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

Rongchang Liu; Jiangyuan Li; Duokui Yan

doi:10.3934/dcds.2018090

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2018, Volume 38, Issue 4: 2187-2206. Doi: 10.3934/dcds.2018090

This issue Previous Article Theory of rotated equations and applications to a population model Next Article Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data

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
* Corresponding author: Duokui Yan

Received: May 31, 2017

Revised: September 30, 2017

Published: June 2018

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

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Abstract

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 }$.

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

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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.

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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 $Q_E$.

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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 $\mathcal{A}_{col}$.

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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)$

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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)$

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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)$

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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)$

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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)$

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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)$

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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)$

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

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