Normal form and representation theory

Abstract

Representation theory of Lie algebras is called upon to develop a procedure for normalizing a dynamical system with two degrees of freedom in the neighborhood of an equilibrium when the Hamiltonian H(x, y, X, Y) in the coordinates (x, y) and their conjugate momenta (X, Y) is of the type H=(X2+Y2)/2+V(x, y, X, Y), the potential energy V being a sum of homogeneous polynomials in the phase variables of degree strictly greater than 2. The fact that the resulting potential V′ is a polynomial in the new coordinates (x′, y′) and the angular momentum G′=x′Y′−y′X′ implies that the normalization is a rotation in the configuration space from a fixed frame to an ideal frame. The technique is intended for normalizing an Hamiltonian in equilibrium at the origin when the Lie derivative associated with the quadratic part is not semisimple, e.g., the planar restricted problem of three bodies at the equilateral equilibrium L4 when the basic frequencies are equal (Routh’s singular case).