Abstract
In the category of motions preserving the angular momentum direction, Gorringe and Leach exhibited two classes of differential equations having elliptical orbits. After enlarging slightly these classes, we show that they are related by a duality correspondence of the Arnold–Vassiliev type. The specific associated conserved quantities (Laplace–Runge–Lenz vector and Fradkin–Jauch–Hill tensor) are then dual reflections of each other.
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Arnold V.I.: Huygens and Barrow, Newton and Hooke. Birkhäuser, Basel (1990)
Arnold, V.I., Vassiliev, V.A.: Newton’s principia read 300 years later. Not. Am. Math. Soc. 36, 1148–1154 (1989); 37, 144 (addendum) (1990)
Bertrand J.: Théorème relatif au mouvement d’un point attiré vers un centre fixe. C. R. Acad. Sci. 77, 849–853 (1873)
Bohlin K.: Note sur le problème des deux corps et sur une intégration nouvelle dans le problème des trois corps. Bull. Astron. 28, 113–119 (1911)
Collas P.: Equivalent potentials in classical mechanics. J. Math. Phys. 22, 2512–2517 (1981)
Gorringe V.M., Leach P.G.L.: Conserved vectors for the autonomous system \({{(\overset{..}{\bf r}+g(r,\theta) \hat{\bf r}+h(r,\theta )\hat{\theta}={\bf 0})}}\). Phys. D 27, 243–248 (1987)
Gorringe V.M., Leach P.G.L.: Hamilton-like vectors for a class of Kepler problems with a force proportional to the velocity. Celest. Mech. 41, 125–130 (1988)
Gorringe V.M., Leach P.G.L.: Conserved vectors and orbit equations for autonomous systems governed by the equation of motion \({{(\overset {..}{\bf r}+f\overset{.}{\bf r}+g{\bf r} = {\bf 0})}}\). Am. J. Phys. 57, 432–435 (1989)
Gorringe V.M., Leach P.G.L.: Kepler’s third law and the oscillator’s isochronism. Am. J. Phys. 61, 991–995 (1993)
Grandati, Y., Bérard, A., and Mohrbach H.: Bohlin–Arnold–Vassiliev duality and conserved quantities, preprint arXiv 0803.2610v2
Grandati Y., Bérard A., Menas F.: Inverse problem and Bertrand’s theorem. Am. J. Phys. 76, 782–787 (2008)
Hojman S.A., Chayet S., Núñez D., Roque M.A.: An algorithm to relate general solutions of different bidimensional problems. J. Math. Phys. 32, 1491–1497 (1991)
Jezewski D., Mittelman D.: Integrals of motion for the classical two-body problem with drag. Int. J. Nonlinear. Mech. 18, 119–124 (1983)
Kasner, E.:Differential-Geometric Aspects of Dynamics. AMS, Providence, RI (1913)
Leach P.G.L.: The first integrals and orbit equation for the Kepler problem with drag. J. Phys. A Math Gen. 20, 1997–2002 (1987)
Leach P.G.L., Flessas G.P.: Generalisations of the Laplace–Runge–Lenz vector. J. Nonlinear Math. Phys. 10(3), 340–423 (2003)
Levi-Civita T.: Sur la résolution qualitative du problème restreint des trois corps. Acta Math. 30, 305–327 (1906)
Mittelman D., Jezewski D.: An analytic solution to the classical two-body problem with drag. Celest. Mech. 28, 401–413 (1982)
Nersessian A., Ter-Antonyan V., Tsulaia M.: A note on quantum Bohlin transformation. Mod. Phys. Lett. 11, 1605–1610 (1996)
Stiefel E., Scheifele G.: Linear and Regular Celestial Mechanics. Springer, Berlin (1971)
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Grandati, Y., Bérard, A. & Mohrbach, H. Duality properties of Gorringe–Leach equations. Celest Mech Dyn Astr 103, 133–141 (2009). https://doi.org/10.1007/s10569-008-9174-1
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DOI: https://doi.org/10.1007/s10569-008-9174-1