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 - A, 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]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[2]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127

[3]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021002

[4]

Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021004

[5]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[6]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[7]

Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021004

[8]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[9]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[10]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169

[11]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[12]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[13]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[14]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[15]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[16]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[17]

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

[18]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[19]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[20]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]