Abstract
In this paper we consider finite-dimensional constrained Hamiltonian systems of polynomial type. In order to compute the complete set of constraints and separate them into the first and second classes we apply the modern algorithmic methods of commutative algebra based on the use of Gröbner bases. As it is shown, this makes the classical Dirac method fully algorithmic. The underlying algorithm implemented in Maple is presented and some illustrative examples are given.
This work was supported in part by Russian Foundation for Basic Research, grant No. 98-01-00101.
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References
Dirac, P.A.M.: Generalized Hamiltonian Dynamics. Canad. J. Math. 2(1950), 129–148; Lectures on Quantum Mechanics, Belfer Graduate School of Science, Monographs Series, Yeshiva University, New York, 1964
Gitman, D.M., Tyutin, I.V.: Quantization of Fields with Constraints, Springer- Verlag, Bonn, 1990.
Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems, Princeton Uni¬versity Press, Princeton, New Jersey, 1992.
Prokhorov, L.V., Shabanov, S.V.: Hamiltonian Mechanics of Gauge Systems, St. Petersburg University, 1997 (in Russian).
Seiler, W.M.: Numerical Integration of Constrained Hamiltonian Systems Using Dirac Brackets. Math. Comp. 68 (1999) 661–681.
Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Intial-Value Problems in Differential-Algebraic Equations, Classics in Applied Mathematics 14, SIAM, Philadelphia, 1996.
Sundermeyer, K.: Constrained Dynamics, Lecture Notes in Physics 169, Springer-Verlag, New York, Berlin, 1982.
Gogilidze, S.A., Khvedelidze, A.M., Mladenov, D.M., Pavel, H.-P: Hamiltonian Reduction of SU(2) Dirac-Yang-Mills Mechanics, Phys. Rev. D57 (1998) 7488–7500.
Arnold, V.I.: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York, 1978.
Tombal, Ph., Moussiaux, A.: MACSYMA Computation of the Dirac-Bergman Algorithms for Hamiltonian Systems with Constraints. J. Symb. Comp. 1 (1985) 419–421.
Chaichian, M., Martinez, D.L., Lusanna, L.: Dirac’s Constrained Systems: The Classification of Second Class Constraints. Ann. Phys. (N. Y.) 232(1994) 40–60.
Battle, C., Comis, J., Pons, J.M., Roman-Roy, N.: Equivalence Between the Lagrangian and Hamiltonian Formalism for Constrained Systems. J. Math. Phys. 27 (1986) 2953–2962.
Becker, T., Weispfenning, V., Kredel, H.: Gröbner Bases. A Computational Approach to Commutative Algebra, Graduate Texts in Mathematics 141, Springer- Verlag, New York, 1993.
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms, 2nd Edition, Springer-Verlag, New York, 1996.
Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, Graduate Texts in Mathematics 185, Springer-Verlag, New York, 1998.
Seiler, W.M., Tucker, R.W.: Involution and Constrained Dynamics. J. Phys. A. 28 (1995) 4431–4451.
Pommaret, J.F.: Partial Differential Equations and Group Theory. New Perspectives for Applications, Kluwer, Dordrecht, 1994.
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Gerdt, V.P., Gogilidze, S.A. (1999). Constrained Hamiltonian Systems and Gröbner Bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing CASC’99. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60218-4_10
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DOI: https://doi.org/10.1007/978-3-642-60218-4_10
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