Lagrangian System

In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange 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 $Q \rarr \mathbb$ over the time axis $\mathbb$. In particular, $Q = \mathbb \times M$ if a reference frame is fixed. In classical field theory, all field systems are the Lagrangian ones.

Lagrangians and Euler–Lagrange operators

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 Euler–Lagrange operator .

In coordinates

Given bundle coordinates on a fiber bundle and the adapted coordinates , , ) on jet manifolds , a Lagrangian and its Euler–Lagrange operator read

: $L=\mathcal(x^\lambda,y^i,y^i_\Lambda) \, d^nx,$

: $\delta L= \delta_i\mathcal \, dy^i\wedge d^nx,\qquad \delta_i\mathcal =\partial_i\mathcal + \sum_(-1)^ \, d_\Lambda \, \partial_i^\Lambda\mathcal,$

where

: $d_\Lambda=d_\cdots d_, \qquad d_\lambda=\partial_\lambda + y^i_\lambda\partial_i +\cdots,$

denote the total derivatives.

For instance, a first-order Lagrangian and its second-order Euler–Lagrange operator take the form

: $L=\mathcal(x^\lambda,y^i,y^i_\lambda) \, d^nx,\qquad \delta_i L =\partial_i\mathcal - d_\lambda \partial_i^\lambda\mathcal.$

Euler–Lagrange equations

The kernel of an Euler–Lagrange operator provides the Euler–Lagrange equations .

Cohomology and Noether's theorems

Cohomology of the variational bicomplex leads to the so-called
variational formula

: $dL=\delta L + d_H \Theta_L,$

where

: $d_H\Theta_L=dx^\lambda\wedge d_\lambda\phi, \qquad \phi\in O^*_\infty(Y)$

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.

Graded manifolds

Extended to graded manifolds, the variational bicomplex provides description of graded Lagrangian systems of even and odd variables.

Alternative formulations

In a different way, Lagrangians, Euler–Lagrange operators and Euler–Lagrange equations are introduced in the framework of the calculus of variations.

Classical mechanics

In classical mechanics equations of motion are first and second order differential equations on a manifold or various fiber bundles over $\mathbb$. A solution of the equations of motion is called a ''motion''.

See also

* Lagrangian mechanics
*Calculus of variations
* Noether's theorem
*Noether identities
* Jet bundle
* Jet (mathematics)
*Variational bicomplex

References

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External links

*{{cite journal, first=G., last=Sardanashvily, title=Fibre Bundles, Jet Manifolds and Lagrangian Theory. Lectures for Theoreticians, year=2009, arxiv=0908.1886, bibcode=2009arXiv0908.1886S

Differential operators
Calculus of variations
Dynamical systems
Lagrangian mechanics