Acceleration in spherical coordinates

[TJ]

Hi.

 

How is acceleration [math]a[/math] written in spherical coordinates, so that [math]a[/math] is not dependant from time but from the line element [math]ds=\sqrt{dr^2+r^2d\theta^2+r^2\sin^2\theta^2d\phi^2}[/math]

So I would like to have [math]\frac{d}{ds}[/math] instead of [math]\frac{d}{dt}[/math] in the next equation:

 

[swansont]

One possibility is to use the chain rule; one can rewrite acceleration in a Cartesian system

 

[math]a = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v\frac{dv}{dx}[/math]