Are Euler-Lagrange Equations Applicable to All Differential Manifolds?

[Incand]

Hey!
I'm not sure if this belongs better here or in mechanics but while I was doing some mechanics problems I started wondering if Lagrange equations are true for any differential manifold.
Obviously they work for pseudo-riemann ones (general relativity) but do they work for others (all)?

I got no real knowledge of the math behind at all just wondered, since they work for relativistic particles, in what geometry they do and don't work.

 

[homeomorphic]

Lagrange's equations are a local thing, so yes, because locally, in a manifold, it's no different from R^n. Actually, the configuration spaces you see in mechanics tend to be manifolds, so you're presumably already using the fact that they work in manifolds. You don't necessarily care about the geometry, unless that feeds into the Lagrangian somehow (generally, it will, but not always). It all just depends on having local coordinate systems, and that precedes the geometry.

 

[Incand]

Thanks, very well explained!