On phase spaces and the variational bicomplex (after G. Zuckerman). (English) Zbl 1065.37046

Reyes, Enrique G.

Mladenov, Ivaïlo 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 (ISBN 954-84952-8-7/pbk). 189-202 (2004).

The construction of phase spaces in field theory is discussed. In particular, following a paper by 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 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); I. M. Anderson and 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.
For the entire collection see [Zbl 1048.53002].

Reviewer: Mircea Craioveanu (Timişoara)

MSC:

37K05	Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
70S05	Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
53D05	Symplectic manifolds (general theory)
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, 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