In 2D Euclidean space
straight line
in $(x,y)$ coordinate $x=x(s)$ and $y=y(s)$ satisfy
$$\frac{d^2x}{ds^2}=\frac{d^2y}{ds^2}=0$$
is the Christoffel symbol $\Gamma^a_{bc}=0$ in $(x,y)$ coordinate?
and how can I get the $\Gamma^a_{bc}$ in polar coordinate $x=rcos\theta$ and $y=rsin\theta$ ?
thanks.
Best Answer
The Christoffel symbols are a measure of the first derivatives of the metric tensor. In particular, they will be zero if all derivatives are zero. In a euclidean space this will alway be the cas-e, not only in 2 dimensions!
For another coordinate system you can either use the definition (e.g. from wikipedia), which can be complicated since in 4D for example there are 40 of them.
Or, which seems to be easier most of the time, compute the Lagrangian of a free particle (it is mostly easy in an easier basis), take the Euler-Lagrange-Equation and bring it in the form $\ddot x^i = \Gamma^i_{jk}\dot x^j\dot x^k$
This can be also seen as a definition of the symbols as coefficients it this equation. Now you can read out the desired Christoffel symbols from those coefficients.