Local Lagrangian

Path Lagrangian

Let be a -manifold and be the space of -paths from to , i.e.

A first order local Lagrangian on has the form

where we abuse notation to invoke a -function

so that the action functional has the form

Euler-Lagrange equations

Let be local coördinates for . A path is stationary¹ iff variations

where we denote .

Proof

Let be a variation of . Then

whence

Applying Integration by parts on the latter term, and noting the boundary term vanishes since we are in , we get

so by the Fundamental lemma of variational calculus

as claimed.

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Footnotes

1. i.e. the first variation vanishes. ↩