Intrinsic orbital coordinates on a Hamiltonian surface

Details Intrinsic orbital coordinates on a Hamiltonian surface Intrinsic orbital coordinates on a Hamiltonian surface

Mar 1989

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989cemda..47...57h&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, Volume 47, Issue 1, pp.57-85

Astronomy and Astrophysics

Astronomy

Scientific paper

A reversible dynamical system with two degrees-of-freedom is reduced to a second-order, Hamiltonian system under a change of independent variable. In certain circumstances, the reduced order system may be integrated following an orthogonal curvilinear transformation from Cartesian x,y to intrinsic orbital coordinates ɛ, η. Solutions for the orbit position and true time variables are expressed by: [ x = f(ξ ,η ),{{ }}y = g(ξ ,η ),{{ }}dt = ± left[ {{?_ξ ^{{2}} {ie} + ?_η ^2 ?}/{2(? + U)}} righ ]1446 1040 where U is the potential function, and z is the new independent variable. The functions f, g may be expressed by quadratures when the metric coefficients {er},{ie} are specified. Two second-order, partial differential equations specify {er}, {ie} and Hamiltonian {tH}. Auxiliary conditions are needed because the solutions are underdetermined. For example, both sets of curvilinear coordinate lines are orbits when certain dynamical compatibility conditions between U and {ie} (or {er}) are satisfied. Alternatively, when orbits cross the parametric curves, the auxiliary condition {er} = {ie} specifies a conformal transformation, and the partial differential equation for {tH} may be reduced to an ordinary differential equation for the orbit curve. In either case, integrability is guaranteed for Lionville dynamical systems. Specific applications are presented to illustrate direct solution for the orbit (e.g., two fixed centers) and inverse solution for the potential.

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