×

zbMATH — the first resource for mathematics

Hamiltonian systems with constraints: a geometric approach. (English) Zbl 0697.58019
Let \(M\) be a differential manifold and \(TM\), \(T^*M\) be its tangent and cotangent bundles. Under the Legendre transformation \(FL: TM\to T^*M\), if the image of \(FL\) is a proper submanifold of \(T^*M\), one obtains a Hamiltonian system with constraint. Then the corresponding equation of motion is the so-called Hamiltonian-Dirac equation. The author discusses first the local problem in which the image of \(FL\) is a submanifold of \(T^*M\) defined by the zeros of a finite family of functions. Then he turns to discuss the global problem in which the image of \(FL\) is any submanifold of \(T^*M\). In both cases the author proposes a new algorithm to obtain the constraint submanifold and the dynamical vector field on it. A simple example is given.
Reviewer: Guizhang Tu

MSC:
70H45 Constrained dynamics, Dirac’s theory of constraints
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abraham, R., and Marsden, J. E. (1978).Foundations of Mechanics, Benjamin/Cummings Publishing Company, Reading, Massachusetts. · Zbl 0393.70001
[2] Battle, C., Gomis, J., Pons, J., and Roman Roy, N. (1986). Equivalence between the Lagrangian and Hamiltonian formalism for constrainted systems,Journal of Mathematical Physics,27, 2953-2962. · Zbl 0613.70012 · doi:10.1063/1.527274
[3] De Le?n, M., and Rodrigues, X. X. (1985).Generalized Classical Mechanics and Fields Theory, North-Holland/Elsevier, Amsterdam.
[4] Dirac, P. A. M. (1950). Generalized Hamiltonian dynamics,Canadian Journal of Mathematics,2, 129-148. · Zbl 0036.14104 · doi:10.4153/CJM-1950-012-1
[5] Dirac, P. A. M. (1964).Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University. · Zbl 0141.44603
[6] Gotay, M. J., and Nester, J. M. (1979). Presymplectic Lagrangian systems I,Annales de l’Institute Henri Poincar?,30, 129-142. · Zbl 0414.58015
[7] Gotay, M. J., and Nester, J. M. (1980). Presymplectic Lagranian systems II,Annales de l’Institute Henri Poincar?,32, 1-13. · Zbl 0453.58016
[8] Gotay, M. J., Nester, J. M., and Hinds, G. (1978). Presymplectic manifolds and the Dirac-Bergman theory of constraints,Journal of Mathematical Physics,19, 2388-2399. · Zbl 0418.58010 · doi:10.1063/1.523597
[9] Lichnerowicz, A. (1975). Vari?t? symplectique et dynamique associ?e ? une sous-vari?t?,Comptes Rendus de l’Academie des Science,280A, 523-527. · Zbl 0315.70016
[10] Sternberg, S. (1964).Lectures on Differential Geometry, Prentice-Hall. · Zbl 0129.13102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.