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

Stability of the rhomboidal symmetric-mass orbit

Lennard Bakker 1, and  Skyler Simmons 1,

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 $m$.
Keywords: linear stability, n-body problem, regularization, binary collision, rhomboidal problem..
Mathematics Subject Classification: Primary: 70F16; Secondary: 37N05, 37J2.
Citation: Lennard Bakker, Skyler Simmons. Stability of the rhomboidal symmetric-mass orbit. Discrete & Continuous Dynamical Systems, 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), 136-147. doi: 10.1016/j.jmaa.2012.03.022.  Google Scholar

[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), 271-290. doi: 10.1007/s10569-011-9358-y.  Google Scholar

[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), 459-460. doi: 10.1007/s10569-012-9402-6.  Google Scholar

[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), 147-164. doi: 10.1007/s10569-010-9298-y.  Google Scholar

[5]

A. Bounemoura, Generic super-exponential stability of invariant tori in Hamiltonian systems, Ergodic Theory Dynam. Systems, 31 (2011), 1287-1303. doi: 10.1017/S0143385710000441.  Google Scholar

[6]

M. Hénon, Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261. Google Scholar

[7]

J. Hietarinta and S. Mikkola, Chaos in the one-dimensional gravitational three-body problem, Chaos, 3 (1993), 183-203. doi: 10.1063/1.165984.  Google Scholar

[8]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[9]

R. Martínez, On the existence of doubly symmetric "Schubart-like'' periodic orbits, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943-975. doi: 10.3934/dcdsb.2012.17.943.  Google Scholar

[10]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, 2nd edition, Applied Mathematical Sciences, 90, Springer, New York, 2009.  Google Scholar

[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), 609-620. doi: 10.3934/dcdsb.2008.10.609.  Google Scholar

[12]

T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem, Celestial Mech. Dynam. Astronom., 109 (2011), 229-239. doi: 10.1007/s10569-010-9325-z.  Google Scholar

[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), 1601-1614. doi: 10.1216/RMJ-2012-42-5-1601.  Google Scholar

[14]

G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems, 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284.  Google Scholar

[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), 299-318. doi: 10.1023/A:1011102815021.  Google Scholar

[16]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. doi: 10.1002/asna.19562830105.  Google Scholar

[17]

M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841. doi: 10.1007/s00205-010-0334-6.  Google Scholar

[18]

C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[19]

C. Simó, New families of solutions in $N$-body problems, in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, 2001, 101-115.  Google Scholar

[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), 157-168. doi: 10.1007/s10569-010-9270-x.  Google Scholar

[21]

W. L. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation, Celestial Mech. Dynam. Astronom., 82 (2002), 179-201. doi: 10.1023/A:1014599918133.  Google Scholar

[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), 37-65. doi: 10.1007/s10569-005-2289-8.  Google Scholar

[23]

A. Venturelli, A variational proof of the existence of von Schubart's orbit, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699-717. doi: 10.3934/dcdsb.2008.10.699.  Google Scholar

[24]

J. Waldvogel, The rhomboidal symmetric four-body problem, Celestial Mech. Dynam. Astronom., 113 (2012), 113-123. doi: 10.1007/s10569-012-9414-2.  Google Scholar

[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), 942-951. doi: 10.1016/j.jmaa.2011.10.032.  Google Scholar

show all references

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), 136-147. doi: 10.1016/j.jmaa.2012.03.022.  Google Scholar

[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), 271-290. doi: 10.1007/s10569-011-9358-y.  Google Scholar

[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), 459-460. doi: 10.1007/s10569-012-9402-6.  Google Scholar

[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), 147-164. doi: 10.1007/s10569-010-9298-y.  Google Scholar

[5]

A. Bounemoura, Generic super-exponential stability of invariant tori in Hamiltonian systems, Ergodic Theory Dynam. Systems, 31 (2011), 1287-1303. doi: 10.1017/S0143385710000441.  Google Scholar

[6]

M. Hénon, Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261. Google Scholar

[7]

J. Hietarinta and S. Mikkola, Chaos in the one-dimensional gravitational three-body problem, Chaos, 3 (1993), 183-203. doi: 10.1063/1.165984.  Google Scholar

[8]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[9]

R. Martínez, On the existence of doubly symmetric "Schubart-like'' periodic orbits, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943-975. doi: 10.3934/dcdsb.2012.17.943.  Google Scholar

[10]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, 2nd edition, Applied Mathematical Sciences, 90, Springer, New York, 2009.  Google Scholar

[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), 609-620. doi: 10.3934/dcdsb.2008.10.609.  Google Scholar

[12]

T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem, Celestial Mech. Dynam. Astronom., 109 (2011), 229-239. doi: 10.1007/s10569-010-9325-z.  Google Scholar

[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), 1601-1614. doi: 10.1216/RMJ-2012-42-5-1601.  Google Scholar

[14]

G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems, 27 (2007), 1947-1963. doi: 10.1017/S0143385707000284.  Google Scholar

[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), 299-318. doi: 10.1023/A:1011102815021.  Google Scholar

[16]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. doi: 10.1002/asna.19562830105.  Google Scholar

[17]

M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841. doi: 10.1007/s00205-010-0334-6.  Google Scholar

[18]

C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[19]

C. Simó, New families of solutions in $N$-body problems, in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, 2001, 101-115.  Google Scholar

[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), 157-168. doi: 10.1007/s10569-010-9270-x.  Google Scholar

[21]

W. L. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation, Celestial Mech. Dynam. Astronom., 82 (2002), 179-201. doi: 10.1023/A:1014599918133.  Google Scholar

[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), 37-65. doi: 10.1007/s10569-005-2289-8.  Google Scholar

[23]

A. Venturelli, A variational proof of the existence of von Schubart's orbit, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699-717. doi: 10.3934/dcdsb.2008.10.699.  Google Scholar

[24]

J. Waldvogel, The rhomboidal symmetric four-body problem, Celestial Mech. Dynam. Astronom., 113 (2012), 113-123. doi: 10.1007/s10569-012-9414-2.  Google Scholar

[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), 942-951. doi: 10.1016/j.jmaa.2011.10.032.  Google Scholar

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