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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.
##### MSC:
 70H45 Constrained dynamics, Dirac’s theory of constraints 70Q05 Control of mechanical systems
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