Local Lagrangian

Scalar field Lagrangian

Let be a -manifold and be the space of scalar fields on . A first order local Lagrangian on has the form

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

so that the action functional has the form

Euler-Lagrange equations

Let be local coördinates for and suppose

A field is stationary with respect to variations agreeing on the boundary iff variations

Proof

Let be a variation of agreeing on the boundary. Then

whence

Applying integration by parts we get

so by the Fundamental lemma of variational calculus

as claimed.

If is an oriented semi-Riemannian manifold with Riemannian volume form and , then the above condition becomes

Proof

Since , we have

so applying ^P1 we have

whence the claimed equation.

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