Local Lagrangian

Path Lagrangian

Let 𝑀 be a 𝐶𝛼-manifold and P(𝑎,𝑏,𝑝𝑎,𝑝𝑏) be the space of 𝐶𝛼-paths 𝛾 from 𝑝𝑎 to 𝑝𝑏, i.e.

𝛾:[𝑎,𝑏]→𝑀:𝑎↦𝑝𝑎:𝑏↦𝑝𝑏.

A first order local Lagrangian on P(𝑎,𝑏,𝑝𝑎,𝑝𝑏) has the form

𝐿[𝛾]=𝐿(𝑡,𝛾(𝑡),˙𝛾(𝑡))d𝑡

where we abuse notation to invoke a 𝐶𝛼-function

𝐿:ℝ×𝑇𝑀→ℝ

so that the action functional ℒ :P(𝑎,𝑏,𝑝𝑎,𝑝𝑏) →ℝ has the form

ℒ[𝛾]=∫𝑏𝑎𝐿(𝑡,𝛾(𝑡),˙𝛾(𝑡))d𝑡.

Euler-Lagrange equations

Let 𝑥 :𝑈 →ℝ𝑚 be local coördinates for 𝑀. A path 𝛾 ∈P(𝑎,𝑏,𝑝𝑎,𝑝𝑏) is stationary¹ iff variations

0=𝜕𝐿𝜕𝛾𝜇−dd𝑡𝜕𝐿𝜕˙𝛾𝜇

where we denote 𝛾𝜇 =𝑥𝜇 ∘𝛾.

Proof

Let 𝛼 :( −𝜖0,𝜖0) →P(𝑎,𝑏,𝑝𝑎,𝑝𝑏) be a variation of 𝛾. Then

ℒ[𝛼(𝜖)]=∫𝑏𝑎𝐿(𝑡,𝛼(𝜖;𝑡),˙𝛼(𝜖;𝑡))d𝑡

whence

𝛿ℒ[𝛾;𝛼]=dd𝜖∣𝜖=0∫𝑏𝑎𝐿(𝑡,𝛼(𝜖;𝑡),˙𝛼(𝜖;𝑡))d𝑡=∫𝑏𝑎dd𝜖∣𝜖=0𝐿(𝑡,𝛼(𝜖;𝑡),˙𝛼(𝜖;𝑡))d𝑡=∫𝑏𝑎(𝜕𝐿𝜕𝛾𝜇𝜕𝛼𝜇𝜕𝜖(0;𝑡)+𝜕𝐿𝜕˙𝛾𝜇𝜕˙𝛼𝜇𝜕𝜖(0;𝑡))d𝑡=∫𝑏𝑎(𝜕𝐿𝜕𝛾𝜇𝜕𝛼𝜇𝜕𝜖(0;𝑡)+𝜕𝐿𝜕˙𝛾𝜇𝜕2𝛼𝜇𝜕𝑡 𝜕𝜖(0;𝑡))d𝑡.

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

𝛿ℒ[𝛾;𝛼]=∫𝑏𝑎(𝜕𝐿𝜕𝛾𝜇−dd𝑡𝜕𝐿𝜕˙𝛾𝜇)𝜕𝛼𝜇𝜕𝜖(0;𝑡)d𝑥=0

so by the Fundamental lemma of variational calculus

0=𝜕𝐿𝜕𝛾𝜇−dd𝑡𝜕𝐿𝜕˙𝛾𝜇

as claimed.

---

tidy | en | SemBr

Footnotes

1. i.e. the first variation 𝛿ℒ[𝛾] vanishes. ↩