Lagrangian system

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

In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle $Q\to\mathbb R$ over the time axis $\mathbb R$ (in particular, $Q=\mathbb R\times M$ if a reference frame is fixed). In classical field theory, all field systems are the Lagrangian ones.

A Lagrangian density $L$ (or, simply, a Lagrangian) of order $r$ is defined as an $n$ , $n=$ 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 $O^*_\infty(Y)$ of exterior forms on jet manifolds of $Y\to X$ . The coboundary operator of this bicomplex contains the variational operator $\delta$ which, acting on $L$ , defines the associated Euler–Lagrange operator $\delta L$ . Given bundle coordinates $(x^\lambda,y^i)$ on a fiber bundle $Y$ and the adapted coordinates $(x^\lambda,y^i,y^i_\Lambda)$ ( $\Lambda=(\lambda_1,\ldots,\lambda_k)$ , $|\Lambda|=k\leq r$ ) on jet manifolds $J^rY$ , a Lagrangian $L$ and its Euler–Lagrange operator read

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

\delta L= \delta_i\mathcal{L} \, dy^i\wedge d^nx,\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^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{L}(x^\lambda,y^i,y^i_\lambda) \, d^nx,\qquad \delta_i L =\partial_i\mathcal{L} - d_\lambda \partial_i^\lambda\mathcal{L}.

The kernel of an Euler–Lagrange operator provides the Euler–Lagrange equations $\delta L=0$ .

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

dL=\delta L + d_H \Theta_L,

where

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

is the total differential and $\Theta_L$ is a Lepage equivalent of $L$ . 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, Euler–Lagrange operators and Euler–Lagrange equations are introduced in the framework of the calculus of variations.

References

Olver, P. Applications of Lie Groups to Differential Equations, 2ed (Springer, 1993) ISBN 0-387-94007-3
Giachetta, G., Mangiarotti, L., Sardanashvily, G., New Lagrangian and Hamiltonian Methods in Field Theory (World Scientific, 1997) ISBN 981-02-1587-8 (arXiv: 0908.1886)

External links

Sardanashvily, G., Graded Lagrangian formalism, Int. G. Geom. Methods Mod. Phys. 10 (2013) N5 1350016; arXiv: 1206.2508

Lagrangian system

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References

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