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Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups. (English) Zbl 0799.58069

Summary: Three conservation laws are associated with the dynamics of Hamiltonian systems with symmetry: The total energy, the momentum map associated with the symmetry group, and the symplectic structure are invariant under the flow. Discrete time approximations of Hamiltonian flows typically do not share these properties unless specifically designed to do so. We develop explicit conservation conditions for a general class of algorithms on Lie groups. For the rigid body these conditions lead to a single-step algorithm that exactly preserves the energy, spatial momentum, and symplectic form. For homogeneous nonlinear elasticity, we find algorithms that conserve angular momentum and either the energy or the symplectic form.

MSC:

37C80 Symmetries, equivariant dynamical systems (MSC2010)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems

Software:

Mathematica
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[1] R. Abraham and J. E. Marsden [1978]Foundations of Mechanics, 2nd ed., Addison-Wesley, Reading, MA. · Zbl 0393.70001
[2] V. I. Arnold [1988]Mathematical Methods of Classical Mechanics, Springer-Verlag, New York.
[3] M. Austin, P. S. Krishnaprasad, and L. S. Wan [1992] ”Almost Lie-Poisson Integrators for the Rigid Body,” preprint.
[4] A. Bayliss and E. Isaacson [1975] ”How to Make Your Algorithm Conservative,”American Mathematical Society, A594–A595.
[5] P. J. Channell [1983]Symplectic Integration Algorithms, Los Alamos National Laboratory Internal Report AT-6:ATN-83-9.
[6] P. J. Channell and J. C. Scovel [1989] ”Symplectic Integration of Hamiltonian Systems,” preprint. · Zbl 0704.65052
[7] H. Cohen and R. G. Muncaster [1984 ”The Dynamics of Psuedo-Rigid Bodies: General Structure and Exact Solutions,”J. Elasticity,14, 127–154. · Zbl 0562.73017 · doi:10.1007/BF00041661
[8] H. Cohen and R. G. Muncaster [1988]The Theory of Pseudo-Rigid Bodies, Springer-Verlag, New York. · Zbl 0687.70001
[9] Y. Coquete-Bruhat and C. DeWitt-Morette [1982]Analysis, Manifolds and Physics, North-Holland, Amsterdam. · Zbl 0492.58001
[10] M. L. Curtis [1984]Matrix Groups, Springer-Verlag, Berlin.
[11] R. de Vogelaere [1956] ”Methods of Integration Which Preserve the Contact Transformation Property of Hamiltonian Equations,” Department of Mathematics, University of Notre Dame, Report 4.
[12] Feng Kan [1986] ”Difference Schemes for Hamiltonian Formalism and Symplectic Geometry,”J. Computational Mathematics,4, 279–289. · Zbl 0596.65090
[13] Feng Kan and M. Qin [1987]The Symplectic Methods for the Computation of Hamiltonian Equations, Springer Lect. Notes Math.1297, 1–31, Springer-Verlag, Berlin.
[14] H. Goldstein [1982]Classical Mechanics, Addison-Wesley, Reading, MA.
[15] D. Greenspan [1984] ”Conservative Numerical Methods for ” \(\dot x = f\left( x \right)\) ,”J. Computational Physics,56, 28–41. · Zbl 0561.65056 · doi:10.1016/0021-9991(84)90081-0
[16] R. A. Labudde and D. Greenspan [1976a] ”Energy and Momentum Conserving Methods of Arbitrary Order for the Numerical Integration of Equations of Motion. Part I,”Numerische Mathematik,25, 323–346. · Zbl 0364.65066 · doi:10.1007/BF01396331
[17] R. A. Labudde and D. Greenspan [1976b] ”Energy and Momentum Conserving Methods of Arbitrary Order for the Numerical Integration of Equations of Motion. Part II,”Numerische Mathematik,26, 1–16. · Zbl 0382.65031 · doi:10.1007/BF01396562
[18] F. M. Lasagni [1988] ”Canonical Runge-Kutta methods,”ZAMP,39, 952–953. · Zbl 0675.34010 · doi:10.1007/BF00945133
[19] D. Lewis and J. C. Simo [1990] ”Nonlinear Stability of Pseudo-Rigid Bodies,”Proceedings of the Royal Society of London A,427, 281–319. · Zbl 0697.73045 · doi:10.1098/rspa.1990.0014
[20] R. Maeder [1991]Programming in Mathematica, Addison-Wesley, Redwood City, CA.
