On the coordinate system-dependence of the accuracy of symplectic numerical methods

Donát M. Takács; Tamás Fülöp

doi:10.33993/jnaat542-1577

Authors

Donát M. Takács Department of Energy Engineering, Faculty of Mechanical Engineering, Budapest University of Technology and Economics https://orcid.org/0000-0002-8463-745X
Tamás Fülöp Department of Energy Engineering, Faculty of Mechanical Engineering, Budapest University of Technology and Economics https://orcid.org/0000-0003-2708-7065

DOI:

https://doi.org/10.33993/jnaat542-1577

Keywords:

Symplectic numerical method, processing, Hamiltonian system, coordinate transformation, canonical transformation

Abstract views: 58

Abstract

Symplectic numerical methods have become a widely-used choice for the accurate simulation of Hamiltonian systems in various fields, including celestial mechanics, molecular dynamics and robotics. Even though their characteristics are well-understood mathematically, relatively little attention has been paid in general to the practical aspect of how the choice of coordinates affects the accuracy of the numerical results, even though the consequences can be computationally significant.
The present article aims to fill this gap by giving a systematic overview of how coordinate transformations can influence the results of simulations performed using symplectic methods. We give a derivation for the non-invariance of the modified Hamiltonian of symplectic methods under coordinate transformations, as well as a sufficient condition for the non-preservation of a first integral corresponding to a cyclic coordinate for the symplectic Euler method. We also consider the possibility of finding order-compensating coordinate transformations that improve the order of accuracy of a numerical method. Various numerical examples are presented throughout.

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References

[1] M. J. Duncan, H. F. Levison, and S. M. Budd, The dynamical structure of the Kuiper belt, Astronomical Journal v. 110, p. 3073, 110 (1995) pp. 3073. DOI: https://doi.org/10.1086/117748

[2] M. Chyba, E. Hairer, and G. Vilmart, The role of symplectic integrators in optimal control, Optimal Control Applications and Methods, 30 (July 2008) no. 4, pp. 367–382. DOI: https://doi.org/10.1002/oca.855. DOI: https://doi.org/10.1002/oca.855

[3] J. Laskar et al., La2010: a new orbital solution for the long-term motion of the Earth, Astronomy & Astrophysics, 532 (July 2011) pp. A89. DOI: https://doi.org/10.1051/0004-6361/201116836. DOI: https://doi.org/10.1051/0004-6361/201116836

[4] R. E. Zeebe, OrbitN: A Symplectic Integrator for Planetary Systems Dominated by a Central Mass—Insight into Long-term Solar System Chaos, The Astronomical Journal, 166 (June 2023) no. 1, pp. 1. DOI: https://doi.org/10.3847/1538-3881/acd63b. DOI: https://doi.org/10.3847/1538-3881/acd63b

[5] D. Pekarek, A. D. Ames, and J. E. Marsden. “Discrete mechanics and optimal control applied to the compass gait biped”. 2007 46th IEEE Conference on Decision and Control. 2007, pp. 5376–5382. DOI: https://doi.org/10.1109/CDC.2007.4434296. DOI: https://doi.org/10.1109/CDC.2007.4434296

[6] S. Ober-Blöbaum, O. Junge, and J. E. Marsden, Discrete mechanics and optimal control: An analysis, ESAIM: Control, Optimisation and Calculus of Variations, 17 (Mar. 2010) no. 2, pp. 322–352. DOI: https: doi.org/10.1051/cocv/2010012. DOI: https://doi.org/10.1051/cocv/2010012

[7] Z. Shareef and A. Trächtler, Simultaneous path planning and trajectory optimization for robotic manipulators using discrete mechanics and optimal control, Robotica, 34 (Sept. 2014) no. 6, pp. 1322–1334. DOI: https: doi.org/10.1017/s0263574714002318. DOI: https://doi.org/10.1017/S0263574714002318

[8] X. Wang et al., An iterative framework to solve nonlinear optimal control with proportional delay using successive convexification and symplectic multi-interval pseudospectral scheme, Applied Mathematics and Computation, 435 (2022) pp. 127448. DOI: https://doi.org/10.1016/j.amc.2022.127448

[9] R. E. Zeebe and L. J. Lourens, Solar System chaos and the Paleocene–Eocene boundary age constrained by geology and astronomy, Science, 365 (Aug. 2019) no. 6456, pp. 926–929. DOI: https://doi.org/10.1126/science. aax0612. DOI: https://doi.org/10.1126/science.aax0612

[10] R. E. Zeebe and L. J. Lourens, Geologically constrained astronomical solutions for the Cenozoic era, Earth and Planetary Science Letters, 592 (Aug. 2022) pp. 117595. DOI: https://doi.org/10.1016/j.epsl. 2022.117595. DOI: https://doi.org/10.1016/j.epsl.2022.117595

[11] M. Beck. “Symplectic methods applied to the Lotka–Volterra system”. MSc thesis. McGill University, 2003.

