Path Lagrangian

Let $𝑀$ be a $𝐶 𝛼$ -manifold and $P (𝑎, 𝑏, 𝑝 𝑎, 𝑝 𝑏)$ be the space of $𝐶 𝛼$ -paths $𝛾$ from $𝑝 𝑎$ to $𝑝 𝑏$ , i.e.

𝛾 : [𝑎, 𝑏] \to 𝑀 : 𝑎 \mapsto 𝑝 𝑎 : 𝑏 \mapsto 𝑝 𝑏 .

A first order local Lagrangian on $P (𝑎, 𝑏, 𝑝 𝑎, 𝑝 𝑏)$ has the form

𝐿 [𝛾] = 𝐿 (𝑡, 𝛾 (𝑡), ˙ 𝛾 (𝑡)) d 𝑡

where we abuse notation to invoke a $𝐶 𝛼$ -function

𝐿 : ℝ \times 𝑇 𝑀 \to ℝ

so that the action functional $ℒ : P (𝑎, 𝑏, 𝑝 𝑎, 𝑝 𝑏) \to ℝ$ has the form

ℒ [𝛾] = \int 𝑏 𝑎 𝐿 (𝑡, 𝛾 (𝑡), ˙ 𝛾 (𝑡)) d 𝑡 .

Euler-Lagrange equations

Let $𝑥 : 𝑈 \to ℝ 𝑚$ be local coördinates for $𝑀$ . A path $𝛾 \in P (𝑎, 𝑏, 𝑝 𝑎, 𝑝 𝑏)$ is stationary¹ iff #m/thm/variations

0 = 𝜕 𝐿 𝜕 𝛾 𝜇 - d d 𝑡 𝜕 𝐿 𝜕 ˙ 𝛾 𝜇

where we denote $𝛾 𝜇 = 𝑥 𝜇 \circ 𝛾$ .

Proof

Let $𝛼 : (- 𝜖 0, 𝜖 0) \to P (𝑎, 𝑏, 𝑝 𝑎, 𝑝 𝑏)$ be a variation of $𝛾$ . Then

ℒ [𝛼 (𝜖)] = \int 𝑏 𝑎 𝐿 (𝑡, 𝛼 (𝜖; 𝑡), ˙ 𝛼 (𝜖; 𝑡)) d 𝑡

whence

𝛿 ℒ [𝛾; 𝛼] = d d 𝜖 ∣ 𝜖 = 0 \int 𝑏 𝑎 𝐿 (𝑡, 𝛼 (𝜖; 𝑡), ˙ 𝛼 (𝜖; 𝑡)) d 𝑡 = \int 𝑏 𝑎 d d 𝜖 ∣ 𝜖 = 0 𝐿 (𝑡, 𝛼 (𝜖; 𝑡), ˙ 𝛼 (𝜖; 𝑡)) d 𝑡 = \int 𝑏 𝑎 (𝜕 𝐿 𝜕 𝛾 𝜇 𝜕 𝛼 𝜇 𝜕 𝜖 (0; 𝑡) + 𝜕 𝐿 𝜕 ˙ 𝛾 𝜇 𝜕 ˙ 𝛼 𝜇 𝜕 𝜖 (0; 𝑡)) d 𝑡 = \int 𝑏 𝑎 (𝜕 𝐿 𝜕 𝛾 𝜇 𝜕 𝛼 𝜇 𝜕 𝜖 (0; 𝑡) + 𝜕 𝐿 𝜕 ˙ 𝛾 𝜇 𝜕 2 𝛼 𝜇 𝜕 𝑡 𝜕 𝜖 (0; 𝑡)) d 𝑡 .

Applying Integration by parts on the latter term, and noting the boundary term vanishes since we are in $P (𝑎, 𝑏, 𝑝 𝑎, 𝑝 𝑏)$ , we get

𝛿 ℒ [𝛾; 𝛼] = \int 𝑏 𝑎 (𝜕 𝐿 𝜕 𝛾 𝜇 - d d 𝑡 𝜕 𝐿 𝜕 ˙ 𝛾 𝜇) 𝜕 𝛼 𝜇 𝜕 𝜖 (0; 𝑡) d 𝑥 = 0

0 = 𝜕 𝐿 𝜕 𝛾 𝜇 - d d 𝑡 𝜕 𝐿 𝜕 ˙ 𝛾 𝜇

as claimed.

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