On motion of a rigid body about a fixed point with respect to a rotating frame

It is well-known in the Theoretical Mechanics how using the Routh procedure one can reduce the order of the Lagrange equations describing the motion if the cyclic coordinates expressing the symmetry properties of the mechanical system are known [1], [2]. However, if the equations of motion are written in redundant variables then the procedure of reduction is not always obvious. The idea of the reduction for such systems can be traced back to Lyapunov (see [3], p. 353 -355), who proposed to consider a motion with respect to the rotating, in the general case nonuniformly, specially chosen frame in the problem on figures of equilibria of the rotating fluid. The development of the studies of Lyapunov was given in [4].
As it is known the general method of such reduction for equations of Poincaré–Chetayev was proposed by Chetayev [5], [6], see also [7]. However the realisation of the Chetayev's theorem on reduction is not always simple for real systems. In this paper the analogue of the Routh procedure is considered for the problem on motion of mechanical system consisting of the rigid body with the fixed point. The origine of the concept of the reduced (amended) potential is shown. The problem on motion of the affine-deformable body is considered in details.