Euler--Lagrange Forms and Cohomology Groups on Jet Bundles

Using the language of jet bundles, we generalize the definitions of Euler--Lagrange one-form and the associated cohomology which were introduced by Guo et al. Commun. Theor. Phys. 37(2002)1. Continuous and discrete Lagrange mechanics and field theory are presented. Higher order Euler--Lagrange cohomology groups are also introduced.