Lagrangian formalism on fibre bundles is formulated in a strict
mathematical way. Therefore, it can be applied, e. g. to classical field theory
(#) and non-autonomous classical
mechanics (#) where dynamic
variables are sections of fibre bundles.
Let Y->X be a fibre bundle
over an n-dimensional smooth manifold
X. A Lagrangian density Ld^{n}x (or, simply, a Lagrangian) of
order r is defined as an exterior n-form on the r-order jet manifold J^{r}Y
of Y->X. A Lagrangian can be introduced
as an element of the variational bicomplex of a differential graded algebra of
exterior forms on jet manifolds of Y->X
of all order. The coboundary operator of this bicomplex contains the variational
operator which, acting on a Lagrangian, defines the associated Euler – Lagrange
operator and the corresponding Euler – Lagrange equations.
First and second Noether theorems also are formulated. To study
symmetries of a Lagrangian, one considers vector fields on a fibre bundle Y which are treated as infinitesimal
generators of one-parameter groups of automorphisms of Y. Such a vector field u
is called a symmetry of a Lagrangian L
if the Lie derivative of L along u vanishes. In this case, the first
Noether theorem leads to a conservation law of the corresponding symmetry
current.
References
G.Sardanashvily, Fibre bundles,
jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886v1
