Covariant classical field theory

In recent years, there has been renewed interest in covariant classical field theory. Here, classical fields are represented by sections of fiber bundles and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the variational bicomplex are the correct domain for such a description. The Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory where momenta correspond to derivatives of field variables with respect to all world coordinates. Non-autonomous mechanics is formulated as covariant classical field theory on fiber bundles over the time axis $\mathbb R$.

Derivation and proof

The worksheet provides some of the geometric structure to the covariant formalism of first-order classical field theories. Talk:Covariant classical field theory/workpage

See also

• Classical field theory
• Exterior algebra
• Variational bicomplex
• Quantum field theory

References

• Saunders, D.J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
• Bocharov, A.V. [et al.] "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 082180958X
• De Leon, M., Rodrigues, P.R., "Generalized Classical Mechanics and Field Theory", Elsevier Science Publishing, 1985, ISBN 0-444-87753-3
• Griffiths, P.A., "Exterior Differential Systems and the Calculus of Variations", Boston: Birkhauser, 1983, ISBN 3-7643-3103-8
• Gotay, M.J., Isenberg, J., Marsden, J.E., Montgomery R., Momentum Maps and Classical Fields Part I: Covariant Field Theory, November 2003
• Echeverria-Enriquez, A., Munoz-Lecanda, M.C., Roman-Roy,M., Geometry of Lagrangian First-order Classical Field Theories, May 1995
• Giachetta, G., Mangiarotti, L., Sardanashvily, G., "Advanced Classical Field Theory", World Scientific, 2009, ISBN 9789812838957 (arXiv: 0811.0331v2)