In general given a line element $ds^2=g_{\mu \nu}dx^\mu dx^\nu$ how does one obtain the Lagrangian.
My lecturer said that we should be able to just read off the Lagrangian from the metric but I have seen 3 formulas for obtaining it
$L=g_{\mu \nu}dx^\mu dx^\nu$ For flat Euclidean space
$L=\tfrac{1}{2}g_{\mu \nu}dx^\mu dx^\nu$ For curved space
$L=-g_{\mu \nu}dx^\mu dx^\nu$ for Minkowski space-time.
I see no rhyme nor reason to choosing these, certainly the problem seems more complicated then just reading the Lagrangian from the metric, but no method was given in class .
Could anyone please explain to me how we obtain Lagrangians for given metrics ?
The Lagrangian is always $\mathcal{L}=g_{\mu\nu}dx^\mu dx^\nu$. Multiplying the Lagrangian by a constant does not change the physics and is only done to simplify the algebraic expression.