# Geodesics on a sphere and the Christoffel symbols

• I
• acegikmoqsuwy
In summary, geodesics on a sphere are the shortest paths between two points on its surface, calculated using the arc length formula and Christoffel symbols, which describe the curvature of the surface and its effect on the shortest paths. These symbols are important in studying geodesics on curved surfaces and can be applied to any curved surface in differential geometry.
acegikmoqsuwy
Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula $$\dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0$$ for the geodesic equation, with the metric $$ds^2=d\theta^2+\sin^2\theta d\phi^2$$. After solving for the Christoffel symbols and plugging in, I got the system of differential equations $$\dfrac{d^2\theta}{dt^2}=\sin\theta\cos\theta \left(\dfrac{d\phi}{dt}\right)^2$$ and $$\dfrac{d^2\phi}{dt^2}=-2\cot\theta\left(\dfrac{d\phi}{dt}\dfrac{d\theta}{dt}\right)$$, but when I plug in the formula for a great circle, $$\tan \theta\cos\phi=1$$ by making the parametrization $$t=\cot\theta=\cos\phi$$, it does not satisfy the differential equations. Can anyone explain to me where I've gone wrong?

You cannot pick just any parametrisation to satisfy the geodesic equations. You need a parametrisation which fixes the length of the tangent vector.

How can the equation of a great circle be ##\tan \theta\cos\phi=1##? Surely it should contain some constants for the two points on it?

Orodruin said:
You need a parametrisation which fixes the length of the tangent vector.
Trying to understand this. Is it referring to the parameterisation of the great circle equation? Is it something to do with parallel transport?

George Keeling said:
Trying to understand this. Is it referring to the parameterisation of the great circle equation? Is it something to do with parallel transport?
Yes, an affinely parametrised geodesic has a tangent that is parallel along it.

acegikmoqsuwy said:
Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula $$\dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0$$ for the geodesic equation, with the metric $$ds^2=d\theta^2+\sin^2\theta d\phi^2$$. After solving for the Christoffel symbols and plugging in, I got the system of differential equations $$\dfrac{d^2\theta}{dt^2}=\sin\theta\cos\theta \left(\dfrac{d\phi}{dt}\right)^2$$ and $$\dfrac{d^2\phi}{dt^2}=-2\cot\theta\left(\dfrac{d\phi}{dt}\dfrac{d\theta}{dt}\right)$$, but when I plug in the formula for a great circle, $$\tan \theta\cos\phi=1$$ by making the parametrization $$t=\cot\theta=\cos\phi$$, it does not satisfy the differential equations. Can anyone explain to me where I've gone wrong?

What you have to do is selecting a parametrization with respect to the arc length (s).

What you have to do is selecting a parametrization with respect to the arc length (s).
As already stated by me four years ago when this thread was new. The OP has not been seen for 6 months. Please avoid thread necromancy when the question has been answered.

## 1. What is a geodesic on a sphere?

A geodesic on a sphere is the shortest path between two points on the surface of a sphere. It is the equivalent of a straight line on a flat surface.

## 2. How are geodesics on a sphere calculated?

Geodesics on a sphere are calculated using the arc length formula, which takes into account the radius of the sphere and the angles between points on the surface.

## 3. What are Christoffel symbols?

Christoffel symbols are mathematical objects used to describe the curvature of a surface or manifold. In the context of geodesics on a sphere, they represent the connection between the tangent vectors at different points on the surface.

## 4. Why are Christoffel symbols important in the study of geodesics on a sphere?

Christoffel symbols are important because they help us understand the behavior of geodesics on a curved surface such as a sphere. They provide information about the curvature of the surface and how it affects the shortest paths between points.

## 5. Can Christoffel symbols be used to calculate geodesics on other surfaces?

Yes, Christoffel symbols can be used to calculate geodesics on any curved surface, not just a sphere. They are a fundamental tool in differential geometry, which is the study of curved spaces.

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