# Christoffel symbol in 2D Euclidean-Space

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.

## migrated from physics.stackexchange.comMar 30 '16 at 9:26

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• Would Mathematics be a better home for this question? – Qmechanic Mar 28 '16 at 12:06

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 $\ddot x^i = \Gamma^i_{jk}\dot x^j\dot x^k$