April 2018, 38(4): 2187-2206. doi: 10.3934/dcds.2018090

New periodic orbits in the planar equal-mass three-body problem

School of Mathematics and System Sciences, Beihang University, Beijing 100191, China

* Corresponding author: Duokui Yan

Received  June 2017 Revised  October 2017 Published  January 2018

Fund Project: The authors are supported by NSFC No.11432001 and the Fundamental Research Funds of the Central Universities

It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle $ \mathit{\theta }$.

Citation: Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
References:
[1]

R. Broucke and D. Boggs, Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38. doi: 10.1007/BF01228732.

[2]

K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325.

[3]

K. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441. doi: 10.1007/s00220-009-0769-5.

[4]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.

[5]

A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry, Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294.

[6]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.

[7]

M. Hénon, A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285. doi: 10.1007/BF01228647.

[8]

W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580.

[9]

C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.

[10]

T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76. doi: 10.1016/j.physd.2015.05.015.

[11]

B. ShiR. LiuD. Yan and T. Ouyang, Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835. doi: 10.1515/ans-2017-6028.

[12]

G. Yu, Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935. doi: 10.1007/s00205-017-1116-1.

[13]

G. Yu, Spatial double choreographies of the Newtonian $ 2n$-body problem, arXiv: 1608.07956.

show all references

References:
[1]

R. Broucke and D. Boggs, Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38. doi: 10.1007/BF01228732.

[2]

K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325.

[3]

K. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441. doi: 10.1007/s00220-009-0769-5.

[4]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.

[5]

A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry, Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294.

[6]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.

[7]

M. Hénon, A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285. doi: 10.1007/BF01228647.

[8]

W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580.

[9]

C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.

[10]

T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76. doi: 10.1016/j.physd.2015.05.015.

[11]

B. ShiR. LiuD. Yan and T. Ouyang, Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835. doi: 10.1515/ans-2017-6028.

[12]

G. Yu, Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935. doi: 10.1007/s00205-017-1116-1.

[13]

