Example of computing geodesics with 2D polar coordinates

In summary, the conversation is about finding and solving the geodesics equation for polar coordinates. The speaker starts by defining the Christoffel symbols with expressions involving polar coordinates. They then take the tangent vector basis and find that the Christoffel symbols are not correct. They also use double $ for every symbol, which is not necessary. The final result should be ##r = \theta^2/2##, but the speaker is unsure about this due to possible mistakes in their calculations.
  • #1
fab13
312
6
I am trying to find and solve the geodesics equation for polar coordinates. If I start by the definition of Christoffel symbols with the following expressions :

$$de_{i}=w_{i}^{j}\,de_{j}=\Gamma_{ik}^{j}du^{k}\,de_{j}$$

with $$u^{k}$$ is the k-th component of polar coordinates ($$1$$ is for $$r$$ and $$2$$ is for $$\theta$$).

Now, if I take :

$$de_{r} = d\theta e_{\theta}$$

$$de_{\theta} = -d\theta e_{r}$$

So : $$\Gamma_{12}^{2} = 1$$ and $$\Gamma_{22}^{1} = -1$$

All others Christoffel symbols seem to be zero.

Now, I can write the geodesics equation with :

$$\dfrac{d^{2}u^{i}}{ds^{2}} + \Gamma_{jk}^{i}\dfrac{du^{j}}{ds}\dfrac{du^{k}}{ds}$$

I get :

$$\dfrac{d^{2}r}{ds^{2}} = \dfrac{d\theta}{ds}\dfrac{d\theta}{ds}\,\,\,(1)$$

$$\dfrac{d^{2}\theta}{ds^{2}} = -\dfrac{dr}{ds}\dfrac{d\theta}{ds}\,\,\,(2)$$

By make appearing the logarithmic derivate : $$\dfrac{d\,ln(u)}{ds}=\dfrac{u'}{u}$$, I have :

for (2) : $$\dfrac{\theta'}{\theta} = - \dfrac{dr}{ds} ; \dfrac{d\theta}{ds} = e^{-r}$$

Finally, I have for (2): $$\theta(s)=s\,e^{-r}$$

for (1), taking $$\theta(s)=s\,e^{-r}$$, I have :

$$\dfrac{d^{2}r}{ds^{2}} = \bigg(\dfrac{d\theta}{ds}\bigg)^{2} = e^{-2r}$$

Finally, we get for (1) : $$r(s)=\dfrac{s^{2}}{2}e^{-2r} = \dfrac{1}{2}\theta^{2}$$

By using results from (1) and (2), I could write :

$$r=\theta^{2}/2$$

I don't understand this result knowing $$s$$ may be choose as a linear or curvilinear parameter (like the length on the geodesics).

If I set $$r$$ fixed, I expect to find $$s=r\theta$$. It doesn't seem clear for me. What should I find as final result ?

Surely I have done a mistake in my above calculus.

If someone could see what's wrong, this would be great.

Thanks in advance.
 
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  • #2
fab13 said:
Now, if I take :

$$de_{r} = d\theta e_{\theta}$$

$$de_{\theta} = -d\theta e_{r}$$

So : $$\Gamma_{12}^{2} = 1$$ and $$\Gamma_{22}^{1} = -1$$

This is not correct. You are using the normalised basis vectors of polar coordinates here where you should be using the tangent vector basis ##E_i = \partial \vec x/\partial y^i##. These are not the Christoffel symbols of polar coordinates. Note that you are working in a torsion free space and therefore the Christoffel symbols must be symmetric in the lower indices.

Also note that you should not use double $ for any math you want to write, it should be used for equations that need to stand alone only. If you use it for every small symbol, it will be very disruptive for people reading your text, use double # instead. This produces inline math mode such as the one used in my previous paragraph.
 

1. How are geodesics computed using 2D polar coordinates?

In order to compute geodesics using 2D polar coordinates, we first need to define the geodesic equation in terms of polar coordinates. This can be done by converting the Cartesian coordinates of the geodesic equation to polar coordinates. Once the geodesic equation is defined, we can use numerical methods or analytical solutions to solve for the geodesic path.

2. What is the significance of using polar coordinates in computing geodesics?

Polar coordinates are useful in computing geodesics because they are particularly well-suited for describing circular or curved paths. This makes them an ideal choice for computing geodesics, which are the shortest paths between two points on a curved surface.

3. Can 2D polar coordinates be used to compute geodesics on any type of surface?

Yes, 2D polar coordinates can be used to compute geodesics on any type of surface, as long as the surface can be described using polar coordinates. This includes surfaces such as spheres, cylinders, and cones.

4. Are there any limitations to using 2D polar coordinates in computing geodesics?

While 2D polar coordinates can be used to compute geodesics on a wide range of surfaces, they may not be the most efficient method for computing geodesics on highly complex surfaces. In these cases, other coordinate systems or numerical methods may be more appropriate.

5. Are there any real-world applications of computing geodesics with 2D polar coordinates?

Yes, there are many real-world applications of computing geodesics with 2D polar coordinates. One example is in navigation systems, where geodesics are used to determine the shortest path between two points on a map. Another example is in robotics, where geodesics can be used to plan the most efficient path for a robot to follow in order to reach a certain point.

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