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Dedicated symplectic integrators for rotation motions

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Abstract

We propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the coefficients of integrators on the moments of inertia of the integrated body, we can construct symplectic dedicated integrators with fewer stages than in the general case for a splitting in three parts of the Hamiltonian. We perform numerical tests to compare the developed dedicated fourth-order integrators to the existing reference integrators for the water molecule. We also estimate analytically the accuracy of these new integrators for the set of the rigid bodies and conclude that they are more accurate than the existing ones only for very asymmetric bodies.

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References

  • Blanes, S., Casas, F., Murua, A.: Splitting and composition methods in the numerical integration of differential equations. Boletin de la Sociedad Espanola de Matematica Aplicada SeMA 45, 89–145 (2008)

    MathSciNet  MATH  Google Scholar 

  • Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal). PhD thesis, Mathematical Institute, University of Innsbruck, Austria, English Translation in Journal of Symbolic Computation, Special Issue on Logic, Mathematics, and Computer Science: Interactions. Vol. 41, Number 3–4, pp. 475–511, 2006 (1965)

  • Celledoni, E., Fassò, F., Säfström, N., Zanna, A.: The exact computation of the free rigid body motion and its use in splitting methods. SIAM J. Sci. Comput. 30(4), 2084–2112 (2008)

    Article  MathSciNet  Google Scholar 

  • Dullweber, A., Leimkuhler, B., McLachlan, R.I.: Symplectic splitting methods for rigid body molecular dynamics. J. Chem. Phys. 107(15), 5840–5851 (1997)

    Article  ADS  Google Scholar 

  • Eisenberg, D., Kauzmann, W.: The Structure and Properties of Water. Clarendon Press, Oxford (1969)

    Google Scholar 

  • Farrés, A., Laskar, J., Blanes, S., Casas, F., Makazaga, J., Murua, A.: High precision symplectic integrators for the solar system. Celest. Mech. Dyn. Astron. 116, 141–174 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  • Fassò, F.: Comparison of splitting algorithms for the rigid body. J. Comput. Phys. 189(2), 527–538 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  • Hairer, E., Vilmart, G.: Preprocessed discrete Moser–Veselov algorithm for the full dynamics of a rigid body. J. Phys. A Math. Gen. 39(42), 13225–13235 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  • Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Jacobi, C.G.J.: Sur la rotation d’un corps. Journal für die reine und angewandte Mathematik 1850(39), 293–350 (1850)

    Article  Google Scholar 

  • Koseleff, P.V.: Calcul formel pour les méthodes de Lie en mécanique hamiltonienne. PhD thesis, École Polytechnique (1993)

  • Koseleff, P.V.: Exhaustive search of symplectic integrators using computer algebra. Integr. Algorithms Class. Mech. Fields Inst. Commun. 10, 103–119 (1996)

    MathSciNet  MATH  Google Scholar 

  • McLachlan, R.I.: Explicit Lie–Poisson integration and the Euler equations. Phys. Rev. Lett. 71(19), 3043–3046 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  • McLachlan, R.I.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16(1), 151–168 (1995)

    Article  MathSciNet  Google Scholar 

  • Munthe-Kaas, H., Owren, B.: Computations in a free Lie algebra. Philos. Trans. R. Soc. Lond. Ser. A 357(1754), 957–981 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  • Omelyan, I.P.: Advanced gradientlike methods for rigid-body molecular dynamics. J. Chem. Phys. 127(4), 044102 (2007)

    Article  ADS  Google Scholar 

  • Reich, S.: Momentum conserving symplectic integrators. Physica D 76(4), 375–383 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  • Sheng, Q.: Solving linear partial differential equations by exponential splitting. IMA J. Numer. Anal. 9(2), 199–212 (1989)

    Article  MathSciNet  Google Scholar 

  • Skokos, C., Gerlach, E., Bodyfelt, J.D., Papamikos, G., Eggl, S.: High order three part split symplectic integrators: efficient techniques for the long time simulation of the disordered discrete nonlinear Schrödinger equation. Phys. Lett. A 378, 1809–1815 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  • Sofroniou, M., Spaletta, G.: Derivation of symmetric composition constants for symmetric integrators. Optim. Methods Softw. 20(4–5), 597–613 (2005)

    Article  MathSciNet  Google Scholar 

  • Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319–323 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  • Suzuki, M.: General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32, 400–407 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  • Tang, Y.F.: A note on the construction of symplectic schemes for splitable Hamiltonian. J. Comput. Math. 20(1), 89–96 (2002)

