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 #m/thm/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.

#state/tidy | #lang/en | #SemBr

Footnotes

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