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On phase spaces and the variational bicomplex (after G. Zuckerman). (English) Zbl 1065.37046
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.
MSC:
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
70S05Lagrangian formalism and Hamiltonian formalism
53D05Symplectic manifolds, general
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies