A proof of the Dirac conjecture for a class of finite-dimensional Hamilton-Dirac systems. (English. Russian original) Zbl 1040.70011

Sobolev, A. S.

Math. Notes 71, No. 5, 724-729 (2002); translation from Mat. Zametki 71, No. 5, 793-797 (2002).

This paper is devoted to constrained Hamiltonian systems. By definition constraints in a neighborhood of a point \(x_0\in \mathbb{E}\) is a map \(\phi: \mathbb{E}\to\mathbb{E}_\phi\) satisfying the following conditions:
a) \(\phi\) is continuously differentiable in a neighborhood of the point \(x_0\) and \(\phi(x_0)= 0\);
b) the map \(\phi'(x_0): \mathbb{E}\to \mathbb{E}_\phi\) is surjective.
Here \(\mathbb{E}\), \(\mathbb{E}_\phi\) are the Euclidean spaces. A Dirac Hamiltonian system on the phase space \(\mathbb{E}\) is a set \((H,\phi,\mathbb{E}, I)\), where \(H\in C^\infty(\mathbb{E}, R^1)\), \(\phi: \mathbb{E}\to \mathbb{E}_\phi\) is a constraint, \(\mathbb{E}\) is an even-dimensional Euclidean space, and \(I: \mathbb{E}\to \mathbb{E}\) is a symplectic operator. The author introduces the Dirac class, and within this class proves the classical Dirac conjecture. The proof is based on the theory of control systems.

Reviewer: Messoud Efendiev (Berlin)

MSC:

70H45	Constrained dynamics, Dirac’s theory of constraints
70Q05	Control of mechanical systems

Keywords:

Dirac class; phase space; symplectic operator; control system; constrained Hamiltonian system