Stability of the rhomboidal symmetric-mass orbit

Previous Article
Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics
DCDS Home
This Issue
Next Article

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

Abstract
Related Papers

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 and 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.
[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.
[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.
[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.
[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.
[6]	M. Hénon, Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. 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), 821-841. doi: 10.1007/s00205-010-0334-6.
[18]	C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[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.
[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.
[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.
[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.
[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.
[24]	J. Waldvogel, The rhomboidal symmetric four-body problem, Celestial Mech. Dynam. Astronom., 113 (2012), 113-123. 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), 942-951. doi: 10.1016/j.jmaa.2011.10.032.

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.
[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.
[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.
[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.
[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.
[6]	M. Hénon, Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. 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), 821-841. doi: 10.1007/s00205-010-0334-6.
[18]	C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[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.
[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.
[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.
[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.
[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.
[24]	J. Waldvogel, The rhomboidal symmetric four-body problem, Celestial Mech. Dynam. Astronom., 113 (2012), 113-123. 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), 942-951. doi: 10.1016/j.jmaa.2011.10.032.

[1]	Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313
[2]	Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55
[3]	Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 909-932. doi: 10.3934/dcds.2009.24.909
[4]	Roberto Castelli, Susanna Terracini. On the regularization of the collision solutions of the one-center problem with weak forces. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1197-1218. doi: 10.3934/dcds.2011.31.1197
[5]	Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[6]	G. Bellettini, G. Fusco, G. F. Gronchi. Regularization of the two-body problem via smoothing the potential. Communications on Pure and Applied Analysis, 2003, 2 (3) : 323-353. doi: 10.3934/cpaa.2003.2.323
[7]	Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062
[8]	Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074
[9]	Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems and Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019
[10]	Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic and Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441
[11]	Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523
[12]	Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589
[13]	Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35
[14]	Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[15]	Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems and Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
[16]	Montserrat Corbera, Jaume Llibre. On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1049-1060. doi: 10.3934/dcds.2013.33.1049
[17]	Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137
[18]	Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems and Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[19]	Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169
[20]	Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Lennard Bakker Skyler Simmons
on Google Scholar
Lennard Bakker Skyler Simmons

[Back to Top]