First integrals of affine connections and Hamiltonian systems of hydrodynamic type | Zendy

Felipe Contatto | Zendy; Dunajski Maciej | Zendy

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First integrals of affine connections and Hamiltonian systems of hydrodynamic type

Author(s) -

Felipe Contatto,

Dunajski Maciej

Publication year - 2015

Publication title -

journal of integrable systems

Language(s) - English

Resource type - Journals

ISSN - 2058-5985

DOI - 10.1093/integr/xyw009

Subject(s) - geodesic , mathematics , affine transformation , connection (principal bundle) , affine connection , hamiltonian (control theory) , hamiltonian system , integrable system , pure mathematics , novikov self consistency principle , type (biology) , scalar (mathematics) , ordinary differential equation , differential geometry , mathematical physics , mathematical analysis , differential equation , geometry , mathematical optimization , ecology , biology

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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