# Connection between Foucault pendulum and parallel transport

• I
• Joker93
In summary, the Foucault pendulum uses parallel transport to show that the normal to the plane of oscillation is not always the same vector.
Joker93
Hello!

I try to think about the Foucault pendulum with the concept of parallel transport(if we think of Earth as being a perfect sphere) but I can't quite figure out what the vector that gets parallel transported represents(for example, is it the normal to the plane of oscillation vector?).

In particular, I can't exaplain the following animation https://en.wikipedia.org/wiki/File:Foucault_pendulum_plane_of_swing_semi3D.gif
which is found in this wikipedia article
https://en.wikipedia.org/wiki/Foucault_pendulum
using the concept of parallel transport.

An object in free-fall would be a geodesic, which is related to parallel transport because a geodesic is just a curve that parallel transports it's own tangent vector.

You might look at https://arxiv.org/pdf/0805.1136.pdf. I havaen't really read it yet. My intuition is that we need Fermi-Walker trasnport, and not parallel transport, but I'm not sure if that's what the reference is saying.

Joker93 said:
what the vector that gets parallel transported represents(for example, is it the normal to the plane of oscillation vector?).

In particular, I can't exaplain the following animation https://en.wikipedia.org/wiki/File:Foucault_pendulum_plane_of_swing_semi3D.gif
In the animation its obviously not the normal to the plane of oscillation, but a vector parallel to the oscillation plane and surface. But it doesn't really matter which vector you show, as they have a fixed 90° offset.

A.T. said:
In the animation its obviously not the normal to the plane of oscillation, but a vector parallel to the oscillation plane and surface. But it doesn't really matter which vector you show, as they have a fixed 90° offset.
That's what I thought at first but why does it get parallel transported in this way? At some points that vector is tangent to its trajectory, so wouldn't its parallel transport look something like the attached image?(from Do Carmo)

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pervect said:
An object in free-fall would be a geodesic, which is related to parallel transport because a geodesic is just a curve that parallel transports it's own tangent vector.

You might look at https://arxiv.org/pdf/0805.1136.pdf. I havaen't really read it yet. My intuition is that we need Fermi-Walker trasnport, and not parallel transport, but I'm not sure if that's what the reference is saying.
But is does not follow a geodesic since a geodesic on a sphere is a great circle.
Also, I do not know about Fermi-Walker transport. I will check out the pdf file though. Thanks!

## 1. What is a Foucault pendulum?

A Foucault pendulum is a device that demonstrates the rotation of the Earth by using a swinging pendulum. The pendulum's plane of oscillation appears to rotate clockwise or counterclockwise over time due to the Earth's rotation underneath it.

## 2. What is parallel transport?

Parallel transport is a mathematical concept in differential geometry that describes the transportation of a vector along a curve without changing its direction. It is often used to study curved surfaces and understand the effects of curvature on objects moving within that space.

## 3. What is the connection between Foucault pendulum and parallel transport?

The connection between the two lies in the demonstration of the Coriolis effect through the pendulum's oscillations. The pendulum's motion can be seen as a parallel transport of the pendulum's plane of oscillation, which is affected by the Earth's rotation, demonstrating the curvature of the Earth's surface.

## 4. How does the Coriolis effect relate to parallel transport in a Foucault pendulum?

The Coriolis effect, which is the apparent deflection of objects moving in a straight path on a rotating surface, is demonstrated through the parallel transport of the pendulum's motion. As the pendulum swings, the Earth's rotation causes the plane of oscillation to rotate, showcasing the effects of curvature on the object's path.

## 5. What other scientific principles are involved in understanding the connection between Foucault pendulum and parallel transport?

Other principles involved include the laws of motion, specifically inertia and the conservation of angular momentum. These laws explain the pendulum's tendency to maintain its plane of oscillation and the rotation of the Earth's surface, respectively.

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