    MathSciNet  MATH  Google Scholar 

  • Touma, J., Wisdom, J.: Lie–Poisson integrators for rigid body dynamics in the solar system. Astron. J. 107, 1189–1202 (1994)

    Article  ADS  Google Scholar 

  • Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150(5–7), 262–268 (1990)

    Article  ADS  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. ASD, IMCCE-CNRS UMR8028, Observatoire de Paris, PSL Université, Sorbonne Université, 77 Av. Denfert-Rochereau, 75014, Paris, France

    Jacques Laskar & Timothée Vaillant

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  1. Jacques Laskar
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  2. Timothée Vaillant
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Correspondence to Jacques Laskar.

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Solutions for the fourth-order integrators N

Solutions for the fourth-order integrators N

For each fourth-order integrator N of Sect. 3.2, the values of the coefficients \(a_i\), \(b_i\), \(c_i\) are given by the following equations for moments of inertia determined by the values of \(1+x=I_1/I_2\) and \(1+y=I_1/I_3\).

1.1 N1: ABABCBABA

$$\begin{aligned} f_0+a_1f_1+a_1^{2}f_2+a_1^{3}f_3= & {} 0 \nonumber \\ g_0+b_1g_1+a_1g_2= & {} 0 \end{aligned}$$
(78)
$$\begin{aligned} f_0= & {} -1-3y-3y^{2}-2xy^{2}-3x^{2}y-12x^{2}y^{2}-12x^{2}y^{3}-4x^{2}y^{4}\\ f_1= & {} 6+18y+24y^{2}-12xy+12xy^{2}-18x^{2}-78x^{2}y-72x^{2}y^{2}-24x^{2}y^{3}\\ f_2= & {} -12-36y-72y^{2}-48x-72xy-168xy^{2}-48xy^{3}-36x^{2}+12x^{2}y \\ {} f_3= & {} -\,24-72y-144xy-48xy^{2}+72x^{2}+24x^{2}y \\ g_0= & {} -1-y-4x-6xy-2xy^{2}-3x^{2}-5x^{2}y-2x^{2}y^{2} \\ g_1= & {} 4+6y+10x+18xy+4xy^{2}+6x^{2}+12x^{2}y+4x^{2}y^{2} \\ g_2= & {} -2-6y-12xy-4xy^{2}+6x^{2}+2x^{2}y \end{aligned}$$

1.2 N2: ABACACABA

$$\begin{aligned} f_0+a_1f_1+a_1^{2}f_2= & {} 0 \nonumber \\ g_0+a_2g_1+a_1g_2= & {} 0 \end{aligned}$$
(79)
$$\begin{aligned} f_0= & {} 1+3y-3y^{3}-8xy^{2}-12xy^{3}+x^{2}y-3x^{2}y^{2}-9x^{2}y^{3}+x^{2}y^{4}-4x^{3}y^{3}+x^{4}y^{2} \\ f_1= & {} -\,6-30y-18y^{2}+18y^{3}-48xy-48xy^{2}+48xy^{3}+6x^{2}-30x^{2}y-78x^{2}y^{2} \\&+\,30x^{2}y^{3}-48x^{3}y^{2}+12x^{4}y \\ f_2= & {} 36y+36y^{2}-36y^{3}-36x+36xy+180xy^{2}-36xy^{3}-72x^{2}-144x^{2}y \\&+\,144x^{2}y^{2}-144x^{3}y+36x^{4} \\ g_0= & {} -1-3y-3y^{2}-2xy^{2}+x^{2}y \\ g_1= & {} 6+12y+6y^{2} \\ g_2= & {} 6y^{2}-12xy+6x^{2} \end{aligned}$$

