Research On The Estimation Of Large-Scale Periodic Orbits On Hamiltonian Systems

The homotopy classes of the periodic orbits of an energy level surface ofHamiltonian systems can describe the classes of the large-scale periodic orbits of thesystems. This need the calculation of the fundamental group π1 (K ) on the energylevel surfaces. It's so difficult to calculate the fundamental group π1 (K ) .We takeplace of it with the first homology group H1 (K ) .The topology properties of anenergy level surface are determined by the that of the phase space and the large- scaleproperties of the Hamiltonian. These properties are used for estimate of the majorantof the classes of the large-scale periodic orbits of the Hamiltonian. We make the proofexistent improved with the instrument aboved, and gain the new proof of the basictheorem of the paper. The example of the application of the theorem is the motion ofthe rigid body that there is external potential (Kovalevskaya case).We give a simplemethod to judge the index of the nondegenerate critical points and get the materialestimate of the majorant of the classes of the large-scale periodic orbits of theHamiltonian. Linked with the estimate of the minorant of the numbers of the steadyperiodic root locus, we gain the material estimate of the numbers of the large-scalesteady periodic root locus of the motion of the rigid body of Kovalevskaya case. Thefirst innovation of the paper is in the improvement on the existent proof of thetheorem. The second innovation of the paper is in the calculation of the numbers ofthe classes of the large-scale periodic orbits of the the motion of the rigid body ofKovalevskaya case .