Figure 1.
The coordinates for Broucke's Orbit
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August
2021, 41(8): 3759-3779.
doi: 10.3934/dcds.2021015
Stability of Broucke's isosceles orbit
Mathematics Department, Utah Valley University, 800 W University Parkway, Orem, UT 84058, USA |
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.
Mathematics Subject Classification:
Primary: 70F16; Secondary: 37N05, 37J25.
Citation:
Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete and Continuous Dynamical Systems,
2021, 41
(8)
: 3759-3779.
doi: 10.3934/dcds.2021015
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
M. Hénon,
Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261.
doi: 10.1007/BF01228465. |
| [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. |
| [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. |
| [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. |
| [11] |
R. McGehee,
Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.
doi: 10.1007/BF01390175. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
J. Schubart,
Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
M. Hénon,
Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261.
doi: 10.1007/BF01228465. |
| [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. |
| [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. |
| [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. |
| [11] |
R. McGehee,
Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.
doi: 10.1007/BF01390175. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
J. Schubart,
Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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
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)
Figure 6.
A plot of the value of $ e $ from Equation (11) corresponding to stability in the 2DF setting
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^- $
Figure 8.
A plot of the second non-trivial eigenvalue of $ K $
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] $
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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