Covariant classical field theory::worksheet

This is a worksheet for Covariant classical field theory

Notation

The notation follows that of introduced in the article on jet bundles. Also, let Γ̄(π) denote the set of sections of π  with compact support.

The action integral

A classical field theory is mathematically described by

• A fibre bundle (ℰ,π,ℳ), where ℳ denotes an n -dimensional spacetime.
• A Lagrangian form Λ : J¹π → ΛⁿM

Let  ⋆ 1  denote the volume form on M , then Λ = L ⋆ 1  where L : J¹π → ℝ is the Lagrangian function. We choose fibred co-ordinates {xⁱ, u^α, u_i^α}  on J¹π , such that

 ⋆ 1 = dx¹ ∧ … ∧ dxⁿ

The action integral is defined by

S(σ) = ∫_σ(ℳ)(j¹σ)^*Λ 

where σ ∈ Γ̄(π) and is defined on an open set σ(ℳ) , and j¹σ  denotes its first jet prolongation.

Variation of the action integral

The variation of a section σ ∈ Γ̄(π)  is provided by a curve σ_t = η_t ∘ σ , where η_t  is the flow of a π -vertical vector field V  on ℰ , which is compactly supported in ℳ . A section σ ∈ Γ̄(π)  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

∫_ℳ(j¹σ)^*ℒ_V¹Λ = 0 

where V¹  denotes the first prolongation of V , by definition of the Lie derivative. Using Cartan's formula, ℒ_X = i_Xd + di_X , Stokes' theorem and the compact support of σ , we may show that this is equivalent to

∫_ℳ(j¹σ)^*i_V¹dΛ = 0 

The Euler-Lagrange equations

Considering a π -vertical vector field on ℰ

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

where β^α = β^α(x,u) . Using the contact forms θ^j = du^j − u_i^jdxⁱ  on J¹π , 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 γ_i^α = γ_i^α(x,u^α,u_i^α) . 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 σ , the criticality condition becomes

and since the β^α  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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