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Two‐point characteristic function for the Kepler–Coulomb problem

### Abstract

Hamilton’s two‐point characteristic function *S* (*q* _{2} *t* _{2},*q* _{1} *t* _{1}) designates the extremum value of the action integral between two space–time points. It is thus a solution of the Hamilton–Jacobi equation in two sets of variables which fulfils the interchange condition *S* (*q* _{1} *t* _{1},*q* _{2} *t* _{2}) =−*S* (*q* _{2} *t* _{2},*q* _{1} *t* _{1}). Such functions can be used in the construction of quantum‐mechanical Green’s functions. For the Kepler–Coulomb problem, rotational invariance implies that the characteristic function depends on three configuration variables, say *r* _{1},*r* _{2},*r* _{12}. The existence of an extra constant of the motion, the Runge–Lenz vector, allows a reduction to two independent variables: *x*≡*r* _{1}+*r* _{2}+*r* _{12} and *y*≡*r* _{1}+*r* _{2}−*r* _{12}. A further reduction is made possible by virtue of a scale symmetry connected with Kepler’s third law. The resulting equations are solved by a double Legendre transformation to yield the Kepler–Coulomb characteristic function in implicit functional form. The periodicity of the characteristic function for elliptical orbits can be applied in a novel derivation of Lambert’s theorem.

© 1975 American Institute of Physics

Published online 03 September 2008

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http://aip.metastore.ingenta.com/content/aip/journal/jmp/16/10/10.1063/1.522430

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2008-09-03

2016-09-26

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