Article Contents
Article Contents

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

• * Corresponding author: Duokui Yan
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 $\mathit{\theta }$.

Mathematics Subject Classification: Primary: 70F07; Secondary: 70F10, 53D12.

 Citation:

• 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 $\theta \in [0.181\pi, \, 0.183 \pi]$.

 $\mathbf{\theta_0= 0.182 \pi, \, \, \, \, \, \theta \in [0.181 \pi, \, 0.183 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.1109, \, 0 )$ $( -1.1189,\, 0 )$ $0.1$ $( 0.11378134, \, 0.035120592)$ $( -1.1144897, \, 0.089194621)$ $0.2$ $(0.12212758, \, 0.068373015 )$ $( -1.1012708, \,0.17787543 )$ $0.3$ $(0.13510199, \, 0.098115021 )$ $(-1.0792782, \, 0.26552094 )$ $0.4$ $( 0.15147483, \, 0.12309328 )$ $( -1.0485685, \, 0.35159462 )$ $0.5$ $( 0.16980434, \, 0.14251378 )$ $( -1.0092177, \, 0.43553762)$ $0.6$ $( 0.18860029, \, 0.15603011)$ $(-0.96132044, \, 0.51676178)$ $0.7$ $(0.20644197, \, 0.16368275 )$ $( -0.90499096, \, 0.59464214)$ $0.8$ $( 0.22204784, \, 0.16582253 )$ $(-0.84036636, \, 0.66850889)$ $0.9$ $( 0.23430842, \, 0.16304077 )$ $( -0.76761388, \, 0.73763832)$ $1$ $(0.28822237, \, 0 ) R(\theta)$ $(-0.144111185, \, 1.0455201)R(\theta)$

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 $\theta \in [0.176\pi, \, 0.181 \pi]$.

 $\mathbf{\theta_0= 0.18 \pi, \, \, \, \, \, \theta \in [0.176 \pi, \, 0.181 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $( 0.1198, \, 0 )$ $(-1.1166, \, 0 )$ $0.1$ $( 0.12300003, \, 0.038069081 )$ $(-1.1122108, \, 0.087838457 )$ $0.2$ $(0.13225041, \, 0.074032720 )$ $( -1.0990555, \, 0.17518597 )$ $0.3$ $(0.14657647, \, 0.10606411 )$ $( -1.0771702, \, 0.26154328 )$ $0.4$ $(0.16456742, \, 0.13280675 )$ $( -1.0466117, \, 0.34639491 )$ $0.5$ $(0.18460176, \, 0.15344287 )$ $(-1.0074550, \, 0.42920169 )$ $0.6$ $( 0.20503978, \, 0.16765957 )$ $( -0.95979063, \, 0.50939364 )$ $0.7$ $(0.22435124, \, 0.17556129 )$ $( -0.90372458, \, 0.58636242 )$ $0.8$ $( 0.24118113, \, 0.17757208 )$ $(-0.83938055, \, 0.65945294 )$ $0.9$ $( 0.25437250, \, 0.17435316 )$ $(-0.76690612, \, 0.72795349 )$ $1$ $( 0.31146657, \, 0 ) R(\theta)$ $(-0.155733285, \, 1.0357709)R(\theta)$

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 $\theta \in [0.165\pi, \, 0.176 \pi]$.

 $\mathbf{\theta_0= 0.173 \pi, \, \, \, \, \, \theta \in [0.165 \pi, \, 0.176 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $( 0.1454,\, 0 )$ $( -1.1064,\, 0 )$ $0.1$ $( 0.14967657,\, 0.046886901 )$ $(-1.1020407,\, 0.083918190 )$ $0.2$ $(0.16194195,\, 0.090810510 )$ $(-1.0889788,\, 0.16740323 )$ $0.3$ $(0.18067059,\, 0.12932570 )$ $( -1.0672593,\, 0.25001086 )$ $0.4$ $(0.20377305,\, 0.16080303 )$ $( -1.0369499,\, 0.33127610 )$ $0.5$ $( 0.22901187,\, 0.18445175 )$ $( -0.99813266,\, 0.41070550 )$ $0.6$ $(0.25429011,\, 0.20016641 )$ $( -0.95089814,\, 0.48777071 )$ $0.7$ $(0.27778726,\, 0.20832504 )$ $( -0.89534096,\, 0.56190183 )$ $0.8$ $(0.29798679,\, 0.20961764 )$ $( -0.83155952,\, 0.63247939 )$ $0.9$ $(0.31364961,\, 0.20493038 )$ $(-0.75965878,\, 0.69882326 )$ $1$ $( 0.37739476,\, 0 ) R(\theta)$ $( -0.18869738,\, 1.0021646)R(\theta)$

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 $\theta \in [0.146\pi, \, 0.165 \pi]$.

 $\mathbf{\theta_0= 0.16 \pi, \, \, \, \, \, \theta \in [0.146 \pi, \, 0.165 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.1803, \, 0 )$ $( -1.0860,\, 0 )$ $0.1$ $( 0.18658246,\, 0.059576674 )$ $(-1.0816238,\, 0.078768710)$ $0.2$ $( 0.20429715,\, 0.11438956 )$ $(-1.0685224,\, 0.15716562)$ $0.3$ $( 0.23056861,\, 0.16092393 )$ $( -1.0467705,\, 0.23480016)$ $0.4$ $( 0.26186968,\, 0.19737903 )$ $( -1.0164725,\, 0.31125003)$ $0.5$ $( 0.29489886,\, 0.22338233 )$ $( -0.97774453,\, 0.38605495)$ $0.6$ $(0.32695826,\, 0.23945199 )$ $( -0.93070180,\, 0.45871449)$ $0.7$ $(0.35597786,\, 0.24655415 )$ $( -0.87545128,\, 0.52868620)$ $0.8$ $(0.38039590,\, 0.24583791 )$ $( -0.81208876,\, 0.59538118)$ $0.9$ $(0.39901814,\, 0.23851003 )$ $( -0.74069823,\, 0.65815496 )$ $1$ $(0.46894463,\, 0) R(\theta)$ $( -0.234472315,\, 0.94630060)R(\theta)$

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 $\theta \in [0.12\pi, \, 0.146 \pi]$.

