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So, let me review a few things that we did today.

So we talked about simplectic manifolds, both in finite and in infinite dimensions.

I showed you some pitfalls, what's happening,

and then we discussed typical examples

of canonical transformations.

and one of them was the cotangent lift, you have a map between two manifolds, you take the tangent map and then you dualize it.

And then it reverses the arrow, it goes from the cotangent bundle of the target manifold to the cotangent bundle of the start manifold.

And this is automatically symplectic because it does much more, it preserves the canonical one-form,

and conversely if you have a diffeomorphism that preserves the canonical one-forms, it must be the lift of somebody.

That's one thing. The other thing that we did is we discussed what happens if you shift the momentum.

This is this situation here with the one-form.

And then we have seen that what happens with the pullback.

And if dA is equal to zero, then it's canonical. Then what we have done, we discussed what happens with the two-form,

and we formed a magnetic symplectic form, which was Ωb,

and we discussed about what happens relative to it in terms of symplectic diffeomorphism.

Then we justify this, that all of this comes in fact from the Lorentz force law, and this is why it's called a magnetic term.

This is a standard terminology now. These are called magnetic cotangent bundles, even though this is the only example in which it's really a magnetic field,

but it was the first one, and even today it's the most important one.

Okay, then we have seen what happens if you ask the question F equal to ma, in what sense is it Hamiltonian?

And the result was that you had two answers that were possible.

One answer was you take the magnetic form and you use the Hamiltonian that is given by the kinetic energy of the particle.

The other answer was you can use the canonical form, but then you have to change the kinetic energy by a shift,

because P, Q and P or X and P are the canonically conjugate variables, but P is not MV in this case.

It has this expression. You have shifted it by E over C times A.

Okay, then we started discussing what is happening relative to the infinite dimensional system, so we started giving examples.

I had some questions, so I should say this, because I didn't write it.

We are going to come back to this, because I'm using it all the time, over and over.

The question was what was this delta over whatever.

So here is the definition. If you have two spaces in some kind of duality, for example, real,

so that means that the least you want for this to be weakly non-degenerate, what does this mean?

It means that if EF is equal to zero for all F, you must conclude that E is zero, and if EF is equal to zero for all E,

you must conclude that F is equal to zero. So this is called a weakly non-degenerate pairing.

In the example that's on the screen, it's simply functions and densities. These are naturally duals to each other.

So, for example, if E is some Banach space and F is its dual, this is completely natural. This is a natural pairing.

And then if you have a function between E and from E to F, then you can take its, I give you a function, H,

that goes, for example, from E to the reals, and now I'm taking its derivative, the differential derivative, DE, at a point, let's call it U.

So this is a linear map from E into the reals. So that's E star. So these are linear functionals on E.

But of course there are other linear functionals on E that you can imagine from here, namely these.

This is also a linear functional. So what I want, I want to represent DEU, DH, I want to represent it in terms of somebody.

So what does this mean? That if I apply it to an E, I want this to be something like this, E with DEU, DH.

So this is the DEU. I'm enforcing the Riesz representation theorem.

It's not, I can't guarantee you that this exists. There is no way, because this is only weakly non-degenerate. This is all I have.

I'm not saying that E star is isomorphic to F. If I could, it would be fantastic. But in realistic examples I don't have it.

And this is the del H del U. So in this case, it's exactly E is functions and F is the densities.

So this is what I meant by this delta H delta U. And you know, if you take functions to be E, smooth functions, you take F to be also smooth functions,

and you take the pairing, the L to pairing, then del function, del U is exactly the one that you know from your formulas in physics.

Exactly the same one. Okay, so the Hamiltonian vector field was given like this, and then we started doing the examples, and we did one example.

And the first example that we did was the wave equation. So I'm not going to repeat this. This is just an example.

So let's do another one, Schrodinger equation. So I'm taking H, the complex Hilbert space, for example, of complex valued functions from R3 to C,