[12] I. Faragó and G. Svantnerné Sebestyén, Operator splitting methods for the Lotka–Volterra equations, Electronic Journal of Qualitative Theory of Differential Equations, 2018 (2018) no. 48, pp. 1–19. DOI: https://doi.org/10.14232/ejqtde.2018.1.48

[13] É. Forest, Geometric integration for particle accelerators, Journal of Physics A: Mathematical and General, 39 (Apr. 2006) no. 19, pp. 5321–5377. DOI: https://doi.org/10.1088/0305-4470/39/19/s03. DOI: https://doi.org/10.1088/0305-4470/39/19/S03

[14] R. D. Ruth, A canonical integration technique, IEEE Trans. Nucl. Sci. 30 (1983) no. CERN-LEP-TH-83-14, pp. 2669–2671. DOI: https://doi.org/10.1109/TNS.1983.4332919

[15] E. Forest. “Canonical integrators as tracking codes (or how to integrate perturbation theory with tracking)”. AIP Conf. Proc. Vol. 184. CONF8707208-. SSC Central Design Group, c/o Lawrence Berkeley Laboratory, One Cyclotron. 1989. DOI: https://doi.org/10.1063/1.38062

[16] B. Erdélyi and M. Berz, Optimal Symplectic Approximation of Hamiltonian Flows, Physical Review Letters, 87 (Aug. 2001) no. 11. DOI: https: doi.org/10.1103/physrevlett.87.114302. DOI: https://doi.org/10.1103/PhysRevLett.87.114302

[17] C. Chen et al., Asymptotically-preserving large deviations principles by stochastic symplectic methods for a linear stochastic oscillator, SIAM Journal on Numerical Analysis, 59 (2021) no. 1, pp. 32–59. DOI: https://doi.org/10.1137/19M1306919

[18] R. D’Ambrosio and S. Di Giovacchino, Strong backward error analysis of symplectic integrators for stochastic Hamiltonian systems, Applied Mathematics and Computation, 467 (2024) pp. 128488. DOI: https://doi.org/10.1016/j.amc.2023.128488

[19] P. J. Morrison, Structure and structure-preserving algorithms for plasma physics, Physics of Plasmas, 24 (2017) no. 5. DOI: https://doi.org/ 10.1063/1.4982054. DOI: https://doi.org/10.1063/1.4982054

[20] M. Uzunca, B. Karasözen, and A. Aydın, Global energy preserving model reduction for multi-symplectic PDEs, Applied Mathematics and Computation, 436 (2023) pp. 127483. DOI: https://doi.org/10.1016/j.amc.2022.127483

[21] D. M. Takács, Á. Pozsár, and T. Fülöp, Thermodynamically extended symplectic numerical simulation of viscoelastic, thermal expansion and heat conduction phenomena in solids, Continuum Mechanics and Thermodynamics, 36 (Feb. 2024) pp. 525–538. DOI: https://doi.org/10.1007/s00161-024-01280-w. DOI: https://doi.org/10.1007/s00161-024-01280-w

[22] G. S. Jones and C. Trenchea, Discrete energy balance equation via a symplectic second-order method for two-phase flow in porous media, Applied Mathematics and Computation, 480 (2024) pp. 128909. DOI: https://doi.org/10.1016/j.amc.2024.128909

[23] G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of nearto-the identity symplectic mappings with application to symplectic integration algorithms, Journal of Statistical Physics, 74 (1994) pp. 1117– 1143. DOI: https://doi.org/10.1007/BF02188219. DOI: https://doi.org/10.1007/BF02188219

[24] E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration, Springer-Verlag, Berlin, 2006.

[25] D. M. Takács and T. Fülöp, Improving the accuracy of the Newmark method through backward error analysis, Computational Mechanics (2024). DOI: https://doi.org/10.1007/s00466-024-02580-3. DOI: https://doi.org/10.1007/s00466-024-02580-3

[26] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge University Press, 2005. DOI: https://doi.org/10.1017/CBO9780511614118

[27] K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer Berlin Heidelberg, 2010. DOI: https://doi.org/10.1007/978-3-642-01777-3

[28] K. Feng and D.-l. Wang, A note on conservation laws of symplectic difference schemes for Hamiltonian systems, Journal of Computational Mathematics, 9 (1991) no. 3, pp. 229–237.

