First integrals of affine connections and Hamiltonian systems of hydrodynamic type

TL;DR

This paper establishes conditions for affine connection geodesic flows to have linear first integrals and explores obstructions to Hamiltonian formulations in hydrodynamic systems, with applications to Zoll connections and Frobenius manifolds.

Contribution

It provides necessary and sufficient conditions for first integrals of affine connections and identifies obstructions to Hamiltonian structures in hydrodynamic systems.

Findings

Conditions expressed via scalar invariants of differential orders 3 and 4.

Explicit obstructions to Hamiltonian formulations for hydrodynamic systems.

Examples include Zoll connections and systems from Frobenius manifolds.

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.