  The first part of the article is, in fact, the classical Routh method
delivered in the language of contemporary theory of Lagrangian systems. But the
Routh method deals only with concrete equations and, therefore, can be applied
only in the case when the configuration spaces of the initial and the reduced
systems are open submanifolds in Euclidean spaces. The global approach gives a
possibility to find the structure of these manifolds in the general case and
also to reveal some properties of the reduced system, first of all, the
existence for this system of a global Lagrange function. We use the notion of a
mechanical system introduced by S. Smale. The described method is applied to
the global reduction in the problem of the motion of a rigid body having a
fixed point in the potential force field with an axial symmetry. We present the
complete proof of the theorem formulated by G.V. Kolosov on the equivalence of
the reduced system in this case to the problem of the motion of a material
point over an ellipsoid and also some corollaries of this theorem based on the
results of L.A. Lyusternik and L.G. Shnirelman.
