Abstract
A generalized constraint algorithm is developed which provides necessary and sufficient conditions for the solvability of the canonical equations of motion associated to presymplectic classical systems. This constraint algorithm is combined with a presymplectic extension of Tulczjew's description of constrained dynamical systems in terms of special symplectic manifolds. The resultant theory provides a unified geometric description as well as a complete solution of the problems of constrained and a priori presymplectic classical systems in both the finite and infinite dimensional cases.
Keywords
- Symplectic Manifold
- Cotangent Bundle
- Legendre Transformation
- Canonical System
- Canonical Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Notes and References
W. Tulczyjew, Symposia Mathematica 14, 247 (1974).
W. Tulczyjew, in Differential Geometric Methods in Mathematical Physics, Lecture Notes in Math., #570, 457, 464 (Springer-Verlag, Berlin, 1977).
W. Tulczyjew, Acta Phys. Polon., B8, 431 (1977).
M. Menzio and W. Tulczyjew, Ann. Inst. H. Poincaré, A28, 349 (1978).
J. Kijowski and W. Tulczyjew, A Symplectic Framework for Field Theories, to appear (Springer-Verlag)
P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monograph Series #2 (1964).
M.J. Gotay, J.M. Nester and G. Hinds, J. Math. Phys. 19, 2388 (1978).
M.J. Gotay and J.M. Nester, Presymplectic Hamilton and Lagrange Systems, Gauge Transformations and the Dirac Theory of Constraints, to appear (Proc. of the VII Int'l. Colloq. on Group Theoretical Methods in Physics, Austin, 1978).
M.J. Gotay, Presymplectic Manifolds, Geometric Constraint Theory, and the Dirac-Bergmann Theory of Constraints, Dissertation, University of Maryland, 1979.
M.J. Gotay and J.M. Nester, Ann. Inst. H. Poincare, A30, 129 (1979).
J. Kijowski, Commun. Math. Phys., 30, 99 (1973).
J. Kijowski and W. Szczyrba, Commun. Math. Phys., 46, 183 (1976).
J. Sniatycki, Ann. Inst. H. Poincaré, A20, 365 (1974).
R. Abraham and J. Marsden, Foundations of Mechanics, second ed., (Benjamin-Cummings, New York, 1978).
P. Chernoff and J. Marsden, Properties of Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Mathematics #425 (Springer-Verlag, Berlin, 1974).
J.-M. Souriau, Structures des Systemes Dynamiques, (Dunod, Paris, 1970).
H.P. Künzle, J. Math. Phys., 13, 739 (1972).
A. Lichnerowicz, C.R. Acad. Sci. Paris, A280, 523 (1975).
A. Lagrangian subspace TN of TM is necessarily closed, so that if N is Lagrangian, then N must be a Banach submanifold of M.
S. Lang, Differential Manifolds, (Addison-Wesley, Reading, Mass., 1972).
A. Weinstein, Adv. Math., 6, 329 (1971).
It is assumed that all of the spaces appearing in this paper are in fact generalized submanifolds in the sense of the Appendix (cf. §VII).
J. Sniatycki and W. Tulczyjew, Indiana U. Math. J., 22, 267 (1972).
In finite-dimensions, condition (ii) is tantamount to requiring that dim {ker ω 1 ∩ TM l-1 } be constant on M l .
J.M. Nester and M.J. Gotay, The Dynamics of Singular Presymplectic Systems, (work in progress).
J.M. Nester and M.J. Gotay, Presymplectic Lagrangian Systems II: The Second-Order Equation Problem, University of Maryland Preprint #79-141, (1979) (to be published).
See also Exercise 5.3L of [14].
This is equivalent to the almost regularity of L (cf. ref. [10]).
Equivalently, FL*α1 = −dE.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1980 Springer-Verlag
About this paper
Cite this paper
Gotay, M.J., Nester, J.M. (1980). Generalized constraint algorithm and special presymplectic manifolds. In: Kaiser, G., Marsden, J.E. (eds) Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092022
Download citation
DOI: https://doi.org/10.1007/BFb0092022
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09742-6
Online ISBN: 978-3-540-38571-4
eBook Packages: Springer Book Archive
