Dep. of Mech. and Math., Moscow Lomonosov State University
119 899 Moscow, Russia.

E-mails: nekhoros@mech.math.msu.su
             nekhoroshev@berlioz.mat.unimi.it

Generalizations of Gordon's theorem

Abstract:
Gordon's theorem claims that given Hamiltonian system all of whose solutions are periodic, the period of solution depend only on the value of the Hamiltonian function on the trajectory of this solution. Generalizations are obtained for the case of invariant isotropic tori of arbitrary dimension k (rather than k=1), which fiber either all phase space or some submanifold of this. One supposes that system has some collection of k first integrals in involution, such that the corresponding vectorfields are tangent to these tori. Then frequencies of quasiperiodic motion on such a torus are depend only of values of these first integrals on the torus. This is true also for the circular action functions, but sufficient conditions in last case are essentially more weak.