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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.