Abstract
We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four.
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Fernandes, R.L. Completely integrable bi-Hamiltonian systems. J Dyn Diff Equat 6, 53–69 (1994). https://doi.org/10.1007/BF02219188
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DOI: https://doi.org/10.1007/BF02219188