# American Institute of Mathematical Sciences

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

## Computing collinear 4-Body Problem central configurations with given masses

 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

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.
Citation: Eduardo Piña. Computing collinear 4-Body Problem central configurations with given masses. Discrete & 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, (2005).   Google Scholar [2] F. R. Moulton, The straight line solutions of the problem of N-bodies,, Annals of Mathematics, 12 (1910), 1.  doi: 10.2307/2007159.  Google Scholar [3] R. Lehmann-Filhés, Ueber swei Fälle des Vielkörpersproblems,, Astr. Nachr, 127 (1891), 137.   Google Scholar [4] E. Piña, Algorithm for planar Four-Body Problem central configurations with given masses,, preprint, ().   Google Scholar [5] E. Piña, New coordinates for the four-body problem,, Rev. Mex. Fis., 56 (2010), 195.   Google Scholar [6] E. Piña and P. Lonngi, Central configurations for the planar Newtonian four-body problem,, Cel. Mech. & Dyn. Astr., 108 (2010), 73.  doi: 10.1007/s10569-010-9291-5.  Google Scholar [7] L. Landau and E. Lifshitz, "Mechanics,", Pergamon Press, (1960).   Google Scholar [8] J. L. Lagrange, Solutions analytiques de quelques problèmes sur les piramides triangulaires,, Nouv. Mem. Acad. Sci. Berlin, (1773), 149.   Google Scholar [9] N. A. Court, Notes on the orthocentric tetrahedra,, The American Mathematical Monthly, 41 (1934), 499.  doi: 10.2307/2300415.  Google Scholar [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.  doi: 10.1007/s12346-010-0009-6.  Google Scholar [11] C. Simó, El conjunto de bifurcación en el problema espacial de tres cuerpos,, in, (1975), 211.   Google Scholar [12] J. V. Jose and E. J. Saletan, "Classical Mechanics, A Contemporary Approach,", Cambridge University Press, (1998).  doi: 10.1017/CBO9780511803772.  Google Scholar [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.   Google Scholar [14] E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", $4^{th}$ edition, (1937).   Google Scholar [15] E. Piña, Rotations with Rodrigues' vector,, Eur. J. Phys., 32 (2011), 1171.  doi: 10.1088/0143-0807/32/5/005.  Google Scholar [16] K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,", $2^{nd}$ edition, (2009).   Google Scholar

show all references

##### References:
 [1] D. G. Saari, "Collisions, Rings, and Other Newtonian N-Body Problems,", American Mathematical Society, (2005).   Google Scholar [2] F. R. Moulton, The straight line solutions of the problem of N-bodies,, Annals of Mathematics, 12 (1910), 1.  doi: 10.2307/2007159.  Google Scholar [3] R. Lehmann-Filhés, Ueber swei Fälle des Vielkörpersproblems,, Astr. Nachr, 127 (1891), 137.   Google Scholar [4] E. Piña, Algorithm for planar Four-Body Problem central configurations with given masses,, preprint, ().   Google Scholar [5] E. Piña, New coordinates for the four-body problem,, Rev. Mex. Fis., 56 (2010), 195.   Google Scholar [6] E. Piña and P. Lonngi, Central configurations for the planar Newtonian four-body problem,, Cel. Mech. & Dyn. Astr., 108 (2010), 73.  doi: 10.1007/s10569-010-9291-5.  Google Scholar [7] L. Landau and E. Lifshitz, "Mechanics,", Pergamon Press, (1960).   Google Scholar [8] J. L. Lagrange, Solutions analytiques de quelques problèmes sur les piramides triangulaires,, Nouv. Mem. Acad. Sci. Berlin, (1773), 149.   Google Scholar [9] N. A. Court, Notes on the orthocentric tetrahedra,, The American Mathematical Monthly, 41 (1934), 499.  doi: 10.2307/2300415.  Google Scholar [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.  doi: 10.1007/s12346-010-0009-6.  Google Scholar [11] C. Simó, El conjunto de bifurcación en el problema espacial de tres cuerpos,, in, (1975), 211.   Google Scholar [12] J. V. Jose and E. J. Saletan, "Classical Mechanics, A Contemporary Approach,", Cambridge University Press, (1998).  doi: 10.1017/CBO9780511803772.  Google Scholar [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.   Google Scholar [14] E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", $4^{th}$ edition, (1937).   Google Scholar [15] E. Piña, Rotations with Rodrigues' vector,, Eur. J. Phys., 32 (2011), 1171.  doi: 10.1088/0143-0807/32/5/005.  Google Scholar [16] K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,", $2^{nd}$ edition, (2009).   Google Scholar
 [1] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [2] Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 [3] Yongming Luo, Athanasios Stylianou. On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3455-3477. doi: 10.3934/dcdsb.2020239 [4] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 [5] Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 [6] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [7] Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems, 2020  doi: 10.3934/dcds.2020401 [8] Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 [9] Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 [10] Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205 [11] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453 [12] Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031 [13] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [14] Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021037 [15] Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 [16] Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030 [17] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

2019 Impact Factor: 1.338