Covariant classical field theory::worksheet
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This is a worksheet for Covariant classical field theory
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[edit] Notation
The notation follows that of introduced in the article on jet bundles. Also, let denote the set of sections of with compact support.
[edit] The action integral
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.
[edit] Variation of the action integral
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
[edit] The Euler-Lagrange equations
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.