When does a Lagrangian have (1/2)?

[Ai52487963]

Homework Statement

Not really a homework question: just a general query. About half the time when working examples, I see a (1/2) thrown into a Lagrangian for use with Euler-Lagrange, but I can't seem to find out why. Is the (1/2) present (or not?) only for the case of a non-symmetric metric?

Like if I had:

$$ds^2 = dx^2 + 2dxdy + dy^2$$,

would the Lagrangian then be:

$$L = \frac{1}{2}(\dot{x}+...)$$?

Whereas, the Lagrangian for the Schwartzschild metric doesn't have the (1/2):

$$L = -(1 - \frac{2m}{r})\dot{t} + ...$$

because the metric that describes Schwartzschild space time is symmetric?

 

[Chopin]

Usually when I've seen a 1/2 in a Lagrangian, it's because it's quadratic in the variable of interest, so when you take the derivative of it as part of the E-L equation, you'll pull in a factor of 2. Adding the 1/2 therefore gives you equations of motion with no numeric coefficient.