Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian

Abstract

We state and prove that a certain class of smooth functions, said to be BH-separable, is a meagre subset for the Fréchet 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 + 𝑎𝑥³+𝑏𝑥²𝑦+𝑐𝑥𝑦²+𝑑𝑦³ are BH-separable.