# American Institute of Mathematical Sciences

January  2015, 35(1): 1-23. doi: 10.3934/dcds.2015.35.1

## Stability of the rhomboidal symmetric-mass orbit

 1 275 TMCB, Brigham Young University, Provo, UT 84602, United States, United States

Received  August 2013 Revised  June 2014 Published  August 2014

We study the rhomboidal symmetric-mass $1$, $m$, $1$, $m$ four-body problem in the four-degrees-of-freedom setting, where $0 < m \leq 1$. Under suitable changes of variables, isolated binary collisions at the origin are regularizable. Analytic existence of the orbit in the four-degrees-of-freedom setting is established. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is analytically reduced to computing three entries of a $4 \times 4$ matrix related to the monodromy matrix. Additionally, it is shown that the four-degrees-of-freedom setting has a two-degrees-of-freedom invariant set, and linear stability results in the subset comes for free'' from the calculation in the full space. The final numerical analysis shows that the four-degrees-of-freedom orbit is linearly unstable except for a very small interval about $m = 0.4$, whereas the two-degrees-of-freedom orbit is linearly stable for all but very small values of $m$.
Citation: Lennard Bakker, Skyler Simmons. Stability of the rhomboidal symmetric-mass orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 1-23. doi: 10.3934/dcds.2015.35.1
##### References:
 [1] L. F. Bakker, S. Mancuso and S. C. Simmons, Linear stability for some symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem,, J. Math. Anal. Appl., 392 (2012), 136. doi: 10.1016/j.jmaa.2012.03.022. [2] L. F. Bakker, T. Ouyang, D. Yan and S. Simmons, Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem,, Celestial Mech. Dynam. Astronom., 110 (2011), 271. doi: 10.1007/s10569-011-9358-y. [3] L. F. Bakker, T. Ouyang, D. Yan and S. Simmons, Erratum to: Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem [mr2821623],, Celestial Mech. Dynam. Astronom., 112 (2012), 459. doi: 10.1007/s10569-012-9402-6. [4] L. F. Bakker, T. Ouyang, D. Yan, S. Simmons and G. E. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem,, Celestial Mech. Dynam. Astronom., 108 (2010), 147. doi: 10.1007/s10569-010-9298-y. [5] A. Bounemoura, Generic super-exponential stability of invariant tori in Hamiltonian systems,, Ergodic Theory Dynam. Systems, 31 (2011), 1287. doi: 10.1017/S0143385710000441. [6] M. Hénon, Stability of interplay oribts,, Cel. Mech., 15 (1977), 243. [7] J. Hietarinta and S. Mikkola, Chaos in the one-dimensional gravitational three-body problem,, Chaos, 3 (1993), 183. doi: 10.1063/1.165984. [8] Y. Long, Index Theory for Symplectic Paths with Applications,, Progress in Mathematics, (2002). doi: 10.1007/978-3-0348-8175-3. [9] R. Martínez, On the existence of doubly symmetric "Schubart-like'' periodic orbits,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943. doi: 10.3934/dcdsb.2012.17.943. [10] K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem,, 2nd edition, (2009). [11] R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 609. doi: 10.3934/dcdsb.2008.10.609. [12] T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem,, Celestial Mech. Dynam. Astronom., 109 (2011), 229. doi: 10.1007/s10569-010-9325-z. [13] T. Ouyang, D. Yan and S. Simmons, Periodic solutions with singularities in two dimensions in the $n$-body problem,, Rocky Mtn. J. Math., 42 (2012), 1601. doi: 10.1216/RMJ-2012-42-5-1601. [14] G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem,, Ergodic Theory Dynam. Systems, 27 (2007), 1947. doi: 10.1017/S0143385707000284. [15] A. E. Roy and B. A. Steves, The Caledonian symmetrical double binary four-body problem. I. Surfaces of zero-velocity using the energy integral,, Celestial Mech. Dynam. Astronom., 78 (2000), 299. doi: 10.1023/A:1011102815021. [16] J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem,, Astr. Nachr., 283 (1956), 17. doi: 10.1002/asna.19562830105. [17] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem,, Arch. Ration. Mech. Anal., 199 (2011), 821. doi: 10.1007/s00205-010-0334-6. [18] C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics,, Classics in Mathematics, (1995). [19] C. Simó, New families of solutions in $N$-body problems,, in European Congress of Mathematics, (2000), 101. [20] A. Sivasankaran, B. A. Steves and W. L. Sweatman, A global regularisation for integrating the Caledonian symmetric four-body problem,, Celestial Mech. Dynam. Astronom., 107 (2010), 157. doi: 10.1007/s10569-010-9270-x. [21] W. L. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation,, Celestial Mech. Dynam. Astronom., 82 (2002), 179. doi: 10.1023/A:1014599918133. [22] W. L. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem,, Celestial Mech. Dynam. Astronom., 94 (2006), 37. doi: 10.1007/s10569-005-2289-8. [23] A. Venturelli, A variational proof of the existence of von Schubart's orbit,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699. doi: 10.3934/dcdsb.2008.10.699. [24] J. Waldvogel, The rhomboidal symmetric four-body problem,, Celestial Mech. Dynam. Astronom., 113 (2012), 113. doi: 10.1007/s10569-012-9414-2. [25] D. Yan, Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem,, J. Math. Anal. Appl., 388 (2012), 942. doi: 10.1016/j.jmaa.2011.10.032.

