Talk:Covariant classical field theory/workpage

This is a worksheet for Covariant classical field theory

Notation

The notation follows that of introduced in the article on jet bundles. Also, let ${\bar {\Gamma }}(\pi )$ denote the set of sections of $\pi \,$ with compact support.

The action integral

A classical field theory is mathematically described by

• A fibre bundle $({\mathcal {E}},\pi ,{\mathcal {M}})$, where ${\mathcal {M}}$ denotes an $n\,$-dimensional spacetime.
• A Lagrangian form $\Lambda :J^{1}\pi \rightarrow \Lambda ^{n}M$

Let $\star 1\,$ denote the volume form on $M\,$, then $\Lambda =L\star 1\,$ where $L:J^{1}\pi \rightarrow \mathbb {R}$ is the Lagrangian function. We choose fibred co-ordinates $\{x^{i},u^{\alpha },u_{i}^{\alpha }\}\,$ on $J^{1}\pi \,$, such that

$\star 1=dx^{1}\wedge \ldots \wedge dx^{n}$

The action integral is defined by

$S(\sigma )=\int _{\sigma ({\mathcal {M}})}(j^{1}\sigma )^{*}\Lambda \,$

where $\sigma \in {\bar {\Gamma }}(\pi )$ and is defined on an open set $\sigma ({\mathcal {M}})\,$, and $j^{1}\sigma \,$ denotes its first jet prolongation.

Variation of the action integral

The variation of a section $\sigma \in {\bar {\Gamma }}(\pi )\,$ is provided by a curve $\sigma _{t}=\eta _{t}\circ \sigma \,$, where $\eta _{t}\,$ is the flow of a $\pi \,$-vertical vector field $V\,$ on ${\mathcal {E}}\,$, which is compactly supported in ${\mathcal {M}}\,$. A section $\sigma \in {\bar {\Gamma }}(\pi )\,$ is then stationary with respect to the variations if

$\left.{\frac {d}{dt}}\right|_{t=0}\int _{\sigma ({\mathcal {M}})}(j^{1}\sigma _{t})^{*}\Lambda =0\,$

This is equivalent to

$\int _{\mathcal {M}}(j^{1}\sigma )^{*}{\mathcal {L}}_{V^{1}}\Lambda =0\,$

where $V^{1}\,$ denotes the first prolongation of $V\,$, by definition of the Lie derivative. Using Cartan's formula, ${\mathcal {L}}_{X}=i_{X}d+di_{X}\,$, Stokes' theorem and the compact support of $\sigma \,$, we may show that this is equivalent to

$\int _{\mathcal {M}}(j^{1}\sigma )^{*}i_{V^{1}}d\Lambda =0\,$

The Euler-Lagrange equations

Considering a $\pi \,$-vertical vector field on ${\mathcal {E}}$

$V=\beta ^{\alpha }{\frac {\partial }{\partial u^{\alpha }}}\,$

where $\beta ^{\alpha }=\beta ^{\alpha }(x,u)\,$. Using the contact forms $\theta ^{j}=du^{j}-u_{i}^{j}dx^{i}\,$ on $J^{1}\pi \,$, we may calculate the first prolongation of $V\,$. We find that

$V^{1}=\beta ^{\alpha }{\frac {\partial }{\partial u^{\alpha }}}+\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}u_{i}^{j}\right){\frac {\partial }{\partial u_{i}^{\alpha }}}\,$

where $\gamma _{i}^{\alpha }=\gamma _{i}^{\alpha }(x,u^{\alpha },u_{i}^{\alpha })\,$. From this, we can show that

$i_{V^{1}}d\Lambda =\left[\beta ^{\alpha }{\frac {\partial L}{\partial u^{\alpha }}}+\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}u_{i}^{j}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\right]\star 1\,$

and hence

$(j^{1}\sigma )^{*}i_{V^{1}}d\Lambda =\left[(\beta ^{\alpha }\circ \sigma ){\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}\circ \sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}\circ \sigma \right){\frac {\partial \sigma ^{j}}{\partial x^{i}}}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right]\star 1\,$

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

$\int _{\mathcal {M}}(j^{1}\sigma )^{*}i_{V^{1}}d\Lambda \,$	$=\int _{\mathcal {M}}\left[{\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma -{\frac {\partial }{\partial x^{i}}}\left({\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right)\right](\beta ^{\alpha }\circ \sigma )\star 1\,$
	$=0\,$

and since the $\beta ^{\alpha }\,$ are arbitrary functions, we obtain

${\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma -{\frac {\partial }{\partial x^{i}}}\left({\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right)=0\,$

These are the Euler-Lagrange Equations.