Families of planar orbits generated by homogeneous potentials

Details Families of planar orbits generated by homogeneous potentials Families of planar orbits generated by homogeneous potentials

Nov 1993

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1993cemda..57..461b&link_type=abstract

Celestial Mechanics and Dynamical Astronomy (ISSN 0923-2958), vol. 57, no. 3, p. 461-472

Astronomy and Astrophysics

Astronomy

6

Orbital Mechanics, Orbits, Particle Motion, Potential Fields, Celestial Mechanics, Centripetal Force, Equations Of Motion, Partial Differential Equations

Scientific paper

The second order partial differential equation which relates the potential V(x, y) to a family of planar orbits f(x,y) = c generated by this potential is applied for the case of homogeneous V(x,y) of any degree m. It is shown that, if the function f(x,y) is also homogeneous, there exists, for each m, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. If f(x,y) is not homogeneous in general, there is not a homogeneous potential which can create the given family; only if gamma = fy/f(sub x satisfies two conditions, a homogeneous potential does not exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.

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