First integrals of affine connections and Hamiltonian systems of hydrodynamic type
Dunajski, Maciej Lukasz
Journal of Integrable Systems
Oxford University Press
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Contatto, F., & Dunajski, M. L. (2016). First integrals of affine connections and Hamiltonian systems of hydrodynamic type. Journal of Integrable Systems, 1 (1. xyw009)https://doi.org/10.1093/integr/xyw009
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 twodimensional Frobenius manifolds.
math.DG, math.DG, hep-th, nlin.SI, affine connections, Hamiltonian systems, hydrodynamic type
Cambridge Commonwealth, European & International Trust and CAPES Foundation (Grant Proc. BEX 13656/13-9) to F.C.
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External DOI: https://doi.org/10.1093/integr/xyw009
This record's URL: https://www.repository.cam.ac.uk/handle/1810/267951
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