Two‐point characteristic function for the Kepler–Coulomb problem
dc.contributor.author | Blinder, S. M. | en_US |
dc.date.accessioned | 2010-05-06T22:50:01Z | |
dc.date.available | 2010-05-06T22:50:01Z | |
dc.date.issued | 1975-10 | en_US |
dc.identifier.citation | Blinder, S. M. (1975). "Two‐point characteristic function for the Kepler–Coulomb problem." Journal of Mathematical Physics 16(10): 2000-2004. <http://hdl.handle.net/2027.42/70864> | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/70864 | |
dc.description.abstract | Hamilton’s two‐point characteristic function S (q2t2,q1t1) 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 (q1t1,q2t2) =−S (q2t2,q1t1). 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 r1,r2,r12. The existence of an extra constant of the motion, the Runge–Lenz vector, allows a reduction to two independent variables: x≡r1+r2+r12 and y≡r1+r2−r12. 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. | en_US |
dc.format.extent | 3102 bytes | |
dc.format.extent | 282043 bytes | |
dc.format.mimetype | text/plain | |
dc.format.mimetype | application/pdf | |
dc.publisher | The American Institute of Physics | en_US |
dc.rights | © The American Institute of Physics | en_US |
dc.title | Two‐point characteristic function for the Kepler–Coulomb problem | en_US |
dc.type | Article | en_US |
dc.subject.hlbsecondlevel | Physics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48104 | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/70864/2/JMAPAQ-16-10-2000-1.pdf | |
dc.identifier.doi | 10.1063/1.522430 | en_US |
dc.identifier.source | Journal of Mathematical Physics | en_US |
dc.identifier.citedreference | See, for example, J. L. Singe, “Classical Dynamics,” in Handbuch der Physik Vol. III∕1, edited by S. Flügge (Springer, Berlin, 1960), p. 117ff. | en_US |
dc.identifier.citedreference | R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw‐Hill, New York, 1965); S. M. Blinder, Foundations of Quantum Dynamics (Academic, London, 1974), Chap. 6; S. M. Blinder, “Configuration‐Space Green’s Functions,” in International Review of Science, Vol. I, Theoretical Chemistry (Butterworths, London, 1975). | en_US |
dc.identifier.citedreference | For the present status of the problem, see M. J. Goovaerts and J. T. Devreese, J. Math. Phys. 13, 1070 (1972); R. G. Storer, J. Math. Phys. 9, 964 (1968). | en_US |
dc.identifier.citedreference | L. Hostler, J. Math. Phys. 5, 591 (1964). The two Green’s functions are related by Fourier transformation as follows: K(r2,r1,t) = limϵ→02π∫−∞∞[G(r2,r1,E+ie)−G(r2,r1,E−iϵ)]e−iEt∕ℏdE.K(r2,r1,t) = limϵ→02π∫−∞∞[G(r2,r1,E+ie)−G(r2,r1,E−iϵ)]e−iEt∕ℏdE. | en_US |
dc.identifier.citedreference | C. Runge, Vector Analysis (Dutton, New York, 1919), p. 79; W. Lenz, Z. Phys. 24, 197 (1924); W. Pauli, Z. Phys. 36, 336 (1926) [English translation in B. L. van der Waerden, Sources of Quantum Mechanics (Dover, New York, 1968), p. 387]. See also articles by H. V. McIntosh (p. 75) and C. E. Wulfman (p. 145) in Group Theory and its Applications, Vol. II, edited by E. M. Loebl (Academic, New York, 1971). | en_US |
dc.identifier.citedreference | The properties of the Runge‐Lenz vector can be developed as follows. Start with Newton’s second law for a particle in a Colulomb field: dpdt = −Ze2r3r. Then L×dpdt = −Ze2r3L×r = −Ze2mr3(r×drdt)×r. This works out to ddt(L×p+Ze2mu) = 0, showing that A is a constant of the motion. The equation of the orbit is obtained from A⋅r = Ar cosθ = −(Ze2m)−1L2+r, r = (Ze2m)−1L2∕(1−A cosθ), which represents a conic section. The vector A is directed towards the aphelion of the orbit; its magnitude equals the eccentricity. | en_US |
dc.identifier.citedreference | L. Hostler, J. Math. Phys. 8, 642 (1967). | en_US |
dc.identifier.citedreference | This also applies w.r.t. the original position variables: S(ζ2r1,ζ2r2,ζ3t) = ζS(r,r2,t). Newton’s second law for a Coulomb force is likewise invariant under the substitution r→ζ2r,r→ζ2r, t→ζ3t.t→ζ3t. This implies Kepler’s third law of planetary motion, that the period of an orbit is proportional to the three‐halves power of its linear dimension. | en_US |
dc.identifier.citedreference | See, for example, E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge, U.P., Cambridge, 1965), 4th Ed., p. 91–92. | en_US |
dc.identifier.citedreference | See, for example, H. Goldstein, Classical Mechanics (Addison‐Wesley, Cambridge, Mass., 1950), p. 299ff. | en_US |
dc.owningcollname | Physics, Department of |
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