On a pendulum motion in multi-dimensional space. Part 1. Dynamical systems

In the proposed cycle of work, we study the equations of the motion of dynamically symmetric fixed $n$-dimensional rigid bodies-pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free $n$-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. In thit work, we derive the general multi-dimensional dynamic equations of the systems under study.