Infinity manifolds of cubic polynomial Hamiltonian vector fields with 2 degrees of freedom

Abstract

Let X be the Hamiltonian vector field with two degrees of freedom associated to the cubic polynomial Hamiltonian H(x,y,z,w). Using the Poincare compactification we show that all the energy levels of X in R-4 reach the infinity in a surface topologically equivalent to the intersection of the 3-dimensional sphere S-3 = {(x, y, z, w) is an element of R-4 : x(2) + y(2) + z(2) +w(2) = 1} with {(x, y, z, w) is an element of R-4 : H-3(x, y, z, w) = 0}, where H-3 denotes the homogeneous part of degree 3 of H. Such a surface is called the Infinity Manifold associated to H. In this paper we describe all possible infinity manifolds of cubic polynomial Hamiltonian vector fields with 2 degrees of freedom. Our method is general, but since actual computations can become very cumbersome, we work out in detail only three out of ten possible cases.