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Dirac brackets in geometric dynamics

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Dirac brackets in geometric dynamics ANNALES DE L’I. H. P., SECTION A JEDRZEJ ´SNIATYCKI Dirac brackets in geometric dynamics Annales de l’I. H. P., section A, tome 20, no 4 (1974), p. 365-372. <> © Gauthier-Villars, 1974, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A », implique l’accord avec les conditions générales d’utilisation (http://www. Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 365 Dirac brackets in geometric dynamics Jedrzej 015ANIATYCKI The University of Calgary, Alberta, Canada. Department of Mathematics Statistics and Computing Science Ann. Inst. Henri Poincaré, Vol. XX, n° 4, 1974, Section A : Physique théorique. ABSTRACT. - Theory of constraints in dynamics is formulated in the framework of symplectic geometry. Geometric significance of secondary constraints and of Dirac brackets is given. Global existence of Dirac brackets is proved. 1. INTRODUCTION The successes of the canonical quantization of dynamical systems with a finite number of degrees of freedom, the experimental necessity of quan- tization of electrodynamics, and the hopes that quantization of the gravi- tational field could resolve difficulties encountered in quantum field theory have given rise to thorough investigation of the canonical structure of field theories. It has been found that the standard Hamiltonian formu- lation of dynamics is inadequate in the physically most interesting cases of electrodynamics and gravitation due to existence of constraints. Methods of dealing with dynamics with constraints have been developed by several authors and it has been realized that the standard Hamiltonian dyn

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