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Stream: deprecated: thermodynamics

Topic: Lagrangian subspaces

Owen Lynch (Feb 21 2022 at 22:11):

If Dirac structures are analogous to symplectic structures, then what type of subspace of a Dirac manifold is analogous to a Lagrangian subspace?

John Baez (Feb 22 2022 at 22:04):

Maybe you mean "what type of submanifold of a Dirac manifold is analogous to a Lagrangian submanifold of a presymplectic manifold?"

John Baez (Feb 22 2022 at 22:06):

Anyway, it sounds a bit tricky. A submanifold $L$ of a presymplectic manifold $X$ is isotropic iff for all points $x \in L$ and tangent vectors $v,w \in T_x L$ we have $\omega(v,w) = 0$. Then $L$ is Lagrangian iff it's a maximal isotropic submanifold.

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

In other words $L$ is isotropic iff the inclusion $i: L \to X$ has $i^\ast \omega = 0$. Since 2-forms pull back, this is a natural condition.

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

But a Poisson manifold is defined using a bivector field $\Pi$, not a 2-form.

John Baez (Feb 22 2022 at 22:10):

And bivector fields push forward. So for a Poisson manifold $X$ it feels more natural to talk about quotient maps $p: X \to M$ with $p_\ast \Pi = 0$

John Baez (Feb 22 2022 at 22:10):

I don't know if people have studied these, but they should have!

John Baez (Feb 22 2022 at 22:11):

Dirac manifolds are sort of a hybrid of these ideas, so it's even more confusing....

Markus Lohmayer (Feb 22 2022 at 22:13):

Maybe I can help a bit? What is the confusing part?

John Baez (Feb 22 2022 at 22:18):

Owen asked this question, more or less:

Lagrangian submanifolds are to symplectic manifolds as what are to Dirac manifolds?

John Baez (Feb 22 2022 at 22:20):

So any answer to that would be helpful. I was just trying to explain why Lagrangian submanifolds are a very natural concept for presymplectic manifolds, but not so natural for Poisson manifolds. Dirac manifolds generalize both presymplectic manifolds and Poisson manifolds.

Markus Lohmayer (Feb 22 2022 at 22:22):

A Dirac structure is a Lagrangian submanifold, but it is not (co)isotripic wrt to some symplectic form, but rather with respect to some symmetric indefinite form. I guess you are aware of this. Therefore I am not so sure about the above question.

John Baez (Feb 22 2022 at 22:26):

Owen and I are thinking of Dirac manifolds (= manifolds with Dirac structure) as a simultaneous generalization of presymplectic and Poisson manifolds. So, like presymplectic and Poisson manifolds, you can use them to describe "phase spaces" in physics. From this viewpoint it makes sense to take as many concepts from symplectic geometry as one can, and generalize them to Dirac manifolds. One of these is "Lagrangian submanifold".

The fact that a Dirac structure has something (co)isotropic about its definition is not the main point here, I think.

John Baez (Feb 22 2022 at 22:27):

It's possible that I'm quite confused.

Markus Lohmayer (Feb 22 2022 at 22:27):

Ah okay I see.

Markus Lohmayer (Feb 22 2022 at 22:29):

So, do you know what a Lagrangian submanifold in symplectic geometry means in terms of a physical example?

Markus Lohmayer (Feb 22 2022 at 22:30):

I don't. But I know why Dirac structures show up, and what (co)isotropy (wrt the indefinite form) means physically for them

John Baez (Feb 22 2022 at 22:38):

Markus Lohmayer said:

So, do you know what a Lagrangian submanifold in symplectic geometry means in terms of a physical example?

Yes. Most importantly for us perhaps, given symplectic manifolds $X$ and $Y$ thought of as phase spaces, a Lagrangian submanifold of $\overline{X} \times Y$ describes a physically realistic relation between states of $X$ and states of $Y$. Here $\overline{X}$ is $X$ with its symplectic structure multiplied by $-1$.

For example $X$ could be the space of position-momentum pairs of a particle at time 0, and $Y$ could be the space of position-momentum pairs of a particle at time 1. Then time evolution via a Hamiltonian gives a relation between points in $X$ and points in $Y$, and this relation is a Lagrangian submanifold $L \subseteq \overline{X} \times Y$.

John Baez (Feb 22 2022 at 22:40):

Another example is studied a lot in

• Baez and Fong, A compositional framework for passive linear networks.

where we show a large class of electrical circuits give rise to a Lagrangian relation between voltage-current pairs on the input wires and voltage-current pairs on the output wires.

John Baez (Feb 22 2022 at 22:41):

(A Lagrangian relation between symplectic manifolds $X$ and $Y$ is a Lagrangian submanifold of $\overline{X} \times Y$.)

Markus Lohmayer (Feb 22 2022 at 22:45):

Thanks John. I suppose I now understand your motivation. I recommend you the following article: https://arxiv.org/pdf/1804.04951.pdf It explains the interconnection of port-Hamiltonian systems (i.e. the composition of power-preserving relations = Dirac structures) in terms of Dirac categories. You could like it ;)

Markus Lohmayer (Feb 23 2022 at 07:32):

I got sleepy yesterday ;) So, I think the answer to "Lagrangian submanifolds are to symplectic manifolds as what are to Dirac manifolds?" is that Dirac manifolds are to Dirac manifolds what Lagrangian submanifolds are to symplectic manifolds, if that makes any sense :)

Markus Lohmayer (Feb 25 2022 at 16:41):

I also want to mention that I like to think about Dirac structures as structures on vector bundles rather than on the base manifold. E.g. a Dirac structure on $T \mathcal{X} \oplus \mathcal{F}$( where $T \mathcal{X}$ is the tangent space of some manifold of system states $\mathcal{X}$ and $\mathcal{F} \rightarrow \mathcal{X}$ is some (usually trivial) vector bundle of port variables) is a vector subbundle of $T \mathcal{X} \oplus \mathcal{F} \oplus T^* \mathcal{X} \oplus \mathcal{F}^*$ on which the canonical bilinear symmetric indefinite form vanishes (and hence I suppose it can be called a Lagrangian submanifold$ (wrt that form)).

John Baez (Feb 25 2022 at 16:49):

Thanks! Owen is starting to read your papers; maybe you two could talk about this stuff sometime.