Eva Miranda

speaker

1,609 appearances 1 recordings 1 series first heard Jan 2025 last heard Jan 2025

Eva Miranda’s voice in public audio — every appearance, attributed to the second.

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What's magic about this example here

Is that there is a this kind of duality of position momenta, and now it's the height function with respect to these rotations on the circle.

And this is an integrable system in dimension two, because in dimension two, which is the sphere.

Remember the definition of integral system is I need as much

First integrals as half of the dimension.

of the manifold.

This is one in this case.

So in dimension two, one function defines an integrable system.

Now, the surprise is that this is always the case.

So if I have an integrable system on a bigger dimensional manifold, I I have a kind of product situation of these spheres.

Indeed, I have these sparred coordinates, disposition momenta, are called action angle coordinates.

Because I have the action

which is this momenta, and the angle, which is these rotations.

And indeed, if I take uh

Look here, if I take the integrable system, the level sets are a circle.

in higher dimensions there will be products of circles.

This is what we call a torus.

And the magic thing is that the symplectic form can be written in this simple way.

In a neighborhood.

of the sir of this uh of the of what we call the fiber of the of this moment.

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