Hamiltonian systems with constraints: a geometric approach. (English) Zbl 0697.58019

Muñoz Lecanda, M. C.

Int. J. Theor. Phys. 28, No. 11, 1405-1417 (1989).

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

Keywords:

Hamiltonian system; Hamiltonian-Dirac equation

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