How do the Euler-Lagrange equations generalise to an arbitrary manifold?

Question

So every formalism for the EL equations I have seen relies on choosing a coordinate chart. However, if we had say, a field on a sphere, then we can’t have global coordinates.

How, in principle, would one get the eom of objects on a manifold that doesn’t have global coordinates?

You can use the variational bicomplex : https://en.wikipedia.org/wiki/Variational_bicomplex — Slereah, Mar 03 '20 at 14:54

score 0 · Answer 1 · answered Mar 03 '20 at 14:56

There are in principle no problems with the Euler-Lagrange (EL) equations themselves since they are local in spacetime and target space, and can hence be expressed using local coordinates. Rather the problem could be that the action functional is not globally defined, but only defined within a local coordinate chart. For an example of the latter, see this Phys.SE post.

How do the Euler-Lagrange equations generalise to an arbitrary manifold?

1 Answers1