[29] C. Kane et al., Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, International Journal for Numerical Methods in Engineering, 49 (2000) no. 10, pp. 1295–1325. DOI: https://doi.org/10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W. DOI: https://doi.org/10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.3.CO;2-N

[30] P. C. Moan, On modified equations for discretizations of ODEs, Journal of Physics A: Mathematical and General, 39 (2006) no. 19, pp. 5545. DOI: https://doi.org/10.1088/0305-4470/39/19/S13. DOI: https://doi.org/10.1088/0305-4470/39/19/S13

[31] M. Takahashi and M. Imada, Monte Carlo calculation of quantum systems. II. Higher order correction, Journal of the Physical Society of Japan, 53 (1984) no. 11, pp. 3765–3769. DOI: https://doi.org/10.1143/JPSJ.53.3765. DOI: https://doi.org/10.1143/JPSJ.53.3765

[32] G. Rowlands, A numerical algorithm for Hamiltonian systems, Journal of Computational Physics, 97 (Nov. 1991) no. 1, pp. 235–239. DOI: https://doi.org/10.1016/0021-9991(91)90046-n. DOI: https://doi.org/10.1016/0021-9991(91)90046-N

[33] M. A. López-Marcos, J. M. Sanz-Serna, and R. D. Skeel, Explicit Symplectic Integrators Using Hessian–Vector Products, SIAM Journal on Scientific Computing, 18 (Jan. 1997) no. 1, pp. 223–238. DOI: https:// doi.org/10.1137/s1064827595288085. DOI: https://doi.org/10.1137/S1064827595288085

[34] R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numerica, 11 (Jan. 2002) pp. 341–434. DOI: https://doi.org/10.1017/ s0962492902000053. DOI: https://doi.org/10.1017/S0962492902000053

[35] F. Casas, On processed splitting methods and high-order actions in pathintegral Monte Carlo simulations, The Journal of Chemical Physics, 133 (Oct. 2010) no. 15. DOI: https://doi.org/10.1063/1.3504163. DOI: https://doi.org/10.1063/1.3504163

[36] C. Jacobi, Uber ein neues Integral für den Fall der drei Körper, wenn die Bahn des störenden Planeten kreisförmig angenommen und die Masse des gestörten vernachlässigt wird, Monthly reports of the Berlin Academy (1836).

[37] M. Nakane and C. G. Fraser, The Early History of Hamilton-Jacobi Dynamics 1834–1837, Centaurus, 44 (2002) no. 3-4, pp. 161–227. DOI: https://doi.org/10.1111/j.1600-0498.2002.tb00613.x

[38] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer New York, 1989. DOI: https://doi.org/10.1007/978-1-4757-2063-1

[39] H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, Pearson, Upper Saddle River, NJ, 2001.

[40] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer Berlin Heidelberg, 1996. DOI: https://doi.org/10.1007/978-3-642-05221-7

[41] C. Lubich, On the convergence of multistep methods for nonlinear stiff differential equations, Numerische Mathematik, 58 (Dec. 1990) no. 1, pp. 839–853. DOI: https://doi.org/10.1007/bf01385657. DOI: https://doi.org/10.1007/BF01385657

[42] D. F. Griffiths and J. M. Sanz-Serna, On the Scope of the Method of Modified Equations, SIAM Journal on Scientific and Statistical Computing, 7 (1986) no. 3, pp. 994–1008. DOI: https://doi.org/10.1137/0907067. DOI: https://doi.org/10.1137/0907067

[43] M. A. López-Marcos, J. M. Sanz-Serna, and R. D. Skeel. “Cheap enhancement of symplectic integrators”. Numerical Analysis 1995. Ed. by D. F. Griffiths and G. A. Watson. Vol. 344. Pitman Research Notes in Mathematics Series. Longman, 1996, pp. 107–122.

