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A symplectic integrator for the symmetry reduced and regularised planar 3-body problem with vanishing angular momentum

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Abstract

We construct an explicit reversible symplectic integrator for the planar 3-body problem with zero angular momentum. We start with a Hamiltonian of the planar 3-body problem that is globally regularised and fully symmetry reduced. This Hamiltonian is a sum of 10 polynomials each of which can be integrated exactly, and hence a symplectic integrator is constructed. The performance of the integrator is examined with three numerical examples: The figure eight, the Pythagorean orbit, and a periodic collision orbit.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    Danya Rose & Holger R. Dullin

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  1. Danya Rose
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  2. Holger R. Dullin
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Correspondence to Danya Rose.

Appendix: Integrator stages

Appendix: Integrator stages

Section 3.1 described how to integrate a monomial Hamiltonian, and Sect. 3.2 described the splitting of Eq. (2) into a minimal number of solvable parts and those solutions. Here we use those solutions to build an explicit first order symplectic composition method for (2).

Let the timestep be \(\varDelta \tau \), let \(\mu _j = (m_k + m_l)\), let the values of the system before and after one timestep respectively be \(\varvec{z}_0 = (\alpha _{1,0},\dots ,\pi _{3,0})^T\) and \(\varvec{z}_1 = (\alpha _{1,1},\ldots ,\pi _{3,1})^T\) and intermediate steps be \(\varvec{\xi }_i = (\alpha _{1,.i-1},\ldots ,\pi _{3,.i-1})^T\). Now

