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A proof of the Dirac conjecture for a class of finite-dimensional Hamilton-Dirac systems. (English. Russian original) Zbl 1040.70011
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.
70H45 Constrained dynamics, Dirac’s theory of constraints
70Q05 Control of mechanical systems
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