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 \bar{\Gamma}(\pi) denote the set of sections of \pi\, with compact support.

[edit] The action integral

A classical field theory is mathematically described by

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^{\alpha}_{i}\}\, 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.

[edit] 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 \,

[edit] 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^{j}_{i}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^{j}_{i}\right)\frac{\partial}{\partial u^{\alpha}_{i}}\,

where \gamma^{\alpha}_{i} = \gamma^{\alpha}_{i}(x,u^{\alpha},u^{\alpha}_{i})\,. 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^{j}_{i}\right)\frac{\partial L}{\partial u^{\alpha}_{i}}\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^{\alpha}_{i}} \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^{\alpha}_{i}} \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^{\alpha}_{i}} \circ j^{1}\sigma \right) = 0\,

These are the Euler-Lagrange Equations.

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