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.

We consider some aspects of Hamiltonianicity of two problems of nonholonomic mechanics, namely, the Chaplygin's ball problem and the Veselova problem. Representations for these two problems have been found in the form of generalized Chaplygin systems, integrable with the method of reducing multiplier. We also specify the algebraic form of the Poisson brackets, with which, after appropriate time substitution, the equations of motion for the stated problems can be represented. We consider generalizations of the two stated problems and offer new realizations of nonholonomic constraints. Some nonholonomic systems are shown, which have the invariant measure and a sufcient number of rst integrals; for such systems, the question of Hamiltonianicity is still open, even after the time substitution.