- #1

alejandromeira

## Homework Statement

Hi, We are trying to calculate the Coriolis acceleration from the Cristoffel symbols in spherical coordinates for the flat space. I think this problem is interesting because, maybe it's a good way if we want to do the calculations with a computer.

We start whit the Christoffel symbols.

(r->1; theta->2; varphi->3).

Theta is the angle whit Z-axis (from 0 to pi), and varphi is the angle with the X-axis (from 0 to 2pi)

##\Gamma^{1}_{22}=-R##

## \Gamma^{1}_{33}=-R·(Sin\theta)^{2}##

##\Gamma^{2}_{12}=\Gamma^{2}_{21}=\Gamma^{3}_{13}=\Gamma^{3}_{31}=1/R##

##\Gamma^{2}_{33}=-sin\theta·cos\theta##

##\Gamma^{3}_{23}=\Gamma^{3}_{32}=\frac {cos\theta}{sin\theta}##

All the others Christoffel symbols are zero.

## Homework Equations

To do the calculation I have used equation, with Einstein summation criterium:

$$

a_{k}=\frac{\partial^{2}x_{k}}{\partial{t^{2}}}+\Gamma^{k}_{ij}·\frac{dx_{i}}{dt}·\frac{dx_{j}}{dt}

$$

First of all, I don't know if we can use this equation for solve the problem

## The Attempt at a Solution

Ok. We have obtained the following solutions (we start only the calculations of ##a_{r}##, the rest is very mechanical)

##a_{r}=\frac{\partial^{2}r}{\partial{t^{2}}}+\Gamma^{1}_{11}·\frac{dr}{dt}·\frac{dr}{dt}+\Gamma^{1}_{12}·\frac{dr}{dt}·\frac{d\theta}{dt}+\Gamma^{1}_{13}·\frac{dr}{dt}·\frac{d\varphi}{dt}+...##(1+9 therms in total)

The results we have obtained, in this way, for the accelerations are:

$$a_{r}=\frac{\partial^{2}r}{\partial{t^{2}}}-R·[\dot\theta]^{2}-R·(sin\theta)^{2}·[\dot\varphi]^{2};m/s^{2}$$$$a_{\theta}=\frac{\partial^{2}\theta}{\partial{t^{2}}}+\frac{2}{R}·\dot{R}·\dot\theta-sen\theta·cos\theta·[\dot\varphi]^{2};s^{-2}$$$$a_{\varphi}=\frac{\partial^{2}\varphi}{\partial{t^{2}}}+\frac{2}{R}·\dot{R}·\dot\varphi+2\frac{cos\theta}{sin\theta}·\dot\theta·\dot\varphi;s^{-2}$$

The problem we have is that we don't know how to interpret these result. We suppose that in these results they are: the centrifugal acceleration, the acceleration of Coriolis, and the acceleration of drag, but I don't know how to separate one from each other.

We also don't know if the left side of the equation is the aceleration in the inertial reference frame or not.

Ok we need a lot of help, you can see!