Comparability of the motion of gravitating systems with respect to its recurrence in time (Q1071534)

The isochronism, i.e. the mutual comparability with respect to time recurrence of the kinetic energy and motion of the system of n bodies is the necessary condition for the Lagrange stability of motion of such a system. The motion of an n-body system is determind by a 6n-dimensional vector function, and its kinetic energy by a scalar function depending on the last 3n components of the velocities of motion of the system. The isochronism of the 6n-dimensional vector function g(t,q) and scalar function \(T(g(t,q))=T(t)\) is characterized by the fact that the Lagrange stability is a special property of the motion of an n-body system. Since the Lagrange stability represents one of the possible forms of stability of the motion, it is possible for the kinetic energy to be minimal in some sense along the trajectory of the Lagrange-stable motion of an n- body system. In particular, this is the case for the kinetic energy of a recurrent, almost periodic and periodic motion of an n-body system. In the cases discussed above the kinetic energy is minimal in the Birkhoff sense.