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From Euler-Jacobi to Bogoyavlensky and back

Davide Murari, Nicola Sansonetto

Abstract

This work focuses on two notions of non-Hamiltonian integrable systems: B-integrability and Euler-Jacobi integrability. We first show that the first notion is stronger. We then investigate which possible "non-evident" properties one can add to the Euler-Jacobi Theorem to make the dynamics B-integrable.

From Euler-Jacobi to Bogoyavlensky and back

Abstract

This work focuses on two notions of non-Hamiltonian integrable systems: B-integrability and Euler-Jacobi integrability. We first show that the first notion is stronger. We then investigate which possible "non-evident" properties one can add to the Euler-Jacobi Theorem to make the dynamics B-integrable.

Paper Structure

This paper contains 8 sections, 6 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. The Bogoyavlensky and Euler–Jacobi Theorems
  3. Aims and contributions of the paper
  4. Bogoyavlensky implies Euler–Jacobi
  5. From Euler–Jacobi to Bogoyavlensky
  6. Examples
  7. Conservation of a differential one-form
  8. EJ-integrable systems, Lie-point symmetries, and B-integrability

Key Result

Theorem 1

Let $M$ be a smooth $n$-dimensional manifold and $X$ a smooth vector field on $M$. Assume that for an integer $0<k\leq n$ there exists a surjective submersion with compact and connected fibers, such that: Moreover, assume the existence of $k$ everywhere linearly independent smooth vector fields $Y_1,...,Y_k$ on $M$ such that Then

Theorems & Definitions (14)

  • Theorem 1: Bogoyavlensky
  • Theorem 2: Euler–Jacobi
  • Theorem 3: B-integrability implies EJ-integrability
  • proof
  • Remark 1
  • Example 1: B-integrable on $\mathbb{T}^2$
  • Example 2: B-integrable on $S^1$
  • Example 3: Not B-integrable
  • Proposition 1
  • proof
  • ...and 4 more