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**