Mechanical Systems with Rapidly Vibrating Constraints

Abstract

We consider a natural Lagrangian system on which an additional holonomic rheonomic constraint is imposed; the time dependence is included in this constraint by a parameter performing rapid periodic oscillations. Such a constraint is said to be a vibrating constraint. The equations of motion are obtained for a system with a vibrating constraint 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 time interval. Some examples are discussed.