The integral invariants of n gyrostats (Q911818)

\textit{R. Cid} and \textit{A. Vigueras} working on the problem of n isolated gyrostats have proved [Celestial Mech. 36, 155-162 (1985; Zbl 0573.70004)] that the motion is controlled by nine first integrals and, provided that the relative angular momenta and kinetic energy of the mobile parts of the system are time-independent, by one more integral which is the Jacobi's integral. The present result generalizes an analogous conclusion of \textit{G. N. Duboshin}'s work [e.g. ibid. 4, 423- 441 (1971; Zbl 0237.70017) and 6, 27-39 (1972; Zbl 0239.70003)], on the motion of n rigid bodies, whose elementary particle interact in accordance to Newton's law or more general laws. Since the n gyrostats are acted upon the internal forces only, which are supposed proportional to the product of all masses obtained in pairs, and as functions of the respective distances, only the individual rotor-translatory motion of the gyrostats is considered, the long-time behaviour of the system as a whole being ignored. Based on these integrals we derive an equal number of integral- invariants, whose role in the study of such a system is, just like of any dynamical system, quite reasonable because they can improve the qualitative approach of its motion. Besides they may be used to check the accuracy of the numerical integration of the equations of motion when other criteria fail or do not exist at all. We also give two other integral-invariants of extreme order, i.e. 1 and n, by particularizing into the case of question, the invariance which characterizes the ``circulation'' of the system in the phase space and the volume occupied by all parts of it.