A proof of the Dirac conjecture for a class of finite-dimensional Hamilton-Dirac systems.
type: Item
instance of: scholarly article
title: A proof of the Dirac conjecture for a class of finite-dimensional Hamilton-Dirac systems.
zbMATH Open document ID: 1040.70011
DOI: 10.1023/A:1015804425220
author: A. S. Sobolev
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 70H45
Mathematics Subject Classification ID: 70Q05
zbMATH DE Number: 1928216
zbMATH Keywords: Dirac class
zbMATH Keywords: phase space
zbMATH Keywords: symplectic operator
zbMATH Keywords: control system
zbMATH Keywords: constrained Hamiltonian system
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1928216