Integrable geodesic flows on 2-torus: Formal solutions and variational principle

Misha Bialy, Andrey E. Mironov

Research output: Contribution to journalArticlepeer-review


In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic flow on 2-torus. We proved in Bialy and Mironov (2011) that this is a semi-Hamiltonian system and we show here that the metric associated with the system is a metric of Egorov type. We use this fact in order to prove that in the case of integrals of degree three and four the system is in fact equivalent to a single remarkable equation of order 3 and 4 respectively. Remarkably the equation for the case of degree four has variational meaning: it is Euler-Lagrange equation of a variational principle. Next we prove that this equation for n= 4 has formal double periodic solutions as a series in a small parameter.

Original languageEnglish
Pages (from-to)39-47
Number of pages9
JournalJournal of Geometry and Physics
StatePublished - 1 Jan 2015


  • Conservation laws
  • Geodesic flows
  • Integrable Hamiltonians
  • Semi-Hamiltonian systems


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