This talk addresses the problem of qualitative study of Lagrangian

systems with cyclic coordinates.

It is well known that the addition to the characteristic function

under the integral of the action

 of the total differential does not change the form of the differential equations corresponding to

 the characteristic function. This is equivalent to adding to the characteristic function of the

 total  derivative in time of the function depending on the generalized coordinates.

Thus "extended" Routh functions are proposed to use for the

separation and qualitative study

 of invariant manifolds (IM)  of the corresponding systems of equations of Routh and Lagrange

 on the basis of an approach similar to the method the Routh-Lyapunov.

Consider the following two problems: Invariant manifolds of the

system with a linear reduced system; Invariant manifolds of Solid

body with a fixed point in the case of Euler.