SYMMETRIES AND THE TOPOLOGY OF DYNAMICAL-SYSTEMS WITH 2 DEGREES OF FREEDOM

The problem of geodesic curves on a closed two-dimensional surface and some of its generalizations related with the addition of gyroscopic f orces are considered. The authors study one-parameter groups of symmet ries in the four-dimensional phase space that are generated by vector fields commuting with the original Hamiltonian vector field. If the ge nus of the surface is greater than one, then there are no nontrivial s ymmetries. For a surface of genus one (a two-dimensional torus) it is established that if there is an additional integral polynomial in the velocities, even or odd with respect to each component of the velocity , then there is a polynomial integral of degree one or two. For a surf ace of genus zero examples of nontrivial integrals of degree three and four are given. Fields of symmetries of first and second degree are s tudied. The presence of such symmetries is related to the existence of ignorable cyclic coordinates and separated variables. The influence o f gyroscopic forces on the existence of fields of symmetries with poly nomial components is studied.