[21] A. Marciniak [1984] ”Energy Conserving, Arbitrary Order Numerical Solutions of theN-body Problem,”Numerische Mathemathik,45, 207–218. · Zbl 0527.65057 · doi:10.1007/BF01389466
[22] K. W. Morton [1977] ”Initial Value Problems by Finite and Other Methods,” inThe State of the Art in Numerical Analysis, D. A. H. Jacobs, Ed., pp. 699–756, Academic Press, London.
[23] J. Moser and A. P. Veselov [1991] ”Discrete versions of some classical integrable systems and factorization of matrix polynomials,”Comm. Math. Phys.,139, 217–243. · Zbl 0754.58017 · doi:10.1007/BF02352494
[24] N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw [1980] ”Geometry from a Time Series,”Physical Review Letters,45, No. 9, 712–716. · doi:10.1103/PhysRevLett.45.712
[25] D. I. Pullin and P. G. Saffman [1991] ”Long-time Symplectic Integration: The Example of Four-Vortex Motion,”Proceedings of the Royal Society of London A,432, 481–494. · Zbl 0726.65087 · doi:10.1098/rspa.1991.0027
[26] R. D. Richtmyer and K. W. Morton [1967]Difference Methods for Initial Value Problems, 2nd ed., Wiley-Interscience, New York. · Zbl 0155.47502
[27] J. M. Sanz-Serna [1988] ”Runge-Kutta Schemes for Hamiltonian Systems,”BIT,28, 877–883. · Zbl 0655.70013 · doi:10.1007/BF01954907
[28] J. M. Sanz-Serna [1992] ”The Numerical Integration of Hamiltonian Systems,” Proceedings of the Conference on Computational Differential Equations, Imperial College, London, 3–7 July 1989, in press. · Zbl 0671.65077
[29] C. Scovel [1991] ”Symplectic Numerical Integration of Hamiltonian Systems,” inThe Geometry of Hamiltonian Systems, Proceedings of the Workshop held June 5–15, 1989, Tudor Ratiu, Ed., Springer-Verlag, New York, pp. 463–496.
[30] J. C. Simo and K. K. Wong [1991] ”Unconditionally Stable Algorithms for Rigid Body Dynamics That Exactly Preserve Energy and Momentum,”International J. Numerical Methods in Engineering,31, 19–52. · Zbl 0825.73960 · doi:10.1002/nme.1620310103
[31] J. C. Simo, M. S. Rifai, and D. D. Fox [1991] ”On a Stress Resultant Geometrically Exact Shell Model. Part VI: Conserving Algorithms for Nonlinear Dynamics,”International J. Numerical Methods in Engineering,34, 117–164. · Zbl 0760.73045 · doi:10.1002/nme.1620340108
[32] J. C. Simo and N. Tarnow [1992] ”The Discrete Energy-Momentum Method. Conserving Algorithms for Nonlinear Elastodynamics,”Zeitschrift für Angewandte Mathematik und Physik,43, 757–792. · Zbl 0758.73001 · doi:10.1007/BF00913408
[33] J. C. Simo, N. Tarnow, and K. Wong [1992] ”Exact Energy-Momentum Conserving Algorithms and Symplectic Schemes for Nonlinear Dynamics,”Computer Methods in Applied Mechanics and Engineering,100, 63–116. · Zbl 0764.73096 · doi:10.1016/0045-7825(92)90115-Z
[34] J. C. Simo and R. L. Taylor [1991] ”Finite Elasticity in Principal Stretches: Formulation and Augmented Lagrangian Algorithms,”Computer Methods in Applied Mechanics and Engineering,85, 273–310. · Zbl 0764.73104 · doi:10.1016/0045-7825(91)90100-K
[35] S. S. Smale [1970a] ”Topology and Mechanics, Part I,”Inventiones Mathematicae,10, 161–169. · Zbl 0202.23201 · doi:10.1007/BF01418778
[36] S. S. Smale [1970b] ”Topology and Mechanics, Part II,”Inventiones Mathematicae,11, 45–64. · Zbl 0203.26102 · doi:10.1007/BF01389805
[37] N. Tarnow and J. C. Simo [1992] ”How To Make Your Favorite Second-Order Algorithm Fourth-Order,” preprint.
[38] E. T. Whittaker [1959]A Treatise on the Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies, Cambridge, 1904; 4th ed., 1937; Dover ed., 1959. · JFM 35.0682.01
[39] J. Wisdom and M. Holman [1991] ”Symplectic Maps for theN-body Problem,”The Astronomical Journal,102 (4), 1528–1538. · doi:10.1086/115978
[40] J. Wisdom and M. Holman [1992] ”Symplectic Maps for theN-body Problem: Stability Analysis,” preprint, March 31, 1992.
[41] G. Zhong and J. E. Marsden [1988] ”Lie-Poisson Hamilton-Jacobi Theory and Lie-Poisson Integrators,”Physics Letters A,33:3, 134–139. · Zbl 1369.70038 · doi:10.1016/0375-9601(88)90773-6
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