Small oscillations of a heavy rigid body around a fixed point and certain cases of existence of “linear integrals”: PMM vol. 37, n≗3, 1973, pp. 544/2–547

Abstract

A large amount of literature (for example, see the surveys in [1, 2]) has been devoted to the motion of a heavy rigid body around a fixed point. The present paper is based on a simple concept, permitting us to use the methods of investigating systems with nonlinearly connected oscillators [3–5] for the study of a specific Hamiltonian system with three degrees of freedom, namely, a rigid body moving around a fixed point. This concept is that when no constraints are imposed on the initial conditions, excluding small motions near the equilibrium position (and such motions are all the general cases of integrability: Euler-Poinsot, Lagrange-Poisson, Kowalewska, in which no constraints whatsoever are imposed on the initial conditions.and the majority of the well-known special cases of integrability), we can at first study the small oscillations near the equilibrium point. Then the integrals of the problem of small oscillations can be used as “sprouts” of the integrals for the complete nonlinear problem of obtaining the integrals of the original problem of the motion of the heavy rigid body around a fixed point. We consider the problem of the linear integrals [6–9] from this point of view. The statement of the problem of the existence of conditions for the linear integrals for the equations of motion of a heavy body is due to Chaplygin [6]. The investigation in [6] was developed in [8], and under certain constraints widely used in papers of the dynamics of a rigid body. the question of the existence conditions and the form of the linear integrals received an exhaustive examination in [9]. It is of interest to understand the nature of at least some of those cases when in the complex nonlinear system of Euler-Poisson equations there arise simple linear relations between the variables, which are preserved during the whole time of the motion. It is natural to relate this with some “degeneracy” of the system of equations, arising for specific values of the system parameters (and sometimes also for specific initial conditions). It is well known [10] that if the characteristic equation of the linear (the linearized) system has a zero root, then a linear integral appears. Below we have shown how we can obtain the existence conditions and the form of the linear integrals in certain well known cases from the fact that a zero “frequency” occurs in the equations of small oscillations of the rigid body next to a stable equilibrium position. We make note also of other additional degeneracies (resonance relations) which occur in certain cases. The question being considered is closely related to Poincare s question on the existence conditions for the “fourth algebraic integral” in the problem of motion of a rigid body around a fixed point (see [1, 2]).