Scalar field Lagrangian

Let $𝑀$ be a $𝐶 𝛼$ -manifold and $𝐶 𝛼 (𝑀)$ be the space of scalar fields on $𝑀$ . A first order local Lagrangian on $𝐶 𝛼 (𝑀)$ has the form

𝐿 [𝜑] = 𝐿 (𝑝, 𝜑 | 𝑝, d 𝜑 | 𝑝)

where we abuse notation and invoke a $𝐶 𝛼$ -map to top forms

𝐿 : (𝑇 00 \oplus 𝑇 01) 𝑀 \to Ω 𝑚 (𝑀)

so that the action functional $ℒ : 𝐶 𝛼 (𝑀) \to ℝ$ has the form

ℒ [𝛾] = \int 𝑝 \in 𝑀 𝐿 (𝑝, 𝜑 | 𝑝, d 𝜑 | 𝑝) .

Euler-Lagrange equations

Let $𝑥 : 𝑈 \to ℝ 𝑚$ be local coördinates for $𝑀$ and suppose

𝐿 = ˜ 𝐿 đ 𝑚 𝑥 = ˜ 𝐿 d 𝑥 1 \land \dots \land d 𝑥 𝑚 .

A field $𝜑 \in 𝐶 𝛼 (𝑀)$ is stationary with respect to variations agreeing on the boundary iff variations

0 = 𝜕 ˜ 𝐿 𝜕 𝜑 - 𝜕 𝜕 𝑥 𝜇 𝜕 ˜ 𝐿 𝜕 (d 𝜑 𝜇) .

Proof

Let $𝛼 : (- 𝜖 0, 𝜖 0) \to 𝐶 𝛼 (𝑀)$ be a variation of $𝜑$ agreeing on the boundary. Then
$ℒ [𝛼 (𝜖)] = \int 𝑝 \in 𝑀 ˜ 𝐿 (𝑝, 𝜑 | 𝑝, d 𝜑 | 𝑝) đ 𝑉 = \int 𝑀 đ 𝑚 𝑥 ˜ 𝐿 (𝑥, 𝜑 | 𝑥, d 𝜑 | 𝑥)$
whence
$𝛿 ℒ [𝜑, 𝛼] = d d 𝜖 ∣ 𝜖 = 0 \int 𝑀 đ 𝑚 𝑥 ˜ 𝐿 (𝑥, 𝛼 (𝜖; 𝑥), d 𝛼 (𝜖; 𝑥)) = \int 𝑀 đ 𝑚 𝑥 d d 𝜖 ∣ 𝜖 = 0 ˜ 𝐿 (𝑥, 𝛼 (𝜖; 𝑥), d 𝛼 (𝜖; 𝑥)) = \int 𝑀 đ 𝑚 𝑥 (𝜕 ˜ 𝐿 𝜕 𝜑 𝜕 𝛼 𝜕 𝜖 (𝛼; 𝑥) + 𝜕 ˜ 𝐿 𝜕 (d 𝜑 𝜇) 𝜕 (d 𝛼 𝜇) 𝜕 𝜖 (0; 𝑡)) = \int 𝑀 đ 𝑚 𝑥 (𝜕 ˜ 𝐿 𝜕 𝜑 𝜕 𝛼 𝜕 𝜖 (𝛼; 𝑥) + 𝜕 ˜ 𝐿 𝜕 (d 𝜑 𝜇) 𝜕 2 𝛼 𝜇 𝜕 𝑥 𝜇 𝜕 𝜖 (0; 𝑡)) .$
Applying integration by parts we get
$𝛿 ℒ [𝜑; 𝛼] = \int 𝑀 đ 𝑚 𝑥 𝜕 𝛼 𝜕 𝜖 (𝛼; 𝑥) (𝜕 ˜ 𝐿 𝜕 𝜑 - 𝜕 𝜕 𝑥 𝜇 𝜕 ˜ 𝐿 𝜕 (d 𝜑 𝜇))$
so by the Fundamental lemma of variational calculus
$0 = 𝜕 ˜ 𝐿 𝜕 𝜑 - 𝜕 𝜕 𝑥 𝜇 𝜕 ˜ 𝐿 𝜕 (d 𝜑 𝜇)$
as claimed.

If $𝑀$ is an oriented semi-Riemannian manifold with Riemannian volume form $đ 𝑉$ and $𝐿 = ¯ 𝐿 đ 𝑉$ , then the above condition becomes

0 = 𝜕 ¯ 𝐿 𝜕 𝜑 - \nabla 𝜇 𝜕 ¯ 𝐿 𝜕 (d 𝜑 𝜇) .

Proof

Since $˜ 𝐿 = \sqrt | 𝑔 | ¯ 𝐿$ , we have
$0 = \sqrt | 𝑔 | 𝜕 ¯ 𝐿 𝜕 𝜑 - 𝜕 𝜕 𝑥 𝜇 \sqrt | 𝑔 | 𝜕 ¯ 𝐿 𝜕 (d 𝜑 𝜇)$
so applying ^P1 we have
$0 = \sqrt | 𝑔 | 𝜕 ¯ 𝐿 𝜕 𝜑 - \sqrt | 𝑔 | \nabla 𝜇 𝜕 ¯ 𝐿 𝜕 (d 𝜑 𝜇)$
whence the claimed equation.

tidy | en | SemBr

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Scalar field Lagrangian

Euler-Lagrange equations

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