Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian

TL;DR

This paper proves that, in a generic sense, Arnold-Liouville integrable systems do not admit a bi-Hamiltonian structure because the class of BH-separable Hamiltonians is very limited.

Contribution

It establishes that BH-separable Hamiltonians are meagre in the Fréchet topology, implying Arnold-Liouville systems are generically not bi-Hamiltonian, and classifies polynomial Hamiltonians of a specific form.

Findings

BH-separable functions form a meagre subset in the Fréchet topology.

Most Arnold-Liouville systems cannot be bi-Hamiltonian.

Explicit classification of certain polynomial Hamiltonians that are BH-separable.

Abstract

We state and prove that a certain class of smooth functions said to be BH-separable is a meagre subset for the Fr\'echet topology. Because these functions are the only admissible Hamiltonians for Arnold-Liouville systems admitting a bi-Hamiltonian structure, we get that, generically, Arnold-Liouville systems cannot be bi-Hamiltonian. At the end of the paper, we determine, both as a concrete representation of our general result and as an illustrative list, which polynomial Hamiltonians $H$ of the form $H(x,y)=xy+ax^3+bx^2y+cxy^2+dy^3$ are BH-separable.