Mechanical systems with rapidly vibrating constraints

We consider a natural Lagrangian system on which a supplementary holonomic nonstationary constraint is imposed; the dependence on time is included in this constraint by the parameter performing rapid periodic oscillations. Such a constraint is called a vibrating constraint. The equations of motion of a system with a vibrating constraint are obtained in the form of Hamilton's equations. It is shown that the structure of the Hamiltonian of the system has a special form convenient for deriving the averaged equations. Usage of the averaging method allows us to obtain the limit equations of motion of the system as the frequency of vibrations tends to infinity and to prove the uniform convergence of the solutions of Hamilton's equations to the solutions of the limit equations on a finite interval of time. Some examples are discussed.