Eva Miranda

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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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Theories of Everything with Curt Jaimungal · Eva Miranda: The Mathematical Bridge Between Classical and Quantum · 26 Jan 2025 podcast

And the isomorphism is exactly Hamilton's equations.

It's exactly what we did.

I give you a one form, so I give you a section of the cotangent bundle, and you give me a vector field.

So you give me a section of the tangent bundle.

So this gives me this isomorphism.

And this is why position and momenta are sort of coupled in a

uh in a magic way.

It's a magic uh way.

uh that they are really coupled.

It's like a magic dance between this uh positional momenta between this tangent and cotangent.

And this is thanks to the fa to the fact that they are in a way dual to each other.

And and this is why

And this is why we are going we are this is why Darboux was able to prove this theorem.

Here we have

That in a neighborhood of a point, we can always pair position a moment here, position a moment, we call it we call them x and y's, but it's the same.

We can always pair coefficients in in such an easy way.

We have a two-form such that the coefficients are constants, ones, right?

So this tells us that in contrast to contract to Riemannian geometry, there are no local.

Invariance and in Darbugu coordinates, as we said, the flow of a Hamiltonian vector field is just given by Hamilton's equations.

So locally, every symplectic manifold is a cotangent bundle.

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