Computer visualization for the topology of integrable cases in rigid body dynamics

Abstract

In modern applied mathematics computer visualization is often extremely useful for solving concrete mechanical and physical problems. Some methods of topological modeling for visualization can be found, for example, in the book of T.L. Kunii and A.T. Fomenko (1997). Many problems of modern geometry and topology, mathematical physics and mechanics are reduced to the analysis of symmetries of corresponding differential equations. In cases when the group of symmetries is large, it is usually possible to integrate the differential equations, i.e. to find solutions of physical problems in a "direct way". The remarkable relation of this problem with topological bifurcation theory was recently discovered. It turns out that classification of dynamical systems which have "the maximal symmetry group" can be given in terms of one-dimensional and two-dimensional topological objects. Some of these results were obtained on the basis of computer visualization of the set of bifurcations appearing in integrable Hamiltonian systems. The author illustrates this theory by visual material showing the bifurcations in concrete dynamical systems from classical mechanics.