G. Yu, Spatial double choreographies of the Newtonian $ 2n$-body problem, arXiv: 1608.07956.

Figure 2.  A picture of the periodic orbit extended by the action minimizer $\mathcal{P}_0$ with $\theta = \pi/10$. The three dots represent the starting configuration $Q_S$, and the three crosses represent the ending configuration $Q_E$ of $\mathcal{P}_0$. In the graph, the red curve is the trajectory of body 2, the blue curve is for body 1 and the black curve is for body 3.
Figure 1.  The configurations $Q_S$ and $Q_E$ are shown, where blue dots represent $q_1$, red dots represent $q_2$ and black dots represent $q_3$. In $Q_S$, three masses are on the $x-$axis with an order $q_{2x} \leq q_{1x} \leq q_{3x}$. In $Q_E$, three masses form an isosceles triangle, whose symmetry axis is a counterclockwise $\theta$ rotation of the $x-$axis. $q_1$ is on the symmetry axis and $q_2$ is above the symmetry axis in $Q_E$.
Figure 3.  In each figure, the horizontal axis is $ \theta/\pi$, and the vertical axis is the action value $ \mathcal{A}$. In each subfigure, the black curve is the graph of the test path's action $\mathcal{A}_{test}$; the purple curve is the graph of $\mathcal{A}_{Euler} = g(\theta)$ in (31) and the red curve is the graph of $f_2(\theta)$ in (31), which is the lower bound of $\mathcal{A}_{col}$.
Table 1.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.181\pi, \, 0.183 \pi]$.
$\mathbf{\theta_0= 0.182 \pi, \, \, \, \, \, \theta \in [0.181 \pi, \, 0.183 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1109, \, 0 )$ $( -1.1189,\, 0 )$
$0.1$ $( 0.11378134, \, 0.035120592)$ $ ( -1.1144897, \, 0.089194621)$
$0.2$ $(0.12212758, \, 0.068373015 )$ $( -1.1012708, \,0.17787543 )$
$0.3$ $(0.13510199, \, 0.098115021 )$ $(-1.0792782, \, 0.26552094 )$
$0.4$ $( 0.15147483, \, 0.12309328 )$ $( -1.0485685, \, 0.35159462 ) $
$0.5$ $( 0.16980434, \, 0.14251378 )$ $ ( -1.0092177, \, 0.43553762) $
$0.6$ $( 0.18860029, \, 0.15603011)$ $ (-0.96132044, \, 0.51676178)$
$0.7$ $(0.20644197, \, 0.16368275 )$ $ ( -0.90499096, \, 0.59464214) $
$0.8$ $( 0.22204784, \, 0.16582253 )$ $(-0.84036636, \, 0.66850889) $
$0.9$ $( 0.23430842, \, 0.16304077 )$ $( -0.76761388, \, 0.73763832) $
$1$ $(0.28822237, \, 0 ) R(\theta)$ $ (-0.144111185, \, 1.0455201)R(\theta)$
$\mathbf{\theta_0= 0.182 \pi, \, \, \, \, \, \theta \in [0.181 \pi, \, 0.183 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1109, \, 0 )$ $( -1.1189,\, 0 )$
$0.1$ $( 0.11378134, \, 0.035120592)$ $ ( -1.1144897, \, 0.089194621)$
$0.2$ $(0.12212758, \, 0.068373015 )$ $( -1.1012708, \,0.17787543 )$
$0.3$ $(0.13510199, \, 0.098115021 )$ $(-1.0792782, \, 0.26552094 )$
$0.4$ $( 0.15147483, \, 0.12309328 )$ $( -1.0485685, \, 0.35159462 ) $
$0.5$ $( 0.16980434, \, 0.14251378 )$ $ ( -1.0092177, \, 0.43553762) $
$0.6$ $( 0.18860029, \, 0.15603011)$ $ (-0.96132044, \, 0.51676178)$
$0.7$ $(0.20644197, \, 0.16368275 )$ $ ( -0.90499096, \, 0.59464214) $
$0.8$ $( 0.22204784, \, 0.16582253 )$ $(-0.84036636, \, 0.66850889) $
$0.9$ $( 0.23430842, \, 0.16304077 )$ $( -0.76761388, \, 0.73763832) $
$1$ $(0.28822237, \, 0 ) R(\theta)$ $ (-0.144111185, \, 1.0455201)R(\theta)$