1.3 N3: ABACBCABA

$$\begin{aligned} f_0+a_1f_1+a_1^{2}f_2+a_1^{3}f_3= & {} 0 \nonumber \\ g_0+b_1g_1+a_1g_2+a_1^{2}g_3= & {} 0 \end{aligned}$$
(80)
$$\begin{aligned} f_0= & {} 1+3y-3y^{3}-4xy^{2}-6xy^{3}+x^{2}y^{4} \\ f_1= & {} -\,6-30y-18y^{2}+18y^{3}-24xy-24xy^{2}+24xy^{3}+12x^{2}y^{3} \\ f_2= & {} 60y+72y^{2}-36y^{3}-24x+36xy+144xy^{2}-12xy^{3}+48x^{2}y^{2}\\ f_3= & {} 24-72y^{2}+72x+144xy-24xy^{2}+48x^{2}y \\ g_0= & {} -\,3-18y-39y^{2}-36y^{3}-12y^{4}-2x-15xy-39xy^{2}-41xy^{3}-15xy^{4} \\&-\,2x^{2}y^{2}-3x^{2}y^{3}+x^{2}y^{5} \\ g_1= & {} 2+12y+30y^{2}+36y^{3}+18y^{4}+2x+12xy+34xy^{2}+48xy^{3}+30xy^{4} \\&+\,4x^{2}y^{2}+12x^{2}y^{3}+14x^{2}y^{4}+2x^{3}y^{4} \\ g_2= & {} 2+30y+84y^{2}+78y^{3}+18y^{4}-6x+6xy+78xy^{2}+102xy^{3}+24xy^{4} \\&+\,4x^{2}y+30x^{2}y^{2}+48x^{2}y^{3}+10x^{2}y^{4}+4x^{3}y^{3} \\ g_3= & {} 12+24y-24y^{2}-72y^{3}-36y^{4}+36x+144xy+168xy^{2}+48xy^{3}-12xy^{4} \\&+\,24x^{2}y+48x^{2}y^{2}+24x^{2}y^{3} \end{aligned}$$

1.4 N4: ABCABACBA

$$\begin{aligned} f_0+a_1f_1+a_1^{2}f_2+a_1^{3}f_3= & {} 0 \nonumber \\ g_0+b_1g_1+a_1g_2= & {} 0 \end{aligned}$$
(81)
$$\begin{aligned} f_0= & {} 1+3y-3y^{3}-4xy^{2}-6xy^{3}+x^{2}y^{4} \\ f_1= & {} -\,6-30y-18y^{2}+18y^{3}-24xy-24xy^{2}+24xy^{3}+12x^{2}y^{3} \\ f_2= & {} 60y+72y^{2}-36y^{3}-24x+36xy+144xy^{2}-12xy^{3}+48x^{2}y^{2} \\ f_3= & {} 24-72y^{2}+72x+144xy-24xy^{2}+48x^{2}y \\ g_0= & {} -1-y-2x-3xy-xy^{2} \\ g_1= & {} 2+6y+6y^{2}+2x+6xy+8xy^{2}+2x^{2}y^{2} \\ g_2= & {} 2-6y^{2}+6x+12xy-2xy^{2}+4x^{2}y \end{aligned}$$

1.5 N5: ABCACACBA

$$\begin{aligned} f_0+a_1f_1+a_1^{2}f_2+a_1^{3}f_3= & {} 0 \nonumber \\ g_0+c_1g_1+a_1g_2+a_1^{2}g_3= & {} 0 \end{aligned}$$
(82)
$$\begin{aligned} f_0= & {} 1+3y-6xy^{2}-x^{2}y-6x^{2}y^{2}+x^{4}y^{2} \\ f_1= & {} -\,6-30y-36xy+36xy^{2}-6x^{2}-42x^{2}y+24x^{2}y^{2}+12x^{4}y \\ f_2= & {} 84y-48x+144xy-72xy^{2}-36x^{2}+180x^{2}y-24x^{2}y^{2}+24x^{3}y+36x^{4} \\ f_3= & {} 24-72y+144x-144xy+48xy^{2}+216x^{2}-168x^{2}y+144x^{3} \\ g_0= & {} 1+4y+x+9xy-8xy^{2}-3x^{2}+4x^{2}y-25x^{2}y^{2}-6x^{3}\\&-\,7x^{3}y-30x^{3}y^{2}-2x^{3}y^{3}\\&-\,3x^{4}-9x^{4}y-12x^{4}y^{2}-2x^{4}y^{3}-6x^{5}y+x^{5}y^{2}-3x^{6}y \\ g_1= & {} 2+2y+12x+12xy+30x^{2}+34x^{2}y+4x^{2}y^{2}+36x^{3}\\&+\,48x^{3}y+12x^{3}y^{2}+18x^{4}\\&+\,30x^{4}y+14x^{4}y^{2}+2x^{4}y^{3} \\ g_2= & {} -\,4-30y-6x-114xy+28xy^{2}-6x^{2}-210x^{2}y+66x^{2}y^{2}\\&-\,18x^{3}-210x^{3}y+48x^{3}y^{2}\\&+\,4x^{3}y^{3}-36x^{4}-96x^{4}y-2x^{4}y^{2}-36x^{5}-18x^{6} \\ g_3= & {} -12+36y-96x+144xy-24xy^{2}-264x^{2}+264x^{2}y\\&-\,48x^{2}y^{2}-360x^{3}+240x^{3}y\\&-\,24x^{3}y^{2}-252x^{4}+84x^{4}y-72x^{5} \end{aligned}$$

