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Symplectic phase flow approximation for the numerical integration of canonical systems

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Summary

New methods are presented for the numerical integration of ordinary differential equations of the important family of Hamiltonian dynamical systems. These methods preserve the Poincaré invariants and, therefore, mimic relevant qualitative properties of the exact solutions. The methods are based on a Runge-Kutta-type ansatz for the generating function to realize the integration steps by canonical transformations. A fourth-order method is given and its implementation is discussed. Numerical results are presented for the Hénon-Heiles system, which describes the motion of a star in an axisymmetric galaxy.

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References

  1. Arnol'd V.I. (1978): Mathematical Methods of Classical Mechanics. Springer, Berlin Heidelberg New York

    Google Scholar 

  2. Beltrami, E. (1868/1902–20): Sulla teoria delle linee geodetiche. Rendiconti del Reale Istituto Lombardo (serie II)1, 708–718 (1868)=Beltrami, E.: Opere Matematiche1, 366–373. Hoepli, Milano (1902–1920)

    Google Scholar 

  3. Carathéodory, C. (1935): Variationsrechnung und partielle Differentialgleichungen erster Ordnung. Teubner, Leipzig

    Google Scholar 

  4. Channell, P.J. (1983): Symplectic Integration Algorithms. Internal Report AT-6: ATN-83-9. Los Alamos National Laboratory, Los Alamos

    Google Scholar 

  5. Channell, P.J., Scovel, C. (1990): Symplectic Integration of Hamiltonian Systems. Nonlinearity3, 231–259

    Google Scholar 

  6. De Vogelaere, R. (1956): Methods of Integration which Preserve the Contact Transformation Property of the Hamiltonian Equations. Report No. 4. Department of Mathematics, University of Notre Dame, Notre Dame, Ind.

    Google Scholar 

  7. Fehlberg, E. (1970): Low-Order Classical Runge-Kutta Formulas with Stepsize Control and Their Application to Some Heat Transfer Problems. Computing6, 61–71

    Google Scholar 

  8. Feng Kang (1985): On Difference Schemes and Symplectic Geometry. In: Feng Kang, ed., Computation of Partial Differential Equations. Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, pp. 42–58. Science Press, Beijing

    Google Scholar 

  9. Feng Kang, Qin Meng-zhao (1987): The Symplectic Methods for the Computation of Hamiltonian Equations. In: Zhu You-lan, Guo Ben-yu, eds., Numerical Methods for Partial Differential Equations. Proceedings of a Conference held in Shanghai, 1987. Lecture Notes in Mathematics, Vol. 1297, pp. 1–35. Springer, Berlin Heidelberg New York

    Google Scholar 

  10. Hairer, E. (1979): Unconditionally Stable Methods for Second Order Differential Equations. Numer. Math.32, 373–379

    Google Scholar 

  11. Hamilton, W.R. (1834/1931–40): On a General Method in Dynamics by Which the Study of the Motions of All Free Systems of Attracting or Repelling Points is Reduced to the Search and Differentiation of One Central Relation, or Characteristic Function. Philosophical Transactions of the Royal Society124, 247–308 (1834)=A.W. Conway, J.L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton, Vol. 2, pp. 103–167. Cambridge University Press, Cambridge (1931–1940)

    Google Scholar 

  12. Hamilton, W.R. (1835/1931–40): Second Essay on a General Method in Dynamics. Philosophical Transactions of the Royal Society125, 95–144 (1835)=A.W. Conway, J.L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton, Vol. 2, pp. 162–216. Cambridge University Press, Cambridge (1931–1940)

    Google Scholar 

  13. Hénon, M., Heiles, C. (1964): The Applicability of the Third Integral of Motion: Some Numerical Experiments. Astron. J.69, 73–79

    Google Scholar 

  14. Jacobi, C.G.J. (1837/1881–91): Über die Reduction der Integration der Partiellen Differentialgleichungen erster Ordnung zwischen irgend einer Zahl Variablen auf die Integration eines einzigen Systemes gewöhnlicher Differentialgleichungen. Crelle Journal für die reine und angewandte Mathematik17, 97–162 (1837)=K. Weierstrass, ed., C.G.J. Jacobi's Gesammelte Werke, Vol. 4, pp. 57–127. Reimer, Berlin (1881–1891)

    Google Scholar 

  15. Lasagni, F.M. (1988): Canonical Runge-Kutta Methods. ZAMP39, 952–953

    Google Scholar 

  16. Menyuk, C.R. (1984): Some Properties of the Discrete Hamiltonian Method. Physica D11, 109–129

    Google Scholar 

  17. Miesbach, S. (1989): Symplektische Phasenflußapproximation zur numerischen Integration kanonischer Differentialgleichungen. Diploma Thesis, Department of Mathematics, University of Technology, München

    Google Scholar 

  18. Ruth, R. (1983): A Canonical Integration Technique. IEEE Trans. Nucl. Sci.30, 2669–2671

    Google Scholar 

  19. Sanz-Serna, J.M. (1988): Runge-Kutta Schemes for Hamiltonian Systems. BIT28, 877–883

    Google Scholar 

  20. Stoer, J., Bulirsch, R. (1980): Introduction to Numerical Analysis. Springer, Berlin Heidelberg New York

    Google Scholar 

  21. Stoffer, D.M. (1988): Some Geometric and Numerical Methods for Perturbed Integrable Systems. Dissertation, Department of Mathematics, Swiss Federal Institute of Technology, Zürich

    Google Scholar 

  22. Suris, Y.B. (1988): On the Preservation of the Symplectic Structure in the Course of Numerical Integration of Hamiltonian Systems. In: S.S. Filippov, ed., Numerical Solution of Ordinary Differential Equations, pp. 148–160. Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow [in Russian]

    Google Scholar 

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Miesbach, S., Pesch, H.J. Symplectic phase flow approximation for the numerical integration of canonical systems. Numer. Math. 61, 501–521 (1992). https://doi.org/10.1007/BF01385523

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