# Understanding Geodesic Parametrization on a Sphere

• I
• Jufa
In summary: In your case:##L = \sqrt{\dot{\theta}^2+sin^2(\theta)\dot{\phi}^2}##so:##\dfrac{d}{d\lambda} (\dot{\theta}) = -\dfrac{1}{2} \dfrac{\partial}{\partial \theta} (\dot{\theta}^2 + \sin^2(\theta) \dot{\phi}^2)##and:##\dfrac{d}{d\lambda} (sin^2(\theta) \dot{\phi}) = -\dfrac{1}{2} \dfrac{\partial}{\partial \phi} (\dot{\theta}^2 + \
Jufa
Let us consider a sphere of a unit radius . Therefore, by choosing the canonical spherical coordinates ##\theta## and ##\phi## we have, for the differential length element:

$$dl = \sqrt{\dot{\theta}^2+sin^2(\theta)\dot{\phi}^2}$$

In order to find the geodesic we need to extremize the following:

$$\int_{\lambda_0}^{\lambda_f} {\sqrt{\dot{\theta}^2+sin^2(\theta)\dot{\phi}^2}} d\lambda$$

We can do it so by imposing the Euler-Lagrange equations for the lagrangian ## L = \sqrt{\dot{\theta}^2+sin^2(\theta)\dot{\phi}^2} ## or equivalently, for the lagrangian ## L' = L^2 ##. These equations for L' look like:

$$\ddot{\theta} +sin(\theta)cos(\theta)\dot{\phi}^2 = 0$$

$$\ddot{\phi}+ cot(\theta) \dot{\phi}\dot{\theta}=0$$

I am pretty sure that I am right until here.
What does not make sense to me is the following:
Suppose you choose a curve in which ## \theta = \pi/2## i.e. both its first and second derivative vanish.
Then we get the following condition:

$$\ddot{\phi} = 0$$

But why does that happen? Once we have fixed the angle ## \theta## we have also fixed the curve (geodesic) and the only condition on ##\phi(\lambda)## should be that it is continuous and injective (i.e., it does not make ##\phi## go back and forth).
For example, the parametrization ##\phi(\lambda) = \frac{\lambda^2}{2\pi}## (which does not have a null second derivative) from ##\lambda_0 = 0## to ##\lambda_f =2 \pi## should be as valid as the parametrization ##\phi(\lambda) = \lambda## from ##\lambda_0 = 0## to ## \lambda_f = 2\pi ##.Thanks in advance.

Last edited:
To extremise with Lagrangian ##L## is not equivalent to extremising with ##L^2##. If you use ##L^2## it is the same as using ##L## with the additional requirement that ##L## is constant and doing so will therefore give you an affinely parametrised geodesic - ie, a geodesic with a constant length tangent. This coincides with the geodesic concept introduced by a connection.

Jufa
Orodruin said:
To extremise with Lagrangian ##L## is not equivalent to extremising with ##L^2##. If you use ##L^2## it is the same as using ##L## with the additional requirement that ##L## is constant and doing so will therefore give you an affinely parametrised geodesic - ie, a geodesic with a constant length tangent. This coincides with the geodesic concept introduced by a connection.

It's worth going through the exercise. In general, if ##L## is a "lagrangian" of the form:

##L = \sqrt{g_{ij} \dfrac{dx^i}{d\lambda} \dfrac{dx^j}{d\lambda}}##

then extremizing the action gives:

##\dfrac{d}{d\lambda} (g_{i j} \dfrac{dx^j}{d\lambda}) - \dfrac{1}{2} (\dfrac{\partial}{\partial x^i} g_{kj}) \dfrac{dx^k}{d\lambda} \dfrac{dx^j}{d\lambda} = g_{ij} \dfrac{dx^j}{d\lambda} \dfrac{\frac{dL}{d\lambda}}{L}##

The additional requirement that ##L## is unchanging as a function of ##\lambda## gives the usual form of the geodesic equation.

## What is geodesic parametrization?

Geodesic parametrization is a mathematical method used to describe and analyze the curvature and shape of surfaces in geometry. It involves parametrizing a surface using a set of parameters, such as coordinates, to represent points on the surface.

## What is the purpose of geodesic parametrization?

The purpose of geodesic parametrization is to simplify the study of surfaces by representing them in a way that is easier to analyze and manipulate. It allows for the calculation of important geometric properties, such as curvature and distance, which can be used in various applications in mathematics, physics, and engineering.

## How is geodesic parametrization different from other methods of parametrization?

Geodesic parametrization differs from other methods of parametrization, such as Cartesian coordinates or polar coordinates, in that it takes into account the intrinsic geometry of the surface. This means that the parameters used in geodesic parametrization are related to the surface itself, rather than an external coordinate system.

## What are the benefits of using geodesic parametrization?

There are several benefits to using geodesic parametrization. It allows for a more accurate representation of the surface, as it takes into account its intrinsic geometry. It also simplifies calculations and analysis of the surface, making it easier to study and understand. Additionally, geodesic parametrization can be used in various applications, such as computer graphics and image processing.

## Are there any limitations to geodesic parametrization?

While geodesic parametrization is a useful method for studying surfaces, it does have some limitations. It may not be suitable for highly complex surfaces or those with irregularities, as it relies on smooth and continuous functions. Additionally, the choice of parameters can affect the accuracy and usefulness of geodesic parametrization.

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