Conservation of integrals of motion for small changes of Hamilton's function in some cases of integrability of the equations of motion of a gyrostat: PMM vol. 35. no. 4, 1971, pp. 718–722

Abstract

Motion of a gyrostat is considered. The equations of motion are written in the Hamilton form and the change in the integrals of motion in the cases of Zhukovskii and Lagrange resulting from the Hamilton function undergoing small variations is studied. Let the mechanical system under investigation depend on a set of parameters and let it be integrable for some definite values of these parameters. Study of the motion of this system in the case when the values of the parameters are changed the system is no longer integrable, appears to be of interest. The solution of this problem involves overcoming certain fundamental difficulties connected with the problem of small denominators. In the case when the system is Hamiltonian and the changes in the values of parameters are small, these difficulties have been overcome using the method proposed by Kolmogorov and Arnol'd in [1 and 2]. Arnol'd's solution [3] of the problem of a rapidly rotating, heavy, asymmetric rigid body with a fixed point, serves to illustrate the application of this method to the rigid body dynamics.