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August  2021, 41(8): 3759-3779. doi: 10.3934/dcds.2021015

Stability of Broucke's isosceles orbit

Skyler Simmons

Mathematics Department, Utah Valley University, 800 W University Parkway, Orem, UT 84058, USA

Received  July 2020 Revised  October 2020 Published  January 2021

We extend the result of Yan to Broucke's isosceles orbit with masses $ m_1 $, $ m_1 $, and $ m_2 $ with $ 2m_1 + m_2 = 3 $. Under suitable changes of variables, isolated binary collisions between the two mass $ m_1 $ particles are regularizable. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is 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 the interval $ 0.595 < m_1 < 0.715 $, whereas the two-degrees-of-freedom orbit is linearly stable for a much wider interval.

Keywords: $ n $-body problem, binary collision, regularization, linear stability, Broucke's orbit.
Mathematics Subject Classification: Primary: 70F16; Secondary: 37N05, 37J25.
Citation: Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015
References:
[1]

L. Bakker and S. Simmons, Stability of the rhomboidal symmetric-mass orbit, Discrete Contin. Dyn. Syst., 35 (2015), 1-23.  doi: 10.3934/dcds.2015.35.1.  Google Scholar

[2]

L. F. Bakker, S. C. 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, URL http://dx.doi.org/10.1007/s10569-010-9325-z. doi: 10.1016/j.jmaa.2012.03.022.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

R. Broucke, On the isosceles triangle configuration in the planar general three body problem, Astron. Astrophys., 73 (1979), 303–313, URL https://doi.org/10.3934/dcds.2015.35.1. Google Scholar

[7]

M. Hénon, Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261.  doi: 10.1007/BF01228465.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.  doi: 10.1007/BF01390175.  Google Scholar

[12]

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

[13]

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

[14]

R. Moeckel and R. Montgomery, Symmetric regularization, reduction and blow-up of the planar three-body problem, Pacific J. Math., 262 (2013), 129-189.  doi: 10.2140/pjm.2013.262.129.  Google Scholar

[15]

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

[16]

T. Ouyang, S. C. Simmons and D. Yan, 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

[17]

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

[18]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

D. Yan, Existence of the Broucke periodic orbit and its linear stability, J. Math. Anal. Appl., 389 (2012), 656-664.  doi: 10.1016/j.jmaa.2011.12.024.  Google Scholar

show all references

References:
[1]

L. Bakker and S. Simmons, Stability of the rhomboidal symmetric-mass orbit, Discrete Contin. Dyn. Syst., 35 (2015), 1-23.  doi: 10.3934/dcds.2015.35.1.  Google Scholar

[2]

L. F. Bakker, S. C. 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, URL http://dx.doi.org/10.1007/s10569-010-9325-z. doi: 10.1016/j.jmaa.2012.03.022.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

R. Broucke, On the isosceles triangle configuration in the planar general three body problem, Astron. Astrophys., 73 (1979), 303–313, URL https://doi.org/10.3934/dcds.2015.35.1. Google Scholar

[7]

M. Hénon, Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261.  doi: 10.1007/BF01228465.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.  doi: 10.1007/BF01390175.  Google Scholar

[12]

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

[13]

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

[14]

R. Moeckel and R. Montgomery, Symmetric regularization, reduction and blow-up of the planar three-body problem, Pacific J. Math., 262 (2013), 129-189.  doi: 10.2140/pjm.2013.262.129.  Google Scholar

[15]

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

[16]

T. Ouyang, S. C. Simmons and D. Yan, 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

[17]

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

[18]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

D. Yan, Existence of the Broucke periodic orbit and its linear stability, J. Math. Anal. Appl., 389 (2012), 656-664.  doi: 10.1016/j.jmaa.2011.12.024.  Google Scholar

Figure 1.  The coordinates for Broucke's Orbit
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Figure 2.  Broucke's orbit as it evolves in time. One-quarter period is shown. The remainder of the orbit is obtained by a symmetric extension
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Figure 3.  A plot of the value of $ Q_4(0) $. The value of $ m_1 $ is plotted on the horizontal axis
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Figure 4.  A plot of the numerically-computed upper-left entry $ k_{11} $ from the matrix $ K $ in Equation 11 (vertical) against $ m_1 $ (horizontal)
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Figure 4">Figure 5.  A vertically rescaled plot of the graph from Figure 4
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Figure 6.  A plot of the value of $ e $ from Equation (11) corresponding to stability in the 2DF setting
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Figure 7.  A vertically rescaled plot of the value of $ e $. The curve appears to be asymptotic to the line $ y = 1 $ as $ m_1 \to 0^- $
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Figure 8.  A plot of the second non-trivial eigenvalue of $ K $
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Figure 9.  A vertically rescaled plot of the value of the second non-trivial eigenvalue of $ K $. For the values of $ m_1 \in [0.595, 0.715] $, this second eigenvalue lies within the interval $ [-1, 1] $
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Figure 10.  Both eigenvalue plots superimposed. The two graphs cross at roughly $ m_1 = 0.71 $, where the difference between the two is numerically $ 0.000429 $
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