On the Geometry of Hamiltonian Symmetries

The problem of integrating equations of mechanics is the most important task of mathematics and mechanics. Before Poincare's book "Curves Defined by Differential Equations", integration tasks were considered as analytical problems of finding formulas for solutions of the equation of motion. After the appearance of this book, it became clear that the integration problems are related to the behavior of the trajectories as a whole. This, of course, stimulated methods of qualitative theory of differential equations.

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