Seminar talk, 25 March 2026

Speaker: Alexander Kuleshov

Title: Exact solutions of some problems of rigid body dynamics

Abstract:
Interest in integrable problems in mechanics has never waned. Finding new integrable cases of differential equations of motion for various mechanical systems, as well as finding solutions in quadratures for these cases, is one of the main problem of theoretical mechanics. The problem of exact integration of differential equations of motion has several aspects. The geometric aspect is associated with the qualitative study of the regular behavior of the trajectories of integrable systems. The constructive aspect is associated with finding the conditions under which an algorithm for explicit solving differential equations using quadratures can be specified. In this regard, another important aspect of the range of issues under consideration arises: the explicit solution of systems of differential equations. For certain classes of differential equations, relying on their specific structure, special methods can be used. An example here is the broad and important class of linear differential equations. The study of many problems in mechanics and mathematical physics reduces to solving a second-order linear homogeneous differential equation. If, by changing the independent variable, it is possible to reduce the corresponding second-order linear differential equation to an equation with rational coefficients, then the necessary and sufficient for solvability by quadratures for such an equation are determined by the so-called Kovacic algorithm. In 1986, the American mathematician J. Kovacic presented an algorithm for finding Liouvillian solutions of a second-order linear homogeneous differential equation with rational coefficients. If the differential equation has no Liouvillian solution, the algorithm also allows one to establish this fact.

This talk will discuss the application of the Kovacic algorithm to investigate the existence of Liouvillian solutions in the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane and on a perfectly rough sphere. It will also discuss the application of the algorithm to investigate the existence of Liouvillian solutions in the problem of motion of a heavy homogeneous ball on a fixed perfectly rough surface of revolution. The existence of Liouvillian solutions in the Hess case of the problem of motion of a heavy rigid body with a fixed point is also analyzed.