Table 2.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.176\pi, \, 0.181 \pi]$.
$\mathbf{\theta_0= 0.18 \pi, \, \, \, \, \, \theta \in [0.176 \pi, \, 0.181 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1198, \, 0 )$ $(-1.1166, \, 0 )$
$0.1$ $( 0.12300003, \, 0.038069081 )$ $ (-1.1122108, \, 0.087838457 )$
$0.2$ $(0.13225041, \, 0.074032720 )$ $( -1.0990555, \, 0.17518597 )$
$0.3$ $(0.14657647, \, 0.10606411 )$ $( -1.0771702, \, 0.26154328 )$
$0.4$ $(0.16456742, \, 0.13280675 )$ $( -1.0466117, \, 0.34639491 ) $
$0.5$ $(0.18460176, \, 0.15344287 )$ $ (-1.0074550, \, 0.42920169 ) $
$0.6$ $( 0.20503978, \, 0.16765957 )$ $ ( -0.95979063, \, 0.50939364 )$
$0.7$ $(0.22435124, \, 0.17556129 )$ $ ( -0.90372458, \, 0.58636242 ) $
$0.8$ $( 0.24118113, \, 0.17757208 )$ $(-0.83938055, \, 0.65945294 ) $
$0.9$ $( 0.25437250, \, 0.17435316 )$ $(-0.76690612, \, 0.72795349 ) $
$1$ $( 0.31146657, \, 0 ) R(\theta)$ $ (-0.155733285, \, 1.0357709)R(\theta)$
$\mathbf{\theta_0= 0.18 \pi, \, \, \, \, \, \theta \in [0.176 \pi, \, 0.181 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1198, \, 0 )$ $(-1.1166, \, 0 )$
$0.1$ $( 0.12300003, \, 0.038069081 )$ $ (-1.1122108, \, 0.087838457 )$
$0.2$ $(0.13225041, \, 0.074032720 )$ $( -1.0990555, \, 0.17518597 )$
$0.3$ $(0.14657647, \, 0.10606411 )$ $( -1.0771702, \, 0.26154328 )$
$0.4$ $(0.16456742, \, 0.13280675 )$ $( -1.0466117, \, 0.34639491 ) $
$0.5$ $(0.18460176, \, 0.15344287 )$ $ (-1.0074550, \, 0.42920169 ) $
$0.6$ $( 0.20503978, \, 0.16765957 )$ $ ( -0.95979063, \, 0.50939364 )$
$0.7$ $(0.22435124, \, 0.17556129 )$ $ ( -0.90372458, \, 0.58636242 ) $
$0.8$ $( 0.24118113, \, 0.17757208 )$ $(-0.83938055, \, 0.65945294 ) $
$0.9$ $( 0.25437250, \, 0.17435316 )$ $(-0.76690612, \, 0.72795349 ) $
$1$ $( 0.31146657, \, 0 ) R(\theta)$ $ (-0.155733285, \, 1.0357709)R(\theta)$
Table 3.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.165\pi, \, 0.176 \pi]$.
$\mathbf{\theta_0= 0.173 \pi, \, \, \, \, \, \theta \in [0.165 \pi, \, 0.176 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1454,\, 0 )$ $( -1.1064,\, 0 )$
$0.1$ $( 0.14967657,\, 0.046886901 )$ $ (-1.1020407,\, 0.083918190 )$
$0.2$ $(0.16194195,\, 0.090810510 )$ $(-1.0889788,\, 0.16740323 )$
$0.3$ $(0.18067059,\, 0.12932570 )$ $( -1.0672593,\, 0.25001086 )$
$0.4$ $(0.20377305,\, 0.16080303 )$ $( -1.0369499,\, 0.33127610 ) $
$0.5$ $( 0.22901187,\, 0.18445175 )$ $ ( -0.99813266,\, 0.41070550 ) $
$0.6$ $(0.25429011,\, 0.20016641 )$ $ ( -0.95089814,\, 0.48777071 )$
$0.7$ $(0.27778726,\, 0.20832504 )$ $ ( -0.89534096,\, 0.56190183 ) $
$0.8$ $(0.29798679,\, 0.20961764 )$ $( -0.83155952,\, 0.63247939 ) $
$0.9$ $(0.31364961,\, 0.20493038 )$ $(-0.75965878,\, 0.69882326 ) $
$1$ $( 0.37739476,\, 0 ) R(\theta)$ $ ( -0.18869738,\, 1.0021646)R(\theta)$
$\mathbf{\theta_0= 0.173 \pi, \, \, \, \, \, \theta \in [0.165 \pi, \, 0.176 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $( 0.1454,\, 0 )$ $( -1.1064,\, 0 )$
$0.1$ $( 0.14967657,\, 0.046886901 )$ $ (-1.1020407,\, 0.083918190 )$
$0.2$ $(0.16194195,\, 0.090810510 )$ $(-1.0889788,\, 0.16740323 )$
$0.3$ $(0.18067059,\, 0.12932570 )$ $( -1.0672593,\, 0.25001086 )$
$0.4$ $(0.20377305,\, 0.16080303 )$ $( -1.0369499,\, 0.33127610 ) $
