Lagrangian system
# Lagrangian system

In mathematics, a Lagrangian system is a pair (Y, L), consisting of a smooth fiber bundle Y → X and a Lagrangian density L, which yields the Euler–Lagrange differential operator acting on sections of Y → X.

In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle Q → ℝ over the time axis ℝ. In particular, Q = ℝ × 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 L (or, simply, a Lagrangian) of order r is defined as an n-form, n = dim X, on the r-order jet manifold $J^rY$ of Y.

A Lagrangian L can be introduced as an element of the variational bicomplex of the differential graded algebra $O^∗_∞(Y)$ of exterior forms on jet manifolds of Y → X. The coboundary operator of this bicomplex contains the variational operator δ which, acting on L, defines the associated Euler–Lagrange operator δL.

### In coordinates

Given bundle coordinates $x^λ,$ $y^i$ on a fiber bundle Y and the adapted coordinates $x^λ,$ $y^i,$ $y^i_Λ,$ (Λ = $(λ_1,$ $...,λ_k),$ |Λ| = k ≤ r) on jet manifolds $J^rY,$ a Lagrangian L and its Euler–Lagrange operator read

    $L={\mathcal {L}}(x^{\lambda },y^{i},y_{\Lambda }^{i})\,d^{n}x,$
    $\delta L=\delta _{i}{\mathcal {L}}\,dy^{i}\wedge d^{n}x,\qquad \delta _{i}{\mathcal {L}}=\partial _{i}{\mathcal {L}}+\sum _{|\Lambda |}(-1)^{|\Lambda |}\,d_{\Lambda }\,\partial _{i}^{\Lambda }{\mathcal {L}},$

where

    $d_{\Lambda }=d_{\lambda _{1}}\cdots d_{\lambda _{k}},\qquad d_{\lambda }=\partial _{\lambda }+y_{\lambda }^{i}\partial _{i}+\cdots ,$

denote the total derivatives.

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

    $L={\mathcal {L}}(x^{\lambda },y^{i},y_{\lambda }^{i})\,d^{n}x,\qquad \delta _{i}L=\partial _{i}{\mathcal {L}}-d_{\lambda }\partial _{i}^{\lambda }{\mathcal {L}}.$

### Euler–Lagrange equations

The kernel of an Euler–Lagrange operator provides the Euler–Lagrange equations δL = 0.

## 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 $θ_L$ is a Lepage equivalent of L. 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 M or various fiber bundles Q over ℝ. A solution of the equations of motion is called a motion.

