On Dirac's generalized Hamiltonian dynamics

V. V. Kozlov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 4, pp. 649-681

Résumé

Various aspects of Dirac's generalized Hamiltonian dynamics are under consideration. The starting point is a Hamiltonian system on a symplectic manifold on which, in addition, a distribution of multidimensional tangent planes is defined. The Hamiltonian vector field must be modified so that this distribution becomes invariant under the phase flow of the modified dynamical system. This problem can be solved in various ways. The simplest approach is to project the Hamiltonian vector field onto the planes of the distribution by using the symplectic structure, the closed nondegenerate 2-form on the symplectic manifold (which defines the symplectic geometry on tangent planes to the phase space). If the distribution in question is integrable, then this approach leads to generalized Hamiltonian dynamics developed by Dirac (and other authors) to quantize systems whose Lagrangian is degenerate in velocities. In application to the mechanics of Lagrange's systems with non-integrable constraints, this approach yields classical non-holonomic systems. Another approach is based on the definition of the motion of Hamiltonian systems as an extremal of a variational problem with constraints. Then in the case of non-holonomic constraints we obtain dynamical systems of a quite different type. In application to Lagrange's systems with non-integrable constraints this approach yields the equations of motion of so-called vaconomic dynamics. For example geometric optic is considered, which is based on Fermat's variatonal principle with Largangian which is homogeneous of degree one with respect to the velocities. Bibliography: 34 titles.

Détail

Keywords: Hamiltonian system, symplectic structure, symplectic geometry, Lagrangian, Hamiltonian, primary constraints, secondary constraints, Poincaré–Helmholtz action, non-holonomic systems, vaconomic systems, non-isotropic optic media, Fermat's principle.
Mots-clés : distributions