Talk:Covariant classical field theory/workpage

This is a worksheet for Covariant classical field theory

Notation

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The notation follows that of introduced in the article on jet bundles. Also, let   denote the set of sections of   with compact support.

The action integral

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A classical field theory is mathematically described by

  • A fibre bundle  , where   denotes an  -dimensional spacetime.
  • A Lagrangian form  

Let   denote the volume form on  , then   where   is the Lagrangian function. We choose fibred co-ordinates   on  , such that

 

The action integral is defined by

 

where   and is defined on an open set  , and   denotes its first jet prolongation.

Variation of the action integral

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The variation of a section   is provided by a curve  , where   is the flow of a  -vertical vector field   on  , which is compactly supported in  . A section   is then stationary with respect to the variations if

 

This is equivalent to

 

where   denotes the first prolongation of  , by definition of the Lie derivative. Using Cartan's formula,  , Stokes' theorem and the compact support of  , we may show that this is equivalent to

 

The Euler-Lagrange equations

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Considering a  -vertical vector field on  

 

where  . Using the contact forms   on  , we may calculate the first prolongation of  . We find that

 

where  . From this, we can show that

 

and hence

 

Integrating by parts and taking into account the compact support of  , the criticality condition becomes

   
 

and since the   are arbitrary functions, we obtain

 

These are the Euler-Lagrange Equations.