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##### References:
 [1] L. F. Bakker, S. Mancuso and S. C. Simmons, Linear stability for some symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem,, J. Math. Anal. Appl., 392 (2012), 136. doi: 10.1016/j.jmaa.2012.03.022. [2] L. F. Bakker, T. Ouyang, D. Yan and S. Simmons, Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem,, Celestial Mech. Dynam. Astronom., 110 (2011), 271. doi: 10.1007/s10569-011-9358-y. [3] L. F. Bakker, T. Ouyang, D. Yan and S. Simmons, Erratum to: Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem [mr2821623],, Celestial Mech. Dynam. Astronom., 112 (2012), 459. doi: 10.1007/s10569-012-9402-6. [4] L. F. Bakker, T. Ouyang, D. Yan, S. Simmons and G. E. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem,, Celestial Mech. Dynam. Astronom., 108 (2010), 147. doi: 10.1007/s10569-010-9298-y. [5] A. Bounemoura, Generic super-exponential stability of invariant tori in Hamiltonian systems,, Ergodic Theory Dynam. Systems, 31 (2011), 1287. doi: 10.1017/S0143385710000441. [6] M. Hénon, Stability of interplay oribts,, Cel. Mech., 15 (1977), 243. [7] J. Hietarinta and S. Mikkola, Chaos in the one-dimensional gravitational three-body problem,, Chaos, 3 (1993), 183. doi: 10.1063/1.165984. [8] Y. Long, Index Theory for Symplectic Paths with Applications,, Progress in Mathematics, (2002). doi: 10.1007/978-3-0348-8175-3. [9] R. Martínez, On the existence of doubly symmetric "Schubart-like'' periodic orbits,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943. doi: 10.3934/dcdsb.2012.17.943. [10] K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem,, 2nd edition, (2009). [11] R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 609. doi: 10.3934/dcdsb.2008.10.609. [12] T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem,, Celestial Mech. Dynam. Astronom., 109 (2011), 229. doi: 10.1007/s10569-010-9325-z. [13] T. Ouyang, D. Yan and S. Simmons, Periodic solutions with singularities in two dimensions in the $n$-body problem,, Rocky Mtn. J. Math., 42 (2012), 1601. doi: 10.1216/RMJ-2012-42-5-1601. [14] G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem,, Ergodic Theory Dynam. Systems, 27 (2007), 1947. doi: 10.1017/S0143385707000284. [15] A. E. Roy and B. A. Steves, The Caledonian symmetrical double binary four-body problem. I. Surfaces of zero-velocity using the energy integral,, Celestial Mech. Dynam. Astronom., 78 (2000), 299. doi: 10.1023/A:1011102815021. [16] J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem,, Astr. Nachr., 283 (1956), 17. doi: 10.1002/asna.19562830105. [17] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem,, Arch. Ration. Mech. Anal., 199 (2011), 821. doi: 10.1007/s00205-010-0334-6. [18] C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics,, Classics in Mathematics, (1995). [19] C. Simó, New families of solutions in $N$-body problems,, in European Congress of Mathematics, (2000), 101. [20] A. Sivasankaran, B. A. Steves and W. L. Sweatman, A global regularisation for integrating the Caledonian symmetric four-body problem,, Celestial Mech. Dynam. Astronom., 107 (2010), 157. doi: 10.1007/s10569-010-9270-x. [21] W. L. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation,, Celestial Mech. Dynam. Astronom., 82 (2002), 179. doi: 10.1023/A:1014599918133. [22] W. L. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem,, Celestial Mech. Dynam. Astronom., 94 (2006), 37. doi: 10.1007/s10569-005-2289-8. [23] A. Venturelli, A variational proof of the existence of von Schubart's orbit,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699. doi: 10.3934/dcdsb.2008.10.699. [24] J. Waldvogel, The rhomboidal symmetric four-body problem,, Celestial Mech. Dynam. Astronom., 113 (2012), 113. doi: 10.1007/s10569-012-9414-2. [25] D. Yan, Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem,, J. Math. Anal. Appl., 388 (2012), 942. doi: 10.1016/j.jmaa.2011.10.032.
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