1.6 N6: ABCBABCBA

$$\begin{aligned} f_0+a_1f_1+a_1^{2}f_2+a_1^{3}f_3+a_1^{4}f_4= & {} 0 \nonumber \\ g_0+b_1g_1+a_1g_2+a_1^{2}g_3+a_1^{3}g_4= & {} 0 \end{aligned}$$
(83)
$$\begin{aligned} f_0= & {} 1+3y+3y^{2}-3y^{3}+8xy^{2}-3x^{2}y+3x^{2}y^{2}+3x^{2}y^{3}+x^{2}y^{4} \\ f_1= & {} -\,6-18y-42y^{2}+18y^{3}+48xy-48xy^{2}-18x^{2}+42x^{2}y+18x^{2}y^{2}+6x^{2}y^{3} \\ f_2= & {} 12+36y+180y^{2}-36y^{3}+12x-252xy+132xy^{2}+12xy^{3}+144x^{2}-48x^{2}y \\ f_3= & {} -\,288y^{2}+576xy-288x^{2} \\ f_4= & {} 144y^{2}-288xy+144x^{2} \\ g_0= & {} -\,2y-6y^{2}+4x+11xy-xy^{2}+xy^{3}+xy^{4}+3x^{2}+13x^{2}y+7x^{2}y^{2}+x^{2}y^{3} \\ g_1= & {} -\,4-10y-6y^{2}-6y^{3}-6y^{4}-10x-28xy-16xy^{2}-4xy^{3}-6xy^{4}-6x^{2}\\&-\,18x^{2}y-10x^{2}y^{2}+2x^{2}y^{3} \\ g_2= & {} 4+14y+42y^{2}+6y^{3}+6y^{4}+2x-42xy+14xy^{2}\\&-\,6xy^{3}+24x^{2}-4x^{2}y+4x^{2}y^{2} \\ g_3= & {} -84y^{2}-12y^{3}+168xy+24xy^{2}-84x^{2}-12x^{2}y \\ g_4= & {} 48y^{2}-96xy+48x^{2} \end{aligned}$$

1.7 N7: ABCBCBCBA

$$\begin{aligned} f_0+b_1f_1+b_1^{2}f_2+b_1^{3}f_3= & {} 0 \nonumber \\ g_0+c_1g_1+b_1g_2= & {} 0 \end{aligned}$$
(84)
$$\begin{aligned} f_0= & {} -\,1-3y-3x-15xy-6xy^{2}-17x^{2}y-12x^{2}y^{2}+3x^{3}-3x^{3}y-6x^{3}y^{2}-x^{4}y^{2} \\ f_1= & {} 12+30y+42x+150xy+36xy^{2}+30x^{2}+222x^{2}y+84x^{2}y^{2}-18x^{3}+114x^{3}y \\&+\,60x^{3}y^{2}-18x^{4}+12x^{4}y+12x^{4}y^{2} \\ f_2= & {} -\,48-84y-204x-420xy-72xy^{2}-300x^{2}\\&-\,708x^{2}y-192x^{2}y^{2}-180x^{3}-492x^{3}y \\&-\,168x^{3}y^{2}-36x^{4}-120x^{4}y-48x^{4}y^{2} \\ f_3= & {} 48+72y+216x+360xy+48xy^{2}+360x^{2}\\&+\,648x^{2}y+144x^{2}y^{2}+264x^{3}+504x^{3}y\\&+144x^{3}y^{2}+72x^{4}+144x^{4}y+48x^{4}y^{2} \\ g_0= & {} y-x+3xy+2xy^{2}-3x^{2}-2x^{2}y \\ g_1= & {} 2+2y+6x+6xy+6x^{2}+8x^{2}y+2x^{2}y^{2} \\ g_2= & {} -\,4-6y-10x-18xy-4xy^{2}-6x^{2}-12x^{2}y-4x^{2}y^{2} \end{aligned}$$

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Laskar, J., Vaillant, T. Dedicated symplectic integrators for rotation motions. Celest Mech Dyn Astr 131, 15 (2019). https://doi.org/10.1007/s10569-019-9886-4

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