$0.5$ $( 0.22901187,\, 0.18445175 )$ $ ( -0.99813266,\, 0.41070550 ) $
$0.6$ $(0.25429011,\, 0.20016641 )$ $ ( -0.95089814,\, 0.48777071 )$
$0.7$ $(0.27778726,\, 0.20832504 )$ $ ( -0.89534096,\, 0.56190183 ) $
$0.8$ $(0.29798679,\, 0.20961764 )$ $( -0.83155952,\, 0.63247939 ) $
$0.9$ $(0.31364961,\, 0.20493038 )$ $(-0.75965878,\, 0.69882326 ) $
$1$ $( 0.37739476,\, 0 ) R(\theta)$ $ ( -0.18869738,\, 1.0021646)R(\theta)$
Table 4.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $ \mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.146\pi, \, 0.165 \pi]$.
$\mathbf{\theta_0= 0.16 \pi, \, \, \, \, \, \theta \in [0.146 \pi, \, 0.165 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1803, \, 0 )$ $( -1.0860,\, 0 )$
$0.1$ $( 0.18658246,\, 0.059576674 )$ $ (-1.0816238,\, 0.078768710)$
$0.2$ $( 0.20429715,\, 0.11438956 )$ $(-1.0685224,\, 0.15716562)$
$0.3$ $( 0.23056861,\, 0.16092393 )$ $( -1.0467705,\, 0.23480016)$
$0.4$ $( 0.26186968,\, 0.19737903 )$ $( -1.0164725,\, 0.31125003) $
$0.5$ $( 0.29489886,\, 0.22338233 )$ $ ( -0.97774453,\, 0.38605495) $
$0.6$ $(0.32695826,\, 0.23945199 )$ $ ( -0.93070180,\, 0.45871449)$
$0.7$ $(0.35597786,\, 0.24655415 )$ $ ( -0.87545128,\, 0.52868620) $
$0.8$ $(0.38039590,\, 0.24583791 )$ $( -0.81208876,\, 0.59538118) $
$0.9$ $(0.39901814,\, 0.23851003 )$ $( -0.74069823,\, 0.65815496 ) $
$1$ $(0.46894463,\, 0) R(\theta)$ $ ( -0.234472315,\, 0.94630060)R(\theta)$
$\mathbf{\theta_0= 0.16 \pi, \, \, \, \, \, \theta \in [0.146 \pi, \, 0.165 \pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.1803, \, 0 )$ $( -1.0860,\, 0 )$
$0.1$ $( 0.18658246,\, 0.059576674 )$ $ (-1.0816238,\, 0.078768710)$
$0.2$ $( 0.20429715,\, 0.11438956 )$ $(-1.0685224,\, 0.15716562)$
$0.3$ $( 0.23056861,\, 0.16092393 )$ $( -1.0467705,\, 0.23480016)$
$0.4$ $( 0.26186968,\, 0.19737903 )$ $( -1.0164725,\, 0.31125003) $
$0.5$ $( 0.29489886,\, 0.22338233 )$ $ ( -0.97774453,\, 0.38605495) $
$0.6$ $(0.32695826,\, 0.23945199 )$ $ ( -0.93070180,\, 0.45871449)$
$0.7$ $(0.35597786,\, 0.24655415 )$ $ ( -0.87545128,\, 0.52868620) $
$0.8$ $(0.38039590,\, 0.24583791 )$ $( -0.81208876,\, 0.59538118) $
$0.9$ $(0.39901814,\, 0.23851003 )$ $( -0.74069823,\, 0.65815496 ) $
$1$ $(0.46894463,\, 0) R(\theta)$ $ ( -0.234472315,\, 0.94630060)R(\theta)$
Table 5.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $ \mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.12\pi, \, 0.146 \pi]$.
$\mathbf{\theta_0= 0.132 \pi, \, \, \, \, \, \theta \in [0.12 \pi, \, 0.146\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2306, \, 0)$ $( -1.0421, \, 0)$
$0.1$ $( 0.24197875, \, 0.079876150 )$ $ (-1.0375636, \, 0.072168012 )$
$0.2$ $( 0.27249601, \, 0.14948693 )$ $( -1.0240177, \, 0.14401235 )$
$0.3$ $( 0.31440456, \, 0.20354167 )$ $( -1.0016223, \, 0.21516044 )$
$0.4$ $(0.36051601, \, 0.24151617 )$ $( -0.97057606, \, 0.28517262 ) $
$0.5$ $(0.40588817, \, 0.26512985 )$ $ ( -0.93107630, \, 0.35354922 ) $
$0.6$ $(0.44748116, \, 0.27663491 )$ $ ( -0.88330444, \, 0.41974435 )$
$0.7$ $(0.48345138, \, 0.27815124 )$ $ ( -0.82742450, \, 0.48317696 ) $
$0.8$ $(0.51262863, \, 0.27151295 )$ $( -0.76358450, \, 0.54323643 ) $
