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Zbl 1065.37046
Reyes, Enrique G.
On phase spaces and the variational bicomplex (after G. Zuckerman). (English)
[A] Mladenov, Iva\"ilo M. (ed.) et al., Proceedings of the 5th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 5--12, 2003. Sofia: Bulgarian Academy of Sciences. 189-202 (2004). ISBN 954-84952-8-7/pbk

The construction of phase spaces in field theory is discussed. In particular, following a paper by {\it G. J. Zuckerman} [In: Action principles and global geometry. Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 259--284 (1987; Zbl 0669.58014)], the author reviews a Hamiltonian formulation of Lagrangian field theory -- in terms of the variational bicomplex of a fixed trivial fiber bundle [for the definition of this bicomplex, see for example {\it I. M. Anderson}, Introduction to the variational bicomplex. Mathematical aspects of classical field theory, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Seattle/WA (USA) 1991, Contemp. Math. 132, 51--73 (1992; Zbl 0772.58013); {\it I. M. Anderson} and {\it N. Kamran}, Acta Appl. Math. 41, 135--144 (1995; Zbl 0848.58043)] -- based on an extension to infinite dimensions of J.-M. Souriau's symplectic approach to mechanics. The problem is how to build phase spaces in a covariant way, using directly the Lagrangian and not going through Dirac's theory of constraints. A basic example is presented.
[Mircea Craioveanu (Timişoara)]
MSC 2000:
*37K05 Hamiltonian structures, etc.
70S05 Lagrangian formalism and Hamiltonian formalism
53D05 Symplectic manifolds, general
37K10 Completely integrable systems etc.

Keywords: presymplectic manifold; symplectic manifold; phase space; Hamiltonian system; Lagrangian field theory; Souriau reduction; variational bicomplex; infinite jet bundle; Euler-Lagrange equations

Citations: Zbl 0669.58014; Zbl 0772.58013; Zbl 0848.58043

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Scientific prize winners of the ICM 2010
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Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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