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Stream: deprecated: thermodynamics

Topic: symplectic geometry and Dirac structures

Owen Lynch (Feb 08 2022 at 04:06):

If $X$ is a symplectic manifold, there is a natural dirac structure on $TX \times_X T^\ast X$, and I will demonstrate it here.

Owen Lynch (Feb 08 2022 at 04:09):

The Dirac structure is the graph of the map $T^\ast X \to T X$ that sends a cotangent vector $\phi \in T^\ast_x X$ to the tangent vector $v$ such that $\omega(-,v) = \phi$.

Owen Lynch (Feb 08 2022 at 04:10):

Or really, I suppose it is the other way around.

Owen Lynch (Feb 08 2022 at 04:11):

It is the graph of the map $T X \to T^\ast X$ that sends the tangent vector $v \in T_x X$ to the cotangent vector $\omega(-,v)$.

Owen Lynch (Feb 08 2022 at 04:12):

This is a Dirac structure because $\langle v, \omega(-,v) \rangle = \omega(v,v) = 0$.

Owen Lynch (Feb 08 2022 at 04:13):

And it is of dimension $n$, where $\dim(T_x X \times T_x^\ast X) = 2n$

Owen Lynch (Feb 08 2022 at 04:16):

We can think of this Dirac structure as giving a port-Hamiltonian system with no ports, as it is a relation between $TX \times_X T^\ast X$ and $\mathbb{R}^0 \times (\mathbb{R}^0)^\ast$.

Owen Lynch (Feb 08 2022 at 04:19):

The rules of time-evolution for a port-Hamiltonian system say that $(x'(t), dH_{x(t)})$ must be an element of the Dirac structure for all $t$, where $x : I \to X$ is a differentiable curve, $H : X \to \mathbb{R}$ is the Hamiltonian, and $dH_{x(t)} \in T^\ast_{x(t)} X$ is the differential of $H$ at point $x(t)$.

Owen Lynch (Feb 08 2022 at 04:20):

We can see that this is precisely requiring $\omega(-,x'(t)) = dH_{x(t)}$, which is Hamilton's equation.

Owen Lynch (Feb 08 2022 at 04:22):

Thus, we should not think of the bilinear symmetric 2-form of signature $(n,n)$ as analogous to the symplectic form

Owen Lynch (Feb 08 2022 at 04:22):

The Dirac structure itself is analogous to the symplectic form.

Owen Lynch (Feb 08 2022 at 04:23):

Each symplectic form produces a Dirac structure, though not all Dirac structures may arise from symplectic forms

Owen Lynch (Feb 08 2022 at 04:26):

In fact, there are Dirac structures which cannot arise from symplectic forms.

Owen Lynch (Feb 08 2022 at 04:27):

Because there are Dirac structures even when $X$ is odd-dimensional.

John Baez (Feb 08 2022 at 04:37):

Okay, great. So it sounds like from this point of view we should think of Dirac operators as generalized symplectic structures. That sounds familiar to me.

John Baez (Feb 08 2022 at 04:38):

Does a Poisson structure on $X$ also give a Dirac structure on $TX \times_X T^\ast X$?

John Baez (Feb 08 2022 at 04:40):

A Poisson structure on $X$ is a bivector field, i.e. a section $\Pi$ of the second exterior power $\Lambda^2 TX$, such that if we define

$\{f,g\} = \Pi(df \wedge dg)$

for smooth real-valued functions $f,g$ on $X$, then $\{f,g\}$ obeys all the usual identities that Poisson brackets should.

John Baez (Feb 08 2022 at 04:42):

Poisson structures are strictly more general than symplectic structures: a symplectic structure $\omega$ on $X$ sets up a vector bundle isomorphism $TX \cong T^\ast X$ and we can use this to turn $\omega \in \Lambda^2 T^\ast X$ into a Poisson structure $\Pi \in \Lambda^2 TX$.

Unlike symplectic structures, Poisson structures exist on odd-dimensional manifolds.

Owen Lynch (Feb 08 2022 at 21:48):

John Baez said:

Okay, great. So it sounds like from this point of view we should think of Dirac operators as generalized symplectic structures. That sounds familiar to me.

You mean Dirac structures?

Owen Lynch (Feb 08 2022 at 21:50):

I'll think about the Poisson structure question

Owen Lynch (Feb 08 2022 at 21:52):

So, the Poisson structure gives a map $T^\ast X \to T^{\ast\ast} X$, sending $\phi$ to $\Pi(\phi \wedge -)$.

Owen Lynch (Feb 08 2022 at 21:53):

Thus, it defines a map $T^\ast X \to T X$

Owen Lynch (Feb 08 2022 at 21:54):

And the graph of this map is indeed a Dirac structure, as $\Pi(\phi \wedge \phi) = 0$.

John Baez (Feb 08 2022 at 21:59):

Owen Lynch said:

John Baez said:

Okay, great. So it sounds like from this point of view we should think of Dirac operators as generalized symplectic structures. That sounds familiar to me.

You mean Dirac structures?

Yes. Right now I'm writing a paper on the hydrogen atom and the Dirac operator. :smirk:

John Baez (Feb 08 2022 at 22:02):

Owen Lynch said:

And the graph of this map is indeed a Dirac structure, as $\Pi(\phi \wedge \phi) = 0$.

Okay, that was easy! Nice!

Actually I think there must be some 'integrability condition' in the definition of Dirac structure - some differential equation that has to hold. This is certainly true for the definition of symplectic structure ($d \omega = 0$) and Poisson structure ($[\Pi, \Pi] = 0$, where the bracket is the Schouten bracket, the generalization of the Lie bracket on vector fields to multivector fields). So, there should be some exercise where you prove that $[\Pi, \Pi] = 0$ implies the integrability condition for the Dirac structure (whatever that is).

John Baez (Feb 08 2022 at 22:02):

Someone must have done this already.

Owen Lynch (Feb 09 2022 at 00:33):

Yeah, that sounds about right, I'll look into it

Owen Lynch (Feb 09 2022 at 00:33):

I should do both of these constructions out in more detail

John Baez (Feb 09 2022 at 02:18):

This paper says:

As we shall see shortly, in geometric mechanics, almost Dirac structures provide a simultaneous generalization of both two-forms (not necessarily closed, and possibly degenerate) as well as almost Poisson structures (that is brackets that need not satisfy the Jacobi identity). A Dirac structure, which corresponds in geometric mechanics to assuming the two-form is closed or to assuming Jacobi’s identity for the Poisson tensor, is one that satisfies...

[some differential equation]

John Baez (Feb 09 2022 at 03:06):

A possibly degenerate closed 2-form is called a presymplectic structure. You can't turn all presymplectic structures into Poisson structures, and you can't turn all Poisson structures into presymplectic structures. The intersection of the two are the symplectic structures. The union of the two is included in the Dirac structures.

John Baez (Feb 09 2022 at 03:07):

This makes me happier about Dirac structures, since I've messed with Poisson structures a lot and presymplectic structures a little too.