$0.9$ $(0.53419879, \, 0.25829537 )$ $( -0.69191492, \, 0.59928148 ) $
$1$ $(0.59692629, \, 0) R(\theta)$ $ ( -0.298463145, \, 0.84227284 )R(\theta)$
$\mathbf{\theta_0= 0.132 \pi, \, \, \, \, \, \theta \in [0.12 \pi, \, 0.146\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2306, \, 0)$ $( -1.0421, \, 0)$
$0.1$ $( 0.24197875, \, 0.079876150 )$ $ (-1.0375636, \, 0.072168012 )$
$0.2$ $( 0.27249601, \, 0.14948693 )$ $( -1.0240177, \, 0.14401235 )$
$0.3$ $( 0.31440456, \, 0.20354167 )$ $( -1.0016223, \, 0.21516044 )$
$0.4$ $(0.36051601, \, 0.24151617 )$ $( -0.97057606, \, 0.28517262 ) $
$0.5$ $(0.40588817, \, 0.26512985 )$ $ ( -0.93107630, \, 0.35354922 ) $
$0.6$ $(0.44748116, \, 0.27663491 )$ $ ( -0.88330444, \, 0.41974435 )$
$0.7$ $(0.48345138, \, 0.27815124 )$ $ ( -0.82742450, \, 0.48317696 ) $
$0.8$ $(0.51262863, \, 0.27151295 )$ $( -0.76358450, \, 0.54323643 ) $
$0.9$ $(0.53419879, \, 0.25829537 )$ $( -0.69191492, \, 0.59928148 ) $
$1$ $(0.59692629, \, 0) R(\theta)$ $ ( -0.298463145, \, 0.84227284 )R(\theta)$
Table 6.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.089\pi, \, 0.12 \pi]$.
$\mathbf{\theta_0= 0.1 \pi, \, \, \, \, \, \theta \in [0.089 \pi, \, 0.12\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2680, \, 0)$ $( -0.9999, \, 0 )$
$0.1$ $(0.28699887, \, 0.097087533 )$ $ ( -0.99513360, \, 0.068868347)$
$0.2$ $(0.33390510, \, 0.17420817 )$ $( -0.98094971, \, 0.13739837 )$
$0.3$ $(0.39193700, \, 0.22664276 )$ $( -0.95760880, \, 0.20514818)$
$0.4$ $(0.45044849, \, 0.25815778 )$ $( -0.92539244, \, 0.27156414 ) $
$0.5$ $(0.50439276, \, 0.27354220 )$ $ (-0.88455797, \, 0.33601715 ) $
$0.6$ $(0.55150942, \, 0.27662332 )$ $ ( -0.83534366, \, 0.39783468 )$
$0.7$ $(0.59074871, \, 0.27023456 )$ $ ( -0.77798406, \, 0.45632275 ) $
$0.8$ $(0.62156413, \, 0.25649856 )$ $( -0.71271941, \, 0.51078103 ) $
$0.9$ $(0.64359767, \, 0.23707560 )$ $( -0.63979436, \, 0.56051180 ) $
$1$ $(0.69032914, \, 0) R(\theta)$ $ ( -0.34516457, \, 0.74809598 )R(\theta)$
$\mathbf{\theta_0= 0.1 \pi, \, \, \, \, \, \theta \in [0.089 \pi, \, 0.12\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2680, \, 0)$ $( -0.9999, \, 0 )$
$0.1$ $(0.28699887, \, 0.097087533 )$ $ ( -0.99513360, \, 0.068868347)$
$0.2$ $(0.33390510, \, 0.17420817 )$ $( -0.98094971, \, 0.13739837 )$
$0.3$ $(0.39193700, \, 0.22664276 )$ $( -0.95760880, \, 0.20514818)$
$0.4$ $(0.45044849, \, 0.25815778 )$ $( -0.92539244, \, 0.27156414 ) $
$0.5$ $(0.50439276, \, 0.27354220 )$ $ (-0.88455797, \, 0.33601715 ) $
$0.6$ $(0.55150942, \, 0.27662332 )$ $ ( -0.83534366, \, 0.39783468 )$
$0.7$ $(0.59074871, \, 0.27023456 )$ $ ( -0.77798406, \, 0.45632275 ) $
$0.8$ $(0.62156413, \, 0.25649856 )$ $( -0.71271941, \, 0.51078103 ) $
$0.9$ $(0.64359767, \, 0.23707560 )$ $( -0.63979436, \, 0.56051180 ) $
$1$ $(0.69032914, \, 0) R(\theta)$ $ ( -0.34516457, \, 0.74809598 )R(\theta)$
Table 7.  The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.084\pi, \, 0.089 \pi]$.
$\mathbf{\theta_0= 0.085 \pi, \, \, \, \, \, \theta \in [0.084 \pi, \, 0.089\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2818, \, 0)$ $( -0.9827, \, 0 )$
$0.1$ $( 0.30528479, \, 0.10386817 )$ $ ( -0.97782312, \, 0.068296014 )$