[44] S. Reich, Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36 (1999) no. 5, pp. 1549–1570. DOI: https://doi.org/10.1137/S003614299732979. DOI: https://doi.org/10.1137/S0036142997329797

[45] E. Hairer and C. Lubich. “Asymptotic Expansions and Backward Analysis for Numerical Integrators”. Dynamics of Algorithms. Springer New York, 2000, pp. 91–106. DOI: https://doi.org/10.1007/978-1-4612-1274-4_5. DOI: https://doi.org/10.1007/978-1-4612-1274-4_5

[46] B. Gompertz, XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. FRS, Philosophical transactions of the Royal Society of London (1825) no. 115, pp. 513–583. DOI: https://doi.org/10.1098/rstl.1825.0026

[47] C. P. Winsor, The Gompertz curve as a growth curve, Proceedings of the National Academy of Sciences, 18 (1932) no. 1, pp. 1–8. DOI: https://doi.org/10.1073/pnas.18.1.1

[48] K. M. C. Tjørve and E. Tjørve, The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the UnifiedRichards family, PLOS ONE, 12 (June 2017) no. 6, pp. 1–17. DOI: https://doi.org/10.1371/journal.pone.0178691. DOI: https://doi.org/10.1371/journal.pone.0178691

[49] J. C. Larsen, Models of cancer growth, Journal of Applied Mathematics and Computing, 53 (2017) pp. 613–645. DOI: https://doi.org/10.1007/s12190-016-0985-z

[50] R. de Vogelaere. Methods of integration which preserve the contact transformation property of the Hamilton equations. Tech. rep. Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, USA, 1956.

[51] M. H. Nasab, Partitioned second derivative methods for separable Hamiltonian problems, Journal of Applied Mathematics and Computing, 65 (2021) pp. 831–859. DOI: https://doi.org/10.1007/s12190-020-01417-5

[52] J. C. Butcher. “The effective order of Runge–Kutta methods”. Conference on the Numerical Solution of Differential Equations: Held in Dundee/Scotland, June 23–27, 1969. Springer. 1969, pp. 133–139. DOI: https://doi.org/10.1007/BFb0060019

[53] J. Wisdom, M. Holman, and J. Touma. “Symplectic correctors”. Integrational Algorithms and Classical Mechanics. Ed. by J. E. Marsden, G. W. Patrick, and W. F. Shadwick. Vol. 10. Fields Institute Communications. Providence: American Mathematical Society, 1996, pp. 217–244. isbn:978-0821802595. DOI: https://doi.org/10.1090/fic/010/14

[54] Atmospheric Icing of Power Networks, Springer Netherlands, 2008.

[55] L. E. Kollár, Ice-shedding-induced vibration of conductors with active vibration control, Cold Regions Science and Technology, 196 (2022) pp. 103504. DOI: https://doi.org/10.1016/j.coldregions.2022.103504. DOI: https://doi.org/10.1016/j.coldregions.2022.103504

[56] W. K. Lee, Domains of Attraction of System of Nonlinearly Coupled Ship Motions by Simple Cell Mapping, Journal of Offshore Mechanics and Arctic Engineering, 114 (Feb. 1992) no. 1, pp. 22–27. DOI: https://doi.org/10.1115/1.2919948. DOI: https://doi.org/10.1115/1.2919948

[57] W. K. Lee and H. D. Park, Chaotic dynamics of a harmonically excited spring-pendulum system with internal resonance, Nonlinear Dynamics, 14 (1997) pp. 211–229. DOI: https://doi.org/10.1023/A:1008256920441

[58] J. M. Sanz-Serna and L. Abia, Order Conditions for Canonical Runge–Kutta Schemes, SIAM Journal on Numerical Analysis, 28 (Aug. 1991) no. 4, pp. 1081–1096. DOI: https://doi.org/10.1137/0728058. DOI: https://doi.org/10.1137/0728058

[59] E. Hairer, Backward analysis of numerical integrators and symplectic methods, Annals of Numerical Mathematics, 1 (1994) pp. 107–132.

[60] E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Berlin Heidelberg, 2008.

[61] J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (Dec. 1988) no. 4, pp. 877–883. DOI: https://doi.org/10.1007/ DOI: https://doi.org/10.1007/BF01954907

bf01954907.

[62] M. Calvo and J. Sanz-Serna, Canonical B-series, Numerische Mathematik, 67 (Mar. 1994) no. 2, pp. 161–175. DOI: https://doi.org/10.1007/s002110050022. DOI: https://doi.org/10.1007/s002110050022

[63] P. Chartier, E. Faou, and A. Murua, An Algebraic Approach to Invariant Preserving Integators: The Case of Quadratic and Hamiltonian Invariants, Numerische Mathematik, 103 (Apr. 2006) no. 4, pp. 575–590. DOI: https://doi.org/10.1007/s00211-006-0003-8. DOI: https://doi.org/10.1007/s00211-006-0003-8

[64] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 2017. DOI: https://doi.org/10.1093/oso/9780198794899.001.0001

[65] U. Walter, Astronautics: The Physics of Space Flight, Springer International Publishing, 2018. DOI: https://doi.org/10.1007/978-3-319-74373-8