This paper addresses the problem of the motion of an unbalanced circular foil in the field of a point source. Equations of joint motion of the source and the foil are constructed. It is shown that in the case of a fixed source of constant intensity the equations of motion of the foil are Hamiltonian. In addition, in the case of a balanced circular foil the equations of motion are integrable. A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that the equations of motion of the unbalanced foil are nonintegrable.

This paper investigates the possibility of the motion control of a ball with a pendulum mechanism with non-holonomic constraints using gaits — the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.