Exact circular Morse functions, Morse-type inequalities and integrals of Hamiltonian systems (Q913333)

The circular Morse functions (for which the critical points form non- degenerate critical circles) are investigated. It is pointed out, that such functions appear frequently as integrals of motion of classical Hamiltonian systems. The apparatus of a combinatorial character (graphs associated with circular Morse functions) is worked out. In this way the authors represent substantial aspects of the situation when the decomposition to round handles take place. The inequalities of the Morse type are proved for the Morse functions on an arbitrary smooth manifold (of the dimension \(\geq 2):\) it is shown that the lower bounds of the number of critical circles with index \(\lambda\) of a given circular Morse function are determined by the homology groups of the manifold in question. A new proof of the existence of an exact circular Morse function is presented.