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arXiv · 1510.01906

First integrals of affine connections and Hamiltonian systems of hydrodynamic type

Felipe Contatto·Maciej Dunajski
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Abstract

We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence of a Hamiltonian formulation of Dubrovin--Novikov type for a given one--dimensional system of hydrodynamic type. We give several examples including Zoll connections, and Hamiltonian systems arising from two--dimensional Frobenius manifolds.

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Felipe Contatto, Maciej Dunajski. 2015-10-07. First integrals of affine connections and Hamiltonian systems of hydrodynamic type. https://arxiv.org/abs/1510.01906

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