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Thank you very much. I'm glad to be here. I want to thank the organizers for the invitation.

For me this is quite exciting because this is indeed something very interesting that is happening.

This mix of various subjects.

And I would like to talk to you, it's infinite dimensional geometry, but it's symmetric infinite dimensional conservative dynamics.

So this is going to have a lot of geometry in it. It's going to be Lagrangian system, Hamiltonian system, symmetry reduction, etc.

I don't know how far I'm going to get, but I'll try.

So, forget that. I don't want to scare you with what I wanted to do. So we're just going to go along.

So in conservative mechanics you have two formulations. One is Hamiltonian, the other one is Lagrangian.

The Hamiltonian approach links immediately to geometry, to various kinds of geometry.

It's symplectic, it's Poisson, it's Dirac, for example.

And the Lagrangian approach links to analysis, to variational principles and to hard analysis.

And it's very interesting to play with both of them. In addition to that, both the Hamiltonian and the Lagrangian approach link to numerics.

I don't know if I'm ever going to get there, but I probably will try to at least make you aware that this exists.

Now, under certain hypotheses, which are reasonable in some fields and other fields are not reasonable,

you have what is called hyper-regularity and Lagrangian and Hamiltonian approaches are the same.

And at the beginning I'm going to go like that, just to get my bearings.

And the interesting thing, of course, and this is what I want to get, is if you have certain symmetries, and then you want to eliminate variables.

And when you eliminate variables, this is a process called reduction. And the reduction has various aspects.

Again, I don't know how far I'm going to get there. There is certain things that are known.

In finite dimensions a lot is known, but not everything.

In infinite dimensions very little is known, in a rigorous way.

So let me start with Hamiltonian mechanics. I'm going to go through some very standard material.

And I will emphasize infinite dimensional problems.

And what can you do about them, how you deal with them, what can you possibly imagine that you are going to do with various problems.

So the phase space of a mechanical system is usually, I mean modern mechanics, is a symplectic manifold.

So a symplectic manifold is a manifold, which I'm going to call M, sometimes I'm going to call it P, from phase space.

And omega is a two-form, and it's closed. That means that d omega is equal to zero.

And already here we have problems with the second statement. What do I mean by non-degenerate?

Well, in finite dimensions what you do is, because you have a bilinear form, you have a map, a linear map from the vector space to the dual vector space.

So what do you do? To every vector V that sits at the point M, you compute the symplectic form omega M and plug it into the first slot.

This is my convention, this is my sign convention.

And then you are going to get something in the dual, so in the cotangent bundle.

Now normally in finite dimensions you simply say, well this is an isomorphism.

But of course you could say injective, you could say surjective, it would be absolutely the same thing.

Now in infinite dimensions it's not so simple, and I'm going to come back to this.

You could require that it be injective, you could require that it be an isomorphism of Banach spaces.

It's not the same thing. We are going to come to various problems that this is posing.

Anyway, just in terms of terminology, if omega is allowed to be degenerate, I'm going to call it a presymplectic manifold.

And the Hamiltonian dynamical system is a triple, which is the phase space M, the symplectic form, and a function which I call H, the Hamiltonian.

Where M omega is a symplectic manifold, H is usually a smooth function.

And if it's non-degenerate, then the equation interior product with somebody, which is a vector field, on omega equal to dH, always has a solution.

It exists and it is unique, if it is non-degenerate in this sense.

And this vector field that depends on H, it's going to be called the Hamiltonian vector field.

This is just standard terminology and notation.

As I said, in infinite dimensions you have problems, because you could require it to be injective, in which case you would say weakly non-degenerate,

or if you require it to be an isomorphism, then it is strongly non-degenerate.

Okay, so let's do some examples. So if I just take a vector space, I don't care finite, infinite, dimensional, and I form the product of V with this dual,

I automatically have a symplectic form.

Of course, I already have a typo, there is no capital omega, there should be a little omega there.