The Dirac Theory of Constraints, the Gotay-Nester Theory and Poisson Geometry

HERNÁN CENDRA; MARÍA DEL ROSARIO ETCHECHOURY; SEBASTIÁN J. FERRARO

Anales de la Academia Nacional de Ciencias Exactas, Físicas y Naturales de Buenos Aires

ACADEMIA NACIONAL DE CIENCIAS EXACTAS FÍSICAS Y NATURALES

Lugar: Buenos Aires; Año: 2013 p. 95 - 115

0365-1185

Obs.: se trata de un artículo publicado por invitación en ocasión del ingreso del Dr. H. Cendra como Académico Correspondiente de la ANCEFN. Si bien estos Anales no tienen referato, los resultados de este artículo están incluidos en una versión muy ampliada que se publicó en 2014 en el Journal of Geometric Mechanics (con referato). --------------- The Dirac theory of constraints has been widely studied and applied very successfully by physicists since the original works by Dirac and by Bergmann. From a mathematical standpoint, several aspects of the theory have been exposed rigorously afterwards by many authors. However, many questions related to, for instance, singular or infinite dimensional cases remain open. The work of Gotay and Nester presents a mathematical generalization in terms of presymplectic geometry, which introduces a dual point of view. We present a study of the Dirac theory of constraints emphasizing the duality between the Poisson-algebraic and the geometric points of view, related respectively to the work of Dirac and of Gotay and Nester, under strong regularity conditions. We deal with some questions insufficiently treated in the literature: a study of uniqueness of solution; avoiding almost completely the use of coordinates; the role of the Pontryagin bundle. We also show how one can globalize some results usually treated locally in the literature. For instance, we introduce the global notion of second class submanifold as being tangent to a second class subbundle. A general study of global results for Dirac and Gotay-Nester theories remains an open question in this theory.