Genericity of the non-periodic solutions of the central force problem

Details Genericity of the non-periodic solutions of the central force problem Genericity of the non-periodic solutions of the central force problem

Mar 1990

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1990ap%26ss.165...95n&link_type=abstract

Astrophysics and Space Science (ISSN 0004-640X), vol. 165, no. 1, March 1990, p. 95-99. Research supported by CIRIT.

Physics

Field Theory (Physics), Particle Motion, Energy Levels, Equations Of Motion, Hamiltonian Functions, Liouville Theorem

Scientific paper

The classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power alpha of the distance r is studied. For negative energy and alpha of (0, 2), each energy level I(h) is foliated by the invariant tori I(hc) of constant angular momentum c and, by Liouville-Arnold's theorem, the flow on each I(hc) is conjugated to a linear flow of rotation number rho sub h(c). A well-known result asserts that if rho sub h(c) is required to be rational for every value of h and c, then alpha must be equal to one (Kepler's problem). In this paper, for almost every alpha of (0, 2), rho sub h(c) is a nonconstant continuous function of c, for every h of less than 0. Motion under central potentials is generically nonperiodic.

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