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Integrability of Motions in Galactic Potentials

Published online by Cambridge University Press:  25 April 2016

Paul Cleary
Paul Cleary*
Affiliation:
Department of Mathematics, Monash University
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Abstract

The dynamics exhibited by systems, such as galaxies, are dominated by the isolating integrals of the motion. The most common are the energy and angular momentum integrals. The motions in a system with a full complement of isolating integrals are regular, that is, periodic or quasi-periodic. Such a system is integrable. If there is a deficiency in the number of integrals, then the motions are chaotic. There is a fundamental quantative difference in the motion, depending on the number of integrals. A technique, called Generalised Painlevé analysis, based on complex variable theory allows the user to determine if a system is integrable. Two new integrable cases of the Henon-Heiles system are presented, bringing the total number of such integrable potentials to five. It is highly probable that there are no further integrable cases of the Henon-Heiles potential. Five cases of the quartic Verhulst potential, defined by certain restrictions on the coefficients, which are found to be integrable are summarised.

Type
Contributions
Information
Publications of the Astronomical Society of Australia , Volume 6 , Issue 4 , 1986 , pp. 453 - 458
Copyright
Copyright © Astronomical Society of Australia 1985

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References

Ablowitz, M. J., Ramani, A. and Segur, H., 1980, J. Math. Phys., 21, 175.Google Scholar
Aizawa, Y. and Saitô, N., 1972, J. Phys. Soc. Jpn., 32, 1636.Google Scholar
Bertrand, J., 1852, J. Math.(i), XVII 121.Google Scholar
Bountis, T., Segur, H. and Vivaldi, F., 1982, Phys. Rey., 25A, Third series, 1257.CrossRefGoogle Scholar
Chang, Y. F., Tabor, M. and Weiss, J., 1982, J. Math. Phys., 23(4), 531.Google Scholar
Darboux, G., 1901, Arch. Néerl.(ii), VI, 371.Google Scholar
Dorizzi, B., Grammaticos, B. and Ramani, A., 1983, J. Math. Phys., 24(9), 2289.Google Scholar
Grammaticos, B., Dorizzi, B. and Ramani, A., 1983, J. Math. Phys., 24(9), 2289.Google Scholar
Henon, M. and Heiles, C., 1964, Astron. J., 69, 73.Google Scholar
Henon, M. in: ‘Chaotic Behaviour of Deterministic Systems’, Les Houches, Session XXXVI, 1981. eds. Iooss, G., Helleman, R. H. G. and Stora, R.. (North-Holland Publishing Company, 1983).Google Scholar
Kovalevskaya, S., 1889, Acta. Math., 12, 177; 1889, 14, 81.Google Scholar
Segur, H., ‘Solitons and Inverse Scattering Transform’, Lectures given at the International School of Physics, ‘Enrico Fermi’, Varenna, Italy, July 7–19, 1980.Google Scholar
Tabor, M., The Onset of Chaos in Dynamical Systems, Advances in Chemical Physics, Vol. 46. (Wiley, New York, 1981).Google Scholar
Tabor, M., 1984, Nature, 310, 277.Google Scholar
Tabor, M. and Weiss, J., 1981, Phys. Rev. A., 24, 2157.Google Scholar
Verhulst, F., 1979, Phil. Trans. Roy. Soc. London., 290, 435.Google Scholar
Whittaker, E. T., ‘Analytical Dynamics of Particles’ (Cambridge U. P., Cambridge, 1959), 4th ed., pp. 334–35 (1st ed. 1904).Google Scholar
Yoshida, H., 1983a, Celes. Mech., 31, 363.Google Scholar
Yoshida, H., 1983b, Celes. Mech., 31, 381.Google Scholar