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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  August 2021 Early access  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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