Computing collinear 4-Body Problem central configurations with given masses

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March 2013, 33(3): 1215-1230. doi: 10.3934/dcds.2013.33.1215

Computing collinear 4-Body Problem central configurations with given masses

Eduardo Piña ^1,

1.	Professor "Eugenio Méndez Docurro 2011", de la Escuela Superior de Física y Matemáticas del IPN, Zacatenco, 07738 México, D F, Mexico

Received April 2011 Revised December 2011 Published October 2012

Abstract
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An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented by using an orthocentric tetrahedron, which edge lengths are function of given masses. Each mass is placed at the corresponding vertex of the tetrahedron. The center of mass (and orthocenter) of the tetrahedron is at the origin of coordinates. The initial position of the tetrahedron is placed with two pairs of vertices each in a coordinate plan, the lines joining any pair of them parallel to a coordinate axis, the center of masses of each and the center of mass of the four on one coordinate axis. From this original position the tetrahedron is rotated by two angles around the center of mass until the direction of configuration coincides with one axis of coordinates. The four coordinates of the vertices of the tetrahedron along this direction determine the central configuration by finding the two angles corresponding to it. The twelve possible configurations predicted by Moulton's theorem are computed for a particular mass choice.

Keywords: Four-body problem, collinear central configuration..

Mathematics Subject Classification: Primary: 70F10; Secondary: 70F1.

Citation: Eduardo Piña. Computing collinear 4-Body Problem central configurations with given masses. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1215-1230. doi: 10.3934/dcds.2013.33.1215

References:

[1]	D. G. Saari, "Collisions, Rings, and Other Newtonian N-Body Problems," American Mathematical Society, Providence, Rhode Island, 2005.
[2]	F. R. Moulton, The straight line solutions of the problem of N-bodies, Annals of Mathematics, 12 (1910), 1-17. doi: 10.2307/2007159.
[3]	R. Lehmann-Filhés, Ueber swei Fälle des Vielkörpersproblems, Astr. Nachr, 127 (1891), 137-144.
[4]	E. Piña, Algorithm for planar Four-Body Problem central configurations with given masses, preprint, arXiv:1006.2430
[5]	E. Piña, New coordinates for the four-body problem, Rev. Mex. Fis., 56 (2010), 195-203.
[6]	E. Piña and P. Lonngi, Central configurations for the planar Newtonian four-body problem, Cel. Mech. & Dyn. Astr., 108 (2010), 73-93. doi: 10.1007/s10569-010-9291-5.
[7]	L. Landau and E. Lifshitz, "Mechanics," Pergamon Press, Reading, 1960.
[8]	J. L. Lagrange, Solutions analytiques de quelques problèmes sur les piramides triangulaires, Nouv. Mem. Acad. Sci. Berlin, (1773), 149-176.
[9]	N. A. Court, Notes on the orthocentric tetrahedra, The American Mathematical Monthly, 41 (1934), 499-523. doi: 10.2307/2300415.
[10]	E. Piña and A. Bengochea, Hyperbolic geometry for the binary collision angles of the three-body problem in the plane, Qualitative Theor. of Dyn. Sys., 8 (2009), 399-417 doi: 10.1007/s12346-010-0009-6.
[11]	C. Simó, El conjunto de bifurcación en el problema espacial de tres cuerpos, in "Acta I Asamblea Nacional de Astronomía y Astrofísica," Instituto de Astrofísica. Univ. de la Laguna. Spain, (1975), 211-217.
[12]	J. V. Jose and E. J. Saletan, "Classical Mechanics, A Contemporary Approach," Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511803772.
[13]	A. Bengochea and E. Piña, The dynamics of saturn, janus and epimetheus as a three-body problem in the plane, Rev. Mex. Fis., 55 (2009), 97-105.
[14]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," $4^{th}$ edition, Cambridge University Press, Cambridge, 1937.
[15]	E. Piña, Rotations with Rodrigues' vector, Eur. J. Phys., 32 (2011), 1171-1178. doi: 10.1088/0143-0807/32/5/005.
[16]	K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," $2^{nd}$ edition, Springer-Verlag, New York, 2009.

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References:

[1]	D. G. Saari, "Collisions, Rings, and Other Newtonian N-Body Problems," American Mathematical Society, Providence, Rhode Island, 2005.
[2]	F. R. Moulton, The straight line solutions of the problem of N-bodies, Annals of Mathematics, 12 (1910), 1-17. doi: 10.2307/2007159.
[3]	R. Lehmann-Filhés, Ueber swei Fälle des Vielkörpersproblems, Astr. Nachr, 127 (1891), 137-144.
[4]	E. Piña, Algorithm for planar Four-Body Problem central configurations with given masses, preprint, arXiv:1006.2430
[5]	E. Piña, New coordinates for the four-body problem, Rev. Mex. Fis., 56 (2010), 195-203.
[6]	E. Piña and P. Lonngi, Central configurations for the planar Newtonian four-body problem, Cel. Mech. & Dyn. Astr., 108 (2010), 73-93. doi: 10.1007/s10569-010-9291-5.
[7]	L. Landau and E. Lifshitz, "Mechanics," Pergamon Press, Reading, 1960.
[8]	J. L. Lagrange, Solutions analytiques de quelques problèmes sur les piramides triangulaires, Nouv. Mem. Acad. Sci. Berlin, (1773), 149-176.
[9]	N. A. Court, Notes on the orthocentric tetrahedra, The American Mathematical Monthly, 41 (1934), 499-523. doi: 10.2307/2300415.
[10]	E. Piña and A. Bengochea, Hyperbolic geometry for the binary collision angles of the three-body problem in the plane, Qualitative Theor. of Dyn. Sys., 8 (2009), 399-417 doi: 10.1007/s12346-010-0009-6.
[11]	C. Simó, El conjunto de bifurcación en el problema espacial de tres cuerpos, in "Acta I Asamblea Nacional de Astronomía y Astrofísica," Instituto de Astrofísica. Univ. de la Laguna. Spain, (1975), 211-217.
[12]	J. V. Jose and E. J. Saletan, "Classical Mechanics, A Contemporary Approach," Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511803772.
[13]	A. Bengochea and E. Piña, The dynamics of saturn, janus and epimetheus as a three-body problem in the plane, Rev. Mex. Fis., 55 (2009), 97-105.
[14]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," $4^{th}$ edition, Cambridge University Press, Cambridge, 1937.
[15]	E. Piña, Rotations with Rodrigues' vector, Eur. J. Phys., 32 (2011), 1171-1178. doi: 10.1088/0143-0807/32/5/005.
[16]	K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," $2^{nd}$ edition, Springer-Verlag, New York, 2009.

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