zbMATH — the first resource for mathematics

Symmetry reduction of the three-body problem based on Euler angles. (English) Zbl 1387.70007
Summary: We consider the classical three-body problem with an arbitrary pair potential which depends on the inter-body distance. A general three-body configuration is set by three “radial” and three angular variables which determine the shape and orientation, respectively, of a triangle with the three bodies located at the vertices. The radial variables are given by the distances between a reference body and the other two, and by the angle at the reference body between the other two. Such radial variables set the potential energy of the system, and they are reminiscent of the inter-body distance in the two-body problem. On the other hand, the angular variables are the Euler angles relative to a rigid rotation of the triangle, and they are analogous to the polar and azimuthal angles of the vector between the two bodies in the two-body problem. We show that the rotational symmetry allows us to obtain a closed set of eight Hamilton equations of motion, whose generalized coordinates are the three radial variables and one additional angle, for which we provide the following geometrical interpretation. We consider the plane through a reference body, which is orthogonal to the line between the reference and the second body. We show that the angular variable above is the angle between the plane projection of the angular-momentum vector and the projection of the radius between the reference and the third body.{
©2018 American Institute of Physics}
70F07 Three-body problems
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
Full Text: DOI
[1] Newton, I., Philosophiae Naturalis Principia Mathematica, 1687, (1871), James Maclehouse
[2] Gutzwiller, M. C., Moon-Earth-Sun: The oldest three-body problem, Rev. Mod. Phys., 70, 2, 589, (1998)
[3] Bowditch, N., The American Practical Navigator, (2002), National Imagery and Mapping Agency · JFM 37.0975.10
[4] Maskelyne, N., Maskelyne Notebook: Rough Drafts of Solutions of Various Astronomical Problems, 1750-1772
[5] Maskelyne, N., Journal of Voyage to St Helena, (1761)
[6] Poincaré, H., Les Méthodes Nouvelles de la Mécanique Céleste, (1892), Gauthier-Villars · JFM 24.1130.01
[7] Szebehely, V.; Bettis, D. G.; Lecar, M., Recent developments of integrating the gravitational problem of N-bodies, Gravitational N-Body Problem: Proceedings of the Iau Colloquium No. 10 Held in Cambridge, England August 12-15, 1970, 136, (1972), Reidel Publishing Company
[8] Lagrange, J. L., Essai sur le Problème des Trois Corps, (1873), Gauthier-Villars
[9] Jacobi, P. M., Sur l’élimination des noeuds dans le problème des trois corps, Astron. Nachr., 20, 6, 81, (1843)
[10] Radau, R., Sur une transformation des équations différentielles de la dynamique, Ann. Sci. Ec. Norm. Super., 5, 311, (1868) · JFM 01.0321.05
[11] Bennett, T. L., On the reduction of the problem of n bodies, Mess. Math., 34, 8, 113, (1904)
[12] Van Kampen, E. R.; Wintner, A., On a symmetrical canonical reduction of the problem of three bodies, Am. J. Math., 59, 1, 153, (1937) · JFM 63.0732.01
[13] Malige, F.; Robutel, P.; Laskar, J., Partial reduction in the n-body planetary problem using the angular momentum integral, Celestial Mech. Dyn. Astron., 84, 3, 283, (2002) · Zbl 1026.70016
[14] Hsiang, W.-Y.; Straume, E., Kinematic geometry of triangles and the study of the three-body problem, Lobachevskii J. Math., 25, 9, (2007) · Zbl 1118.53011
[15] Sydnes, L., Geometric reduction of the three-body problem, Lobachevskii J. Math., 34, 4, 332, (2013) · Zbl 1332.70013
[16] Goldstein, H.; Poole, C. P.; Safko, J. L., Classical Mechanics, (2004), Pearson
[17] Poincaré, H., Œuvres, 496-498, (1952), Gauthier-Villars
[18] Morbidelli, A., Modern Celestial Mechanics: Aspects of Solar System Dynamics, (2002), Taylor & Francis · Zbl 1411.70019
[19] Courant, R., Methods of Mathematical Physics, (1966), Interscience
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.