Infinity Manifolds of Cubic Polynomial Hamiltonian Vector Fields with 2 Degrees of Freedom

Details Infinity Manifolds of Cubic Polynomial Hamiltonian Vector Fields with 2 Degrees of Freedom Infinity Manifolds of Cubic Polynomial Hamiltonian Vector Fields with 2 Degrees of Freedom

Oct 2000

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000hscm.conf..138f&link_type=abstract

HAMILTONIAN SYSTEMS AND CELESTIAL MECHANICS (HAMSYS-98). Proceedings of the III International Symposium. Held 7-11 December 1998

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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 Poincaré compactification we show that all the energy levels of X in R4 reach the infinity in a surface topologically equivalent to the intersection of the 3-dimensional sphere S3 = {(x, y, z, w) ∈ R4 : x2 + y2 + z2 + w2 = 1} with {(x, y, z, w) ∈ R4 : H3 (x, y, z,w) = 0}, where H3 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.

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