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


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

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 $$\theta^{j} = du^{j} - u^{j}_{i}dx^{i}\,$$ 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

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.