$$\begin{aligned} \varvec{\xi }_1&= \left( \begin{array}{c} \alpha _{1,0} \\ \alpha _{2,0} \\ \alpha _{3,0} \\ \pi _{1,0} + 2 \alpha _{1,0} \left( \left( 2\alpha _{1,0}^2 + a_{1,0}\right) \left( h a_{1,0} + M_1\right) + m_1 \alpha _{1,0} \mu _1 a_{1,0}\right) \varDelta \tau \\ \pi _{2,0} + 2 \alpha _{2,0} \left( \left( 2\alpha _{2,0}^2 + a_{2,0}\right) \left( h a_{2,0} + M_2\right) + m_2 \alpha _{2,0} \mu _2 a_{2,0}\right) \varDelta \tau \\ \pi _{3,0} + 2 \alpha _{3,0} \left( \left( 2\alpha _{3,0}^2 + a_{3,0}\right) \left( h a_{3,0} + M_3\right) + m_3 \alpha _{3,0} \mu _3 a_{3,0}\right) \varDelta \tau \end{array}\right) \\ \varvec{\xi }_2&= \left( \begin{array}{c} \alpha _{1,.1}\ \exp \left( \frac{1}{4} \left( N_2 \alpha _{2,.1}^2 + N_3 \alpha _{3,.1}^2\right) \alpha _{1,.1}\ \pi _{1,.1}\ \varDelta \tau \right) \\ \alpha _{2,.1} \\ \alpha _{3,.1} \\ \pi _{1,.1}\ \exp \left( -\frac{1}{4} \left( N_2 \alpha _{2,.1}^2 + N_3 \alpha _{3,.1}^2\right) \alpha _{1,.1}\ \pi _{1,.1}\ \varDelta \tau \right) \\ \pi _{2,.1} - \frac{1}{4}\ N_2\ \alpha _{1,.1}^2\ \pi _{1,.1}^2\ \alpha _{2,.1} \varDelta \tau \\ \pi _{3,.1} - \frac{1}{4}\ N_3\ \alpha _{1,.1}^2\ \pi _{1,.1}^2\ \alpha _{3,.1} \varDelta \tau \end{array}\right) \\ \varvec{\xi }_3&= \left( \begin{array}{c} \alpha _{1,.2} \\ \alpha _{2,.2}\ \exp \left( \frac{1}{4} \left( N_3 \alpha _{3,.2}^2 + N_1 \alpha _{1,.2}^2\right) \alpha _{2,.2}\ \pi _{2,.2}\ \varDelta \tau \right) \\ \alpha _{3,.2} \\ \pi _{1,.2} - \frac{1}{4}\ N_1\ \alpha _{2,.2}^2\ \pi _{2,.2}^2\ \alpha _{1,.2} \varDelta \tau \\ \pi _{2,.2}\ \exp \left( -\frac{1}{4} \left( N_3 \alpha _{3,.2}^2 + N_1 \alpha _{1,.2}^2\right) \alpha _{2,.2}\ \pi _{2,.2}\ \varDelta \tau \right) \\ \pi _{3,.2} - \frac{1}{4}\ N_3\ \alpha _{2,.2}^2\ \pi _{2,.2}^2\ \alpha _{3,.2} \varDelta \tau \end{array}\right) \\ \varvec{\xi }_4&= \left( \begin{array}{c} \alpha _{1,.3} \\ \alpha _{2,.3} \\ \alpha _{3,.3}\ \exp \left( \frac{1}{4} \left( N_1 \alpha _{1,.3}^2 + N_2 \alpha _{2,.3}^2\right) \alpha _{3,.3}\ \pi _{3,.3}\ \varDelta \tau \right) \\ \pi _{1,.3} - \frac{1}{4}\ N_1\ \alpha _{3,.3}^2\ \pi _{3,.3}^2\ \alpha _{1,.3} \varDelta \tau \\ \pi _{2,.3} - \frac{1}{4}\ N_2\ \alpha _{3,.3}^2\ \pi _{3,.3}^2\ \alpha _{2,.3} \varDelta \tau \\ \pi _{3,.3}\ \exp \left( -\frac{1}{4} \left( N_1 \alpha _{1,.3}^2 + N_2 \alpha _{2,.3}^2\right) \alpha _{3,.3}\ \pi _{3,.3}\ \varDelta \tau \right) \end{array}\right) \end{aligned}$$
$$\begin{aligned} \varvec{\xi }_5&= \left( \begin{array}{c} \alpha _{1,.4} + \pi _{1,.4} \left( \frac{1}{4} \left( N_2 \alpha _{2,.4}^4 + N_3 \alpha _{3,.4}^4\right) + \frac{1}{2 m_1}\alpha _{2,.4}^2 \alpha _{3,.4}^2\right) \varDelta \tau \\ \alpha _{2,.4} \\ \alpha _{3,.4} \\ \pi _{1,.4} \\ \pi _{2,.4} + \frac{1}{2} \pi _{1,.4}^2 \left( N_2 \alpha _{2,.4}^3 + \frac{1}{m_1} \alpha _{2,.4} \alpha _{3,.4}^2\right) \varDelta \tau \\ \pi _{3,.4} + \frac{1}{2} \pi _{1,.4}^2 \left( N_3 \alpha _{3,.4}^3 + \frac{1}{m_1} \alpha _{3,.4} \alpha _{2,.4}^2\right) \varDelta \tau \end{array}\right) \\ \varvec{\xi }_6&= \left( \begin{array}{c} \alpha _{1,.5} \\ \alpha _{2,.5} + \pi _{2,.5} \left( \frac{1}{4} \left( N_3 \alpha _{3,.5}^4 + N_1 \alpha _{1,.5}^4\right) + \frac{1}{2 m_2}\alpha _{3,.5}^2 \alpha _{1,.5}^2\right) \varDelta \tau \\ \alpha _{3,.5} \\ \pi _{1,.5} + \frac{1}{2} \pi _{2,.5}^2 \left( N_1 \alpha _{1,.5}^3 + \frac{1}{m_2} \alpha _{1,.5} \alpha _{3,.5}^2\right) \varDelta \tau \\ \pi _{2,.5} \\ \pi _{3,.5} + \frac{1}{2} \pi _{2,.5}^2 \left( N_3 \alpha _{3,.5}^3 + \frac{1}{m_2} \alpha _{3,.5} \alpha _{1,.5}^2\right) \varDelta \tau \end{array}\right) \\ \varvec{\xi }_7&= \left( \begin{array}{c} \alpha _{1,.6} \\ \alpha _{2,.6} \\ \alpha _{3,.6} + \pi _{3,.6} \left( \frac{1}{4} \left( N_1 \alpha _{1,.6}^4 + N_2 \alpha _{2,.6}^4\right) + \frac{1}{2 m_3}\alpha _{1,.6}^2 \alpha _{2,.6}^2\right) \varDelta \tau \\ \pi _{1,.6} + \frac{1}{2} \pi _{3,.6}^2 \left( N_1 \alpha _{1,.6}^3 + \frac{1}{m_3} \alpha _{1,.6} \alpha _{2,.6}^2\right) \varDelta \tau \\ \pi _{2,.6} + \frac{1}{2} \pi _{3,.5}^2 \left( N_2 \alpha _{2,.5}^3 + \frac{1}{m_3} \alpha _{2,.5} \alpha _{1,.5}^2\right) \varDelta \tau \\ \pi _{3,.6} \end{array}\right) \\ \varvec{\xi }_8&= \left( \begin{array}{c} \alpha _{1,.7} \left( 1 + \frac{1}{2} \left( \frac{1}{m_3} \alpha _{2,.7} \pi _{2,.7} + \frac{1}{m_2} \alpha _{3,.7} \pi _{3,.7}\right) \alpha _{1,.7}^2 \varDelta \tau \right) ^{-\frac{1}{2}} \\ \alpha _{2,.7} \exp \left( -\frac{1}{4 m_3}\alpha _{1,.7}^3 \pi _{1,.7} \varDelta \tau \right) \\ \alpha _{3,.7} \exp \left( -\frac{1}{4 m_2}\alpha _{1,.7}^3 \pi _{1,.7} \varDelta \tau \right) \\ \pi _{1,.7} \left( 1 + \frac{1}{2} \left( \frac{1}{m_3} \alpha _{2,.7} \pi _{2,.7} + \frac{1}{m_2} \alpha _{3,.7} \pi _{3,.7}\right) \alpha _{1,.7}^2 \varDelta \tau \right) ^{\frac{3}{2}} \\ \pi _{2,.7} \exp \left( \frac{1}{4 m_3}\alpha _{1,.7}^3 \pi _{1,.7} \varDelta \tau \right) \\ \pi _{3,.7} \exp \left( \frac{1}{4 m_2}\alpha _{1,.7}^3 \pi _{1,.7} \varDelta \tau \right) \end{array}\right) \end{aligned}$$
$$\begin{aligned} \varvec{\xi }_9&= \left( \begin{array}{c} \alpha _{1,.8} \exp \left( -\frac{1}{4 m_3}\alpha _{2,.8}^3 \pi _{2,.8} \varDelta \tau \right) \\ \alpha _{2,.8} \left( 1 + \frac{1}{2} \left( \frac{1}{m_1} \alpha _{3,.8} \pi _{3,.8} + \frac{1}{m_3} \alpha _{1,.8} \pi _{1,.8}\right) \alpha _{2,.8}^2 \varDelta \tau \right) ^{-\frac{1}{2}} \\ \alpha _{3,.8} \exp \left( -\frac{1}{4 m_1}\alpha _{2,.8}^3 \pi _{2,.8} \varDelta \tau \right) \\ \pi _{1,.8} \exp \left( \frac{1}{4 m_3}\alpha _{2,.8}^3 \pi _{2,.8} \varDelta \tau \right) \\ \pi _{2,.8} \left( 1 + \frac{1}{2} \left( \frac{1}{m_1} \alpha _{3,.8} \pi _{3,.8} + \frac{1}{m_3} \alpha _{1,.8} \pi _{1,.8}\right) \alpha _{2,.8}^2 \varDelta \tau \right) ^{\frac{3}{2}} \\ \pi _{3,.8} \exp \left( \frac{1}{4 m_1}\alpha _{2,.8}^3 \pi _{2,.8} \varDelta \tau \right) \end{array}\right) \\ \varvec{\xi }_{10}&= \left( \begin{array}{c} \alpha _{1,.9} \exp \left( -\frac{1}{4 m_2}\alpha _{3,.9}^3 \pi _{3,.9} \varDelta \tau \right) \\ \alpha _{2,.9} \exp \left( -\frac{1}{4 m_1}\alpha _{3,.9}^3 \pi _{3,.9} \varDelta \tau \right) \\ \alpha _{3,.9} \left( 1 + \frac{1}{2} \left( \frac{1}{m_2} \alpha _{1,.9} \pi _{1,.9} + \frac{1}{m_1} \alpha _{2,.9} \pi _{2,.9}\right) \alpha _{3,.9}^2 \varDelta \tau \right) ^{-\frac{1}{2}} \\ \pi _{1,.9} \exp \left( \frac{1}{4 m_2}\alpha _{3,.9}^3 \pi _{3,.9} \varDelta \tau \right) \\ \pi _{2,.9} \exp \left( \frac{1}{4 m_1}\alpha _{3,.9}^3 \pi _{3,.9} \varDelta \tau \right) \\ \pi _{3,.9} \left( 1 + \frac{1}{2} \left( \frac{1}{m_2} \alpha _{1,.9} \pi _{1,.9} + \frac{1}{m_1} \alpha _{2,.9} \pi _{2,.9}\right) \alpha _{3,.9}^2 \varDelta \tau \right) ^{\frac{3}{2}} \end{array}\right) = \varvec{z}_1. \end{aligned}$$

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Rose, D., Dullin, H.R. A symplectic integrator for the symmetry reduced and regularised planar 3-body problem with vanishing angular momentum. Celest Mech Dyn Astr 117, 169–185 (2013). https://doi.org/10.1007/s10569-013-9503-x

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