Topological analysis of natural systems with quadratic integrals (Q1085994)

A method for the qualitative investigation of natural systems is considered, which enables integrals that are quadratic in velocities to be used based on a description of the surfaces of integral levels in the phase space. The concept of the normalized quadratic integral and the connection between its presence and the separation of positional variables is established. The method of topological analysis, proposed by \textit{S. Smale} [Invent. Math. 10, 305-331 (1970; Zbl 0202.232)] for systems with linear integrals, is extended to problems containing quadratic integrals. Surfaces of the integral levels, their bifurcations, and the regions of possible motion for given values of the integrals are described. As an example of their application, the motion of a solid in a potential field is considered.