The Hamilton-Ostrogradski generalized principle and its application for damping of oscillations

An elastic system with the center of mass able to move freely along a specified direction is considered. The control force acting along this direction is supposed to excite all the oscillation modes. It is required to determine what this force should be so that the center of mass would move to a specified distance in the specified time, attain the necessary velocity, and start uniform and rectilinear motion. No oscillations should be beyond the specified time interval over all the forms under consideration. The generalized Hamilton-Ostrogradski principle is formulated, and the sought control force is determined based on it.