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

So good morning and welcome back. Welcome to the final lecture of this course.

Today we'll turn to classical mechanics again.

And we already had on the problem sheet some classical mechanics.

I'll very quickly review this and then we'll come to the topic of kinematical and dynamical symmetries

and their distinction from the point of view of the co-tangent bundle.

And that should give you some insight into the Runge-Lenz vector

and what the difference is to the kinematical symmetry generators.

You have, say, on flat space like rotational operators and so on

generated by the angular momentum and other observables.

And so we'll start with 6.3 kinematical and dynamical symmetries.

So you see this is the third lecture that's totally concerned with physics

so that raises the proportion of physics in this course to an unseen 11%.

I wish you to notice this.

Okay. So let's recall from one of the previous problem sheets

the set up of classical mechanics in a symplectic manifold picture.

So we have the definition that a 2n-dimensional manifold, smooth manifold M

together with a two-form omega with an omega

which is a two-form on this 2n-dimensional smooth manifold

for which the following holds.

A, this two-form is closed so that means that the exterior derivative of the two-form is zero.

And the second condition is that the two-form be non-degenerate.

This is closed. And non-degeneracy here means that if you have omega, x equals zero for all y,

well, what is y? Y is a vector field obviously on this manifold.

Then you can already conclude that x is zero.

Okay. And that's a non-degeneracy condition.

And such a pair M with omega satisfying these conditions is called a symplectic manifold.

I say symplectic manifold.

And on a symplectic manifold we have observables.

And those are simply the smooth functions.

Observables are the smooth functions on this 2n-dimensional symplectic manifold.

Now, this makes contact with what we know from classical mechanics via the theorem

that there always exists darbu coordinates in an extended region.

So theorem, for any point of this symplectic manifold M, there exists an open set U

containing P, so it's not the empty set, containing the point in question.

Such that there also exists coordinates.

So, aha. Yeah.

Let's call them shy. I know you hate it, but let's call them shy.

So they are coordinates from U into R 2n.

Why? Because it's 2n-dimensional.

Such that if we consider the omega, this symplectic form as it is called,

in the thus induced coordinate system, so this is the shy.

Let's write capital A, shy, capital B, so I label from A to B, AB from 1 to 2n.

I label the coordinate induced tangent vectors such that this guy,

and that's the statement of the theorem, there exist coordinates on such an extended region,

such that the omega looks like this.

And all the remaining entries are zero, and I indicated here two blocks,

where each block has the size n, so in total this is a 2n by 2n matrix.

And note, this is very special because it's not just a normal form for an antisymmetric 2n by 2n matrix at one point.

It claims that around a point there's always an open region, maybe a small open region,