Symmetries, Topology and Resonances in Hamiltonian Mechanics

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John Hornstein has written about the author’s theorem on nonintegrability of geodesic flows on closed surfaces of genus greater than one: Here is an example of how differential geometry, differential and algebraic topology, and Newton’s laws make music together (Amer. Math. Monthly, November 1989).
Kozlov’s book is a systematic introduction to the problem of exact integration of equations of dynamics. The key to the solution is to find nontrivial symmetries of Hamiltonian systems. After Poincare’s work it became clear that topological considerations and the analysis of resonance phenomena play a crucial role in the problem on the existence of symmetry fields and nontrivial conservation laws.