Abstract
The drift of numerical solution of a dynamical system off the integral and constraint surface is eliminated with the help of a gradient supplement to the equations of motion. This supplement makes the image point in the phase space to move along the normal to a given surface. The solution is compared with those obtained by integrating the generalized Hamilton–Dirac dynamics equations. The calculations were performed in Maple V R5 with the use of the standard subroutine dsolve/numeric for solving ordinary differential equations by the fourth-order Runge–Kutta method.
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REFERENCES
Baumgarte, J., Stabilization of Constraints and Integrals of Motion in Dynamical Systems, Comput. Meth. Appl. Mech. Eng., 1972, vol. 1, pp. 1–16.
Asher, U.M. and Petzold, L.R., Stability of Computational Methods for Constrained Dynamics Systems, SIAM J. Sci. Comput., 1993, vol. 14, no. 1, pp. 95–120.
Leimkuhler, B.J. and Skeel, R.D., Symplectic Numerical Integrators in Constrained Hamiltonian Systems, J. Comput. Phys., 1994, vol. 112, pp. 117–125.
Seiler, W.M., Numerical Integration of Constrained Hamiltonian Systems Using Dirac Brackets, Math. Comput., 1998, vol. 68, no. 226, pp. 661–682.
Quispel, G.R.W. and Turner, G.S., Discrete Gradient Methods for Solving ODEs Numerically with Preserving a First Integral, J. Phys., 1996, vol. A29, pp. L341–L349.
Spravochnik po teorii avtomaticheskogo upravleniya (Handbook on Automatic Control Theory), Krasovskii, A.A., Ed., Moscow: Nauka, 1987.
Kriksin, Yu.A., A Conservative Difference Scheme for a System of Hamiltonian Equations with External Action, Zh. Vychisl. Mat. Mat. Fiz., 1993, vol. 33, pp. 206–218.
Dirac, P.A.M., Lectures on Quantum Mechanics, New York: Yeshiva Univ., 1964. Translated under the title Lektsii po kvantovoi mekhanike, Moscow: Mir, 1968.
Lions, J.-L., Contrôle optimal de systèmes gouvernés par des équations aux derivées partielles, Paris: Dunod Gauthier-Villars, 1968. Translated under the title Optimal'noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Moscow: Mir, 1972.
Davenport, J., Siret, Y., and Tournier, E., Calcul formel. Systèmes et algorithmes de manipulations algébriques, Paris: Masson, 1987. Translated under the title Sistemy i algoritmy algebraicheskikh vychislenii, Moscow: Mir, 1991.
Gerdt, V.P., Computer Algebra, Symmetry Analysis and Integrability of Nonlinear Evolution Equations, Int. J. Modern Phys. C, 1993, vol. 4, no. 2, pp. 279–286.
Gerdt, V.P., Computer Algebra and Constraint Dynamics, in Problems of Modern Physics, JINR D2–99–263, Dubna: JINR, 1999, pp. 164–171.
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal'nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Nauka, 1983.
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Palii, Y.G. Correction of Numerical Integration as an Optimal Control Problem. Programming and Computer Software 28, 84–87 (2002). https://doi.org/10.1023/A:1014876817871
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DOI: https://doi.org/10.1023/A:1014876817871