$0.2$ $( 0.36044290, \, 0.18168802 )$ $( -0.96333464, \, 0.13622832 )$
$0.3$ $(0.42517243, \, 0.23070675 )$ $( -0.93953669, \, 0.20329795 )$
$0.4$ $( 0.48803512, \, 0.25746053 )$ $( -0.90673504, \, 0.26887596 ) $
$0.5$ $(0.54454099, \, 0.26800959 )$ $ (-0.86520333, \, 0.33225532 ) $
$0.6$ $(0.59300170, \, 0.26663595 )$ $ (-0.81520260, \, 0.39268889 )$
$0.7$ $(0.63277000, \, 0.25631044 )$ $ ( -0.75700467, \, 0.44941510 ) $
$0.8$ $(0.66356766, \, 0.23917705 )$ $( -0.69090435, \, 0.50167763 ) $
$0.9$ $(0.68521757, \, 0.21687569 )$ $( -0.61721738, \, 0.54873962 ) $
$1$ $(0.72320338, \, 0) R(\theta)$ $ ( -0.36160169, \, 0.71048767 )R(\theta)$
$\mathbf{\theta_0= 0.085 \pi, \, \, \, \, \, \theta \in [0.084 \pi, \, 0.089\pi ]}$
$t$ $\tilde{q}_1$ $\tilde{q}_2$
$0$ $(0.2818, \, 0)$ $( -0.9827, \, 0 )$
$0.1$ $( 0.30528479, \, 0.10386817 )$ $ ( -0.97782312, \, 0.068296014 )$
$0.2$ $( 0.36044290, \, 0.18168802 )$ $( -0.96333464, \, 0.13622832 )$
$0.3$ $(0.42517243, \, 0.23070675 )$ $( -0.93953669, \, 0.20329795 )$
$0.4$ $( 0.48803512, \, 0.25746053 )$ $( -0.90673504, \, 0.26887596 ) $
$0.5$ $(0.54454099, \, 0.26800959 )$ $ (-0.86520333, \, 0.33225532 ) $
$0.6$ $(0.59300170, \, 0.26663595 )$ $ (-0.81520260, \, 0.39268889 )$
$0.7$ $(0.63277000, \, 0.25631044 )$ $ ( -0.75700467, \, 0.44941510 ) $
$0.8$ $(0.66356766, \, 0.23917705 )$ $( -0.69090435, \, 0.50167763 ) $
$0.9$ $(0.68521757, \, 0.21687569 )$ $( -0.61721738, \, 0.54873962 ) $
$1$ $(0.72320338, \, 0) R(\theta)$ $ ( -0.36160169, \, 0.71048767 )R(\theta)$
[1]

Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158

[2]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85

[3]

Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609

[4]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229

[5]

Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569

[6]

Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062

[7]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519

[8]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[9]

Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169

[10]

Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987

[11]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299

[12]

Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631

[13]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080

[14]

Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074

[15]

Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157

[16]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379

[17]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523

[18]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009

[19]

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 & Continuous Dynamical Systems - A, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137

[20]

Jean-Baptiste Caillau, Bilel Daoud, Joseph Gergaud. Discrete and differential homotopy in circular restricted three-body control. Conference Publications, 2011, 2011 (Special) : 229-239. doi: 10.3934/proc.2011.2011.229

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (63)
  • HTML views (221)
  • Cited by (0)

Other articles
by authors

[Back to Top]