The following content has been provided by the University of Erlangen-Nürnberg.

So I'm going to begin by recalling this theory that I'm going to use.

I did the rigid body last time, now I'm going to the fluids.

And I'm going to show you various surprises.

One of them you probably, well at least in principle you should know about it,

the other one you won't, but I'll explain it.

So we are talking about this semi-direct product reduction theorem with co-cycles.

So we have a group, we have a representation space, we have a co-cycle,

we have a DFI in action, etc.

And here is the theorem.

On one single slide, the four things which are equivalent,

if you freeze the parameter a0, then one and two are equivalent,

which is classical mechanics.

It is Hamilton's variational principles for variations

that are vanishing at the endpoints.

And the equivalence with the other Lagrange equations.

Then comes the generalization of the Poincare theorem.

You have a constrained variational principle.

This time please notice that there are two variables,

xity and at, and in the previous one there was no at.

This is coming extra.

And it holds on g cross v star.

And the variational principle is constrained.

The variation of delta xi are exactly as in Poincare's theorem.

They are of the type d eta dt minus xi eta,

where eta is any curve that vanishes at the endpoints.

And there is also a constraint on the variations on a.

So these are delta a is equal to minus eta dc over eta.

And then the equations of motion are, these are the real equations of motion,

if you wish, d dt delta l delta xi is equal to minus eta xi.

Then there is this diamond operation.

I remind you in a second what it is,

minus the transposition of this dct del l del a.

So this diamond operator is nothing else but the momentum map.

Now you have listened to various things, so I can speak like this,

of the lift of the action on v to its cotangent bundle.

It's literally this.

But of course, if you don't know simpleclic geometry, I gave a formula.

It's very, very easy.

OK.

So this is the theorem.

And what we did last time, we did the heavy top.

And we discussed this.

So now let's do fixed boundary barotropic fluids.

This is one step beyond Arnold.

Arnold did it incompressible.

I'm going to do it compressible.

And I do it barotropic.

I mean, if you are not used to these things, they all have names.

Barotropic simply means that the pressure depends on the density.