 $\mathbf{\theta_0= 0.132 \pi, \, \, \, \, \, \theta \in [0.12 \pi, \, 0.146\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2306, \, 0)$ $( -1.0421, \, 0)$ $0.1$ $( 0.24197875, \, 0.079876150 )$ $(-1.0375636, \, 0.072168012 )$ $0.2$ $( 0.27249601, \, 0.14948693 )$ $( -1.0240177, \, 0.14401235 )$ $0.3$ $( 0.31440456, \, 0.20354167 )$ $( -1.0016223, \, 0.21516044 )$ $0.4$ $(0.36051601, \, 0.24151617 )$ $( -0.97057606, \, 0.28517262 )$ $0.5$ $(0.40588817, \, 0.26512985 )$ $( -0.93107630, \, 0.35354922 )$ $0.6$ $(0.44748116, \, 0.27663491 )$ $( -0.88330444, \, 0.41974435 )$ $0.7$ $(0.48345138, \, 0.27815124 )$ $( -0.82742450, \, 0.48317696 )$ $0.8$ $(0.51262863, \, 0.27151295 )$ $( -0.76358450, \, 0.54323643 )$ $0.9$ $(0.53419879, \, 0.25829537 )$ $( -0.69191492, \, 0.59928148 )$ $1$ $(0.59692629, \, 0) R(\theta)$ $( -0.298463145, \, 0.84227284 )R(\theta)$

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 $\theta \in [0.089\pi, \, 0.12 \pi]$.

 $\mathbf{\theta_0= 0.1 \pi, \, \, \, \, \, \theta \in [0.089 \pi, \, 0.12\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2680, \, 0)$ $( -0.9999, \, 0 )$ $0.1$ $(0.28699887, \, 0.097087533 )$ $( -0.99513360, \, 0.068868347)$ $0.2$ $(0.33390510, \, 0.17420817 )$ $( -0.98094971, \, 0.13739837 )$ $0.3$ $(0.39193700, \, 0.22664276 )$ $( -0.95760880, \, 0.20514818)$ $0.4$ $(0.45044849, \, 0.25815778 )$ $( -0.92539244, \, 0.27156414 )$ $0.5$ $(0.50439276, \, 0.27354220 )$ $(-0.88455797, \, 0.33601715 )$ $0.6$ $(0.55150942, \, 0.27662332 )$ $( -0.83534366, \, 0.39783468 )$ $0.7$ $(0.59074871, \, 0.27023456 )$ $( -0.77798406, \, 0.45632275 )$ $0.8$ $(0.62156413, \, 0.25649856 )$ $( -0.71271941, \, 0.51078103 )$ $0.9$ $(0.64359767, \, 0.23707560 )$ $( -0.63979436, \, 0.56051180 )$ $1$ $(0.69032914, \, 0) R(\theta)$ $( -0.34516457, \, 0.74809598 )R(\theta)$

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 $\theta \in [0.084\pi, \, 0.089 \pi]$.

 $\mathbf{\theta_0= 0.085 \pi, \, \, \, \, \, \theta \in [0.084 \pi, \, 0.089\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2818, \, 0)$ $( -0.9827, \, 0 )$ $0.1$ $( 0.30528479, \, 0.10386817 )$ $( -0.97782312, \, 0.068296014 )$ $0.2$ $( 0.36044290, \, 0.18168802 )$ $( -0.96333464, \, 0.13622832 )$ $0.3$ $(0.42517243, \, 0.23070675 )$ $( -0.93953669, \, 0.20329795 )$ $0.4$ $( 0.48803512, \, 0.25746053 )$ $( -0.90673504, \, 0.26887596 )$ $0.5$ $(0.54454099, \, 0.26800959 )$ $(-0.86520333, \, 0.33225532 )$ $0.6$ $(0.59300170, \, 0.26663595 )$ $(-0.81520260, \, 0.39268889 )$ $0.7$ $(0.63277000, \, 0.25631044 )$ $( -0.75700467, \, 0.44941510 )$ $0.8$ $(0.66356766, \, 0.23917705 )$ $( -0.69090435, \, 0.50167763 )$ $0.9$ $(0.68521757, \, 0.21687569 )$ $( -0.61721738, \, 0.54873962 )$ $1$ $(0.72320338, \, 0) R(\theta)$ $( -0.36160169, \, 0.71048767 )R(\theta)$
•  R. Broucke  and  D. Boggs , Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975) , 13-38.  doi: 10.1007/BF01228732. 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. 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. 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. 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. W. B. Gordon , A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977) , 961-971.  doi: 10.2307/2373993. 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. 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. C. Marchal , How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002) , 325-353.  doi: 10.1023/A:1020128408706. 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. B. Shi , R. Liu , D. 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. 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. G. Yu, Spatial double choreographies of the Newtonian $2n$-body problem, arXiv: 1608.07956.

Figures(3)

Tables(7)