Periodic oscillations in the restricted hip-hop $ 2N+1 $-body problem

Andrés Rivera; Oscar Perdomo; Nelson Castañeda

doi:10.3934/dcdsb.2023062

Article Contents

2023, Volume 28, Issue 11: 5481-5493. Doi: 10.3934/dcdsb.2023062

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Periodic oscillations in the restricted hip-hop $ 2N+1 $-body problem

1.
Dpto. Ciencias Naturales y Matemáticas, Pontificia Universidad Javeriana Cali, Cali, Colombia
2.
Department of Mathematical Sciences, Central Connecticut State University, Connecticut, USA

^*Corresponding author: Andrés Rivera
^*Corresponding author: Andrés Rivera

Received: July 2022

Revised: January 2023

Early access: March 2023

Published: November 2023

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Abstract

This manuscript investigates a dynamical system in which $ 2N $ primary particles of equal masses move in space under Newton's law of gravitation forming the vertices of antiprisms while a particle of negligible mass moves along the common axis of symmetry of the antiprisms. This $ n $-body problem that we call the restricted hip-hop $ (2N + 1) $-body problem is an extension of the generalized Sitnikov problem studied in [17] for which the primaries remain in a plane. This work also relies on an early study [14] where certain families of periodic hip-hop solutions to a $ 2N $–body problem were constructed. We prove the existence of a continuous symmetric family of solutions of the restricted hip-hop $ (2N+1) $-body problem for each family of symmetric and periodic hip-hop solutions of the primaries studied in [14]. The main tools for proving our results are the implicit function theorem and a compactness argument. In addition, we present some numerical periodic solutions to the restricted $ 7 $-body problem.

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Mathematics Subject Classification: Primary: 70F10, 37C27; Secondary: 34C25, 34A12.

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Figure 1. Restricted Hip-hop 5-body problem with four primaries. At any time, the primaries are placed at the vertices of a regular $ 4 $-gonal antiprism and the plus one body moves on the line orthogonal to the two $ 2 $-gons that are the base face of the antiprism

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Figure 2. Graph of the $ 2T_1 $-periodic functions $ d(t) $, $ r(t) $ and $ z(t) $ given by the parameters $ a_1 = 0.581722 $, $ b_1 = 0.81081 $, $ u_1 = 1.96752 $ and $ T_1 = 6.53474 $. The dashed graph is the function $ z(t) $ and the function that starts at $ r_0 = 2 $ is the function $ r(t) $

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Figure 3. Image of the (6+1)-bodies for the solution given by the parameters $ a_1 = 0.581722 $, $ b_1 = 0.81081 $, $ u_1 = 1.96752 $ and $ T_1 = 6.53474 $. From left to right we show the bodies when $ t = T_1/2 $, $ t = 3T_1/4 $, $ t = T_1 $, $ t = 5 T_1/4 $ and $ t = 3T_1/2 $

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Figure 4. Graph of the $ 2T_2 $-periodic functions $ d(t) $, $ r(t) $ and $ z(t) $ given by the parameters $ a_2 = 1.37168 $, $ b_2 = 0.717282 $, $ u_2 = 1.73494 $ and $ T_2 = 6.95831 $. The dashed graph is the function $ z(t) $ and the function that starts at $ r_0 = 2 $ is the function $ r(t) $

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Figure 5. Image of the (6+1)-bodies for the solution given by the parameters $ a_2 = 1.37168 $, $ b_2 = 0.717282 $, $ u_2 = 1.73494 $ and $ T_2 = 6.95831 $. From left to right we show the images when $ t = T_2/2 $, $ t = 3T_2/4 $, $ t = T_2 $, $ t = 5 T_2/4 $ and $ t = 3T_2/2 $

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References

[1]	V. M. Alekseev, Quasirandom dynamical systems II, Mat. Zametki, 6 (1969), 489-498.
[2]	V. I. Arnold, V. Kozlov and A. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 3$^{rd}$ edition, Springer, 2006.
[3]	E. Barrabes and J. Cors, Highly eccentricity hip-hop solutions of the $2N$-body problem, Physica. D, 239 (2010), 214-219. doi: 10.1016/j.physd.2009.10.019.
[4]	E. Barrabes, J. Cors, C. Pinyol and J. Soler, Hip-hop solutions of the $2N$-body problem, Celestial Mech. Dynam. Astronom., 95 (2006), 55-66. doi: 10.1007/s10569-006-9016-y.
[5]	E. Belbruno, J. Llibre and M. Ollé, On the families of periodic orbits which bifurcate from the circular Sitnikov motions, Celestial Mech. Dynam. Astronom., 60 (1994), 99-129. doi: 10.1007/BF00693095.
[6]	A. Chenciner and A. Venturelli, Minima de l'intégrale d'action du problème newtonien de $4$ corps de masses égales dans $\mathbb{R}^3$: orbites "hip-hop", Celestial Mech. Dynam. Astronom., 77 (2000), 139-152. doi: 10.1023/A:1008381001328.
[7]	M. Corbera and J. Llibre, Periodic orbits of the Sitnikov problem via a Poincaré map, Celestial Mech. Dynam. Astronom., 77 (2000), 273-303. doi: 10.1023/A:1011117003713.
[8]	M. Corbera and J. Llibre, On symmetric periodic orbits of the elliptic Sitnikov problem via the analytic continuation method, Contemporay Mathematics, 292 (2002), 91-127. doi: 10.1090/conm/292/04918.
[9]	D. L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.
[10]	L. Jiménez-Lara and A. Escalona-Buendía, Symmetries and bifurcations in the Sitnikov problem, Celestial Mech. Dynam. Astronom., 79 (2001), 97-117. doi: 10.1023/A:1011109827402.
[11]	J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM J. Applied Dynamical Systems, 7 (2008), 561-576. doi: 10.1137/070695253.
[12]	S. Mathlouthi, Periodic orbits of the restricted three-body problem, Trans. Amer. Math. Soc., 350 (1998), 2265-2276. doi: 10.1090/S0002-9947-98-01731-0.
[13]	R. Ortega and A. Rivera, Global bifurcations from the center of mass in the Sitnikov problem, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 719-732. doi: 10.3934/dcdsb.2010.14.719.
[14]	O. Perdomo, A. Rivera, J. Arredondo and N. Castañeda, Periodic oscillations in a $2N$ body problem, to appear, SIAM J. Applied Dynamical Systems.
[15]	O. Perdomo, Embedded constant mean curvature hypersurfaces on spheres, Asian J. Math., 14 (2010), 73-108. doi: 10.4310/AJM.2010.v14.n1.a5.
[16]	S. Terracini and A. Venturelli, Symmetric trayectories for the $2N$-body problem with equal masses, Arch. Rational Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.
[17]	A. Rivera, Periodic solutions in the generalized Sitnikov $N+1$-body problem, SIAM J. Applied Dyn. Syst., 12 (2013), 1515-1540. doi: 10.1137/120883876.
[18]	R. Robinson, Uniform subaharmonic orbit for Sitnikov problem, Discrete Contin. Dyn. Syst. Ser. S., 1 (2008), 647-652. doi: 10.3934/dcdss.2008.1.647.
[19]	D. J. Scheeres and J. Bellerose, The restricted Hill Full $4$-body problem: Application to spacraft motion about binary asteroids, Dyn.Syst.: An International Journal, 20 (2005), 23-44. doi: 10.1080/1468936042000281321.
[20]	K. A. Sitnikov, Existence of oscillating motion for the three-body problem, Dokl. Akad. Nauk, 133 (1960), 303-306 (in Russian).