Nominal impedance

2013-04-09T14:09:11Z

85.228.226.145: Fixed typo

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

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

A Lagrangian density <math>L</math> (or, simply, a [[Lagrangian]]) of order <math>r</math> is defined as an [[exterior form|<math>n</math>-form]], <math>n=</math>dim<math>X</math>, on the <math>r</math>-order [[jet bundle|jet manifold]] <math>J^rY</math> of <math>Y</math>. A Lagrangian <math>L</math> can be introduced as an element of the [[variational bicomplex]] of the [[differential graded algebra]] <math>O^*_\infty(Y)</math> of [[differential form|exterior forms]] on [[jet bundle|jet manifolds]] of <math>Y\to X</math>. The [[cohomology|coboundary operator]] of this bicomplex contains the variational operator <math>\delta</math> which, acting on <math>L</math>, defines the associated Euler–Lagrange operator <math>\delta L</math>. Given bundle coordinates <math>(x^\lambda,y^i)</math> on a fiber bundle <math>Y</math> and the adapted coordinates <math>(x^\lambda,y^i,y^i_\Lambda)</math> (<math>\Lambda=(\lambda_1,\ldots,\lambda_k)</math>, <math>|\Lambda|=k\leq r</math>) on jet manifolds <math>J^rY</math>, a Lagrangian <math>L</math> and its Euler–Lagrange operator read

: <math>L=\mathcal{L}(x^\lambda,y^i,y^i_\Lambda) \, d^nx,</math>

: <math>\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},</math>

where

: <math>d_\Lambda=d_{\lambda_1}\cdots d_{\lambda_k}, \qquad
d_\lambda=\partial_\lambda + y^i_\lambda\partial_i +\cdots,</math>

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

: <math>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}.</math>

The kernel of an Euler–Lagrange operator provides the [[Euler–Lagrange equation]]s <math>\delta L=0</math>.

[[Cohomology]] of the [[variational bicomplex]] leads to the so called
variational formula

: <math>dL=\delta L + d_H \Theta_L,</math>

where

: <math>d_H\phi=dx^\lambda\wedge d_\lambda\phi, \qquad \phi\in
O^*_\infty(Y)</math>

is the total differential and <math>\Theta_L</math> is a Lepage equivalent of <math>L</math>. [[Noether's first theorem]] and [[Noether's second theorem]] are corollaries of this variational formula.

Extended to [[graded manifold]]s, 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]].

== See also ==
*[[Lagrangian]]
*[[Calculus of variations]]
*[[Noether's theorem]]
*[[Noether identities]]
*[[Jet bundle]]
*[[Jet (mathematics)]]
*[[Variational bicomplex]]

== References ==
* Olver, P. ''Applications of Lie Groups to Differential Equations, 2ed'' (Springer, 1993) ISBN 0-387-94007-3
* Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], ''New Lagrangian and Hamiltonian Methods in Field Theory'' (World Scientific, 1997) ISBN 981-02-1587-8 ([http://xxx.lanl.gov/abs/0908.1886 arXiv: 0908.1886])

== External links ==
* [[Gennadi Sardanashvily|Sardanashvily, G.]], Graded Lagrangian formalism, Int. G. Geom. Methods Mod. Phys. '''10''' (2013) N5 1350016; [http://xxx.lanl.gov/abs/1206.2508 arXiv: 1206.2508]

[[Category:Differential operators]]
[[Category:Calculus of variations]]
[[Category:Dynamical systems]]
[[Category:Lagrangian mechanics]]

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Nominal impedance