On the Poincaré problem of the third integral of the equations of rotation of a heavy asymmetric top

The Poincaré problem of the existence of the third integral of the equations of motion of a heavy asymmetric rigid body with a fixed point, which is independent of the integrals of energy and the area integral and which is represented as a series in powers of a small parameter with coefficients in the form of single-valued analytic functions on a six-dimensional phase space, is considered. The small parameter is the ratio of the distance from the center of mass to the suspension point to the characteristic size of the rigid body. This problem was formulated by Poincaré in the fifth chapter of his famous "New Methods of Celestial Mechanics". If we additionally require that the third integral is in involution with the area integral, then the answer to the Poincaré problem is negative (as was shown by the author back in 1975). In this paper, the Poincaré problem is solved in the original general formulation (without the assumption that the Poisson bracket is zero): if the body is dynamically asymmetric, then the third single-valued integral does not exist. The proof uses the Poincaré method, supplemented by some new ideas, as well as a more thorough analysis of the expansion of the perturbing function in a Fourier series in angular variables.

asymmetric top, symmetry group, Hamiltonian system, action-angle variables, secular set, key set, analytic integrals, polynomial integrals, reduces system