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The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if =f
y
/f
x
satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases. 相似文献
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S. Grigoriadou G. Bozis B. Elmabsout 《Celestial Mechanics and Dynamical Astronomy》1999,74(3):211-221
Szebehely’s equation is a first order partial differential equation relating a given family of orbits f (x, y) = q traced
by a unit mass material point, the total energy E=E(f), and the unknown potential V=V (x, y) which produces the family. Although
linear in V, this equation cannot generally be solved. In this paper we develop the reasoning for finding several cases for
which Szebehely’s equation can be solved by quadratures.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
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