Liouville's Equation - Part Three - Symmetries of the Linearized Equation

Details Liouville's Equation - Part Three - Symmetries of the Linearized Equation Liouville's Equation - Part Three - Symmetries of the Linearized Equation

Apr 1989

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989a%26a...214...92s&link_type=abstract

Astronomy and Astrophysics, Vol. 214, NO.1/2/APR(II), P. 92, 1989

Astronomy and Astrophysics

Astrophysics

7

Scientific paper

Let the linearized Liouville-Poisson equation be i∂f/∂t = A f,f = f(q, p), f, p = phase coordinates. A on f's is not a hermitian operator. However, an eigenvalue equation, A fω = ωfω, with real ω's and non-orthogonal eigenfunctions can be set up. For spherically symmetric potentials A and A2 have 0(3) symmetry. There exists an angular momentum operator, Ji, which commutes with A. This classifies the eigenfunctions into classes specified by a pair of eigennumbers (j, m) belonging to {J2,,Jz}. This in turn enables one to separate the dependence of the eigenfunctions on the direction angles of (q, p) and reduce the six dimensional phase space problem into a two dimensional one in terms of the magnitudes (q, p).

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