tsiganov

Abstract

We discuss global tensor invariants of a rigid body motion in the cases of Euler, Lagrange and Kovalevskaya. These invariants are obtained by substituting tensor fields with cubic on variable components into the invariance equation and solving the resulting algebraic equations using computer algebra systems. According to the Poincaré-Cartan theory of invariants, the existence of invariant geometric structures raises the question of using them to study the dynamics.

On tensor invariants for integrable cases of Euler, Lagrange and Kovalevskaya rigid body motion

Abstract

We discuss global tensor invariants of a rigid body motion in the cases of Euler, Lagrange and Kovalevskaya. These invariants are obtained by substituting tensor fields with cubic on variable components into the invariance equation and solving the resulting algebraic equations using computer algebra systems. According to the Poincaré-Cartan theory of invariants, the existence of invariant geometric structures raises the question of using them to study the dynamics.

Paper Structure

This paper contains 12 sections, 8 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. The Jacobi theory of functional determinants
  3. Tensor invariants for system with $n-1$ first integrals
  4. Tensor invariants for system with $n-2$ first integrals
  5. The Euler case
  6. The Lagrange case
  7. Invariant tensor fields of type (2,0)
  8. Invariant tensor fields of type (3,0)
  9. The case of Kovalevskaya
  10. Invariant tensor fields of type (2,0)
  11. Invariant tensor fields of type (3,0)
  12. Conclusion

Key Result

Proposition 1

If $[\![\mathcal{E},\mathcal{E}]\!]=0$, a divergence-free vector field with $n-1$ independent first integrals (z-div) is Ha-mil-to-nian with respect to each of its first integrals The corresponding Poisson invariant bivectors have the form

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8