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So as I promised, I'm going to start today on Poisson manifolds.

So my plan today is I'm going to do, I'm going to recall to you the major theorems in finite dimensional Poisson geometry, just the simple ones.

Then I will go to the Banach space case and I sort of shifted my presentation because some of the Banach space situations are quite complicated later on.

And so I'm going to do something. And you know, I'm always open to suggestions, I can always come back and I can show you.

But some of it is probably too much, there's too much functional analysis at a certain point.

Okay, so here is the definition of a Poisson manifold.

It's a manifold on which the ring of functions, of smooth functions on the manifold has also a Lie algebra structure, which I denote by a bracket.

And the two structures have to talk to each other, otherwise it wouldn't be interesting, right?

The ring structure and the Lie algebra structure, so they talk to each other, otherwise the Leibniz relation, right?

It's a derivation on each factor. And this is what's written there as the fundamental relationship between them.

So the first example that you know are finite dimensional symplectic manifolds because they have Poisson brackets.

However, as I mentioned to you before, when I'm dealing with Banach symplectic manifolds, this is not the case.

Because I cannot guarantee the existence of a Hamiltonian vector field for every function if it's weak.

So this is already a problem. And I'm having to face this somehow. I mean, there's going to come back to bite me, right?

Later when I go into the Banach category. And you'll see what can be done. It's not very satisfactory.

So as opposed to symplectic manifolds, there is a center of the Lie algebra here.

And the elements in the center are called Casimir functions. So these are functions that Poisson commutes with every other function, right?

In the symplectic category, in finite dimensional manifolds, this is impossible, right?

Such a thing doesn't exist. Only the constants commute with everything.

Okay, so now the question is how do you define the Hamiltonian vector field? I clearly cannot define it the way I did it for the symplectic manifolds.

Because there I used in an absolutely crucial way the fact that the map omega-flat was an isomorphism at every single point between Tm and T star m.

Here I cannot do this. I have nothing like this. You'll see in a moment what I have. I have something else. But it's not good.

So the way you handle this is you take advantage of the fact that derivations and vector fields are all in the same thing.

But you do have derivations because if you freeze the second component, for example, the second variable in the Lie algebra,

then the Leibniz identity tells you that you have a derivation on the ring of functions.

And by a standard theorem in manifold theory, all derivations determine a vector field.

So I'm going to say that the vector field that corresponds to the derivation, take the bracket with H in the second component, in the second slot.

This is the Hamiltonian vector field. So this is what I wrote here.

That means that XH acting on F, which is the same thing as the Lie derivative of XH to F, or if you wish, DF acting on XH.

This is all the same thing. It's just different notation. It's FH.

And that defines for me a vector field which I call the Hamiltonian vector field for the function H.

Notice that my conventions are in such a way that Hamilton's equations have exactly the same signs, the ones that I used before, F dot equal to FH.

OK. So example one was symplectic manifolds.

But there is a second example which is equally important, and these are called the Li-Poisson brackets.

The history behind it is very interesting.

Li wrote this explicitly. It's not implicit. It's not between the lines. It's as explicit as you can get.

He wrote the formula in the middle of a page and gave it a number. There is no discussion.

Of course, nobody really understood it and it was forgotten.

Then it was rediscovered by many, many people.

Kirillov-Costand, Arnold, Sudio, etc. More or less at the same time.

And it was not known. In fact, I think it was Deustermatt who pointed out that this is in Li.

It's in his second volume of Continuierliche Transformationsgruppen.

I can give you the page if you want to.

And the formula that he writes without the plus and minuses is the formula that you see there on the blackboard.

The only difference between my formula and his is that he used partial derivatives and he used Cijk as opposed to just writing the bracket down.

But that's it.

Now, I put the plus and the minus in the front and you'll see, not today, later, why.

Because in mechanics, these signs mean something.

They are not innocent.