In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the EulerâÂÂLagrange differential operator acting on sections of .
In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle over the time axis . In particular, if a reference frame is fixed. In classical field theory, all field systems are the Lagrangian ones.
A Lagrangian density (or, simply, a Lagrangian) of order is defined as an -form, , on the -order jet manifold of .
A Lagrangian can be introduced as an element of the variational bicomplex of the differential graded algebra of exterior forms on jet manifolds of . The coboundary operator of this bicomplex contains the variational operator which, acting on , defines the associated EulerâÂÂLagrange operator .
Given bundle coordinates on a fiber bundle and the adapted coordinates , , ) on jet manifolds , a Lagrangian and its EulerâÂÂLagrange operator read
where
denote the total derivatives.
For instance, a first-order Lagrangian and its second-order EulerâÂÂLagrange operator take the form
The kernel of an EulerâÂÂLagrange operator provides the EulerâÂÂLagrange equations .
Cohomology of the variational bicomplex leads to the so-called variational formula
where
is the total differential and is a Lepage equivalent of . Noether's first theorem and Noether's second theorem are corollaries of this variational formula.
Extended to graded manifolds, the variational bicomplex provides description of graded Lagrangian systems of even and odd variables.
In a different way, Lagrangians, EulerâÂÂLagrange operators and EulerâÂÂLagrange equations are introduced in the framework of the calculus of variations.
In classical mechanics equations of motion are first and second order differential equations on a manifold or various fiber bundles over . A solution of the equations of motion is called a motion.