2000 character limit reached
First integrals of affine connections and Hamiltonian systems of hydrodynamic type (1510.01906v2)
Published 7 Oct 2015 in math.DG, hep-th, and nlin.SI
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.