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for discussion at geometry of physics I needed to be able to point to principle of extremal action, so I created a little entry.
Did you ever come to a conclusion on the Maslov-Litvinov idea of classical mechanics using the tropical rig, compared to the quantum use of $\mathbb{C}$? We had quite a discussion 6 years ago.
I never thought really much about this, sorry. Also in my mind this idea is an isolated island. I don’t see where it should connect to. (That is not to say that it doesn’t lead anywhere, but is part of the expalantion why I haven’t thought much about it.)
But there is something else in the context of classical equations of motion, which I keep banging my head against, but which I still cannot quite see:
In the context of extended prequantum field theory the action functinal is the prequantum 0-bundle which is only one end of a whole hierarchy of which the next step is the prequantum bundle itself. In this context it is striking that passing to the critical locus, hence implementing the “principle of extremal action” means precisely passing to the 0-plectic maximal isotropic subspace, the maximal subspace on which the curvature 1-form of the 0-bundle vanishes. A 0-bundle is just a function and its 1-form curvature is just the (variational) derivative.
In view of this, it is compelling to regard Lagrangian subspaces of phase space – which is the maximal locus where the prequantum 1-bundle has vanishing curvature – as “extended loci of solutions of the Euler-Lagrange equations in codimension 1”. And so on for the higher prequantum n-bundles.
It is clear that this is telling me something, but I still need to better listen what it is.
Also, what annoys me is that I have now a beautiful way to produce the critical locus (the ordinary one in codimension 0) in cohesive homotopy type theory. But I don’t know yet how to say “Lagrangian subspace” in cohesive homotopy type theory. Grr.
a beautiful way to produce the critical locus
Can that be shared here?
Oh, sure, I pointed to it before.
At differential cohesion it’s in the section Critical locus,
in the dcct-pdf it is in section 3.10.8
and right now I am working also on the relevant section Equations of Motion at geometry of physics.
So the simple way to do it is: the variational derivative of a function is just postcomposition with the map that modulates the Maurer-Cartan form in cohesion, and then to form the critical locus we form the homotopy fiber of that variational derivative after coreflecting into the petit $\infty$-topos over the domain (which we can do in differential cohesion). This forms the (formally) open subset on which the variational 1-form vanishes, which is the right answer.
But for the same reason this is also a problem when we try to go to higher codimension and think of Lagrangian submanifolds in phase space as being analogous: because these will not be open subspaces, but will be “mid-dimensional manifolds”.
(Well, hm, this is true if we have a functional on a finite dimensional manifold, which of course generically we do not have in field theory. Maybe the infinity-dimensionality saves the naive generalization here? Hm, not sure…)
To record #2 in $n$Lab I created Maslov dequantization.
For #3 Urs – regarding that the tropical geometry (obtained by Maslov dequantization of complex algebraic geometry) is important in homological mirror symmetry it gives us a hope to think that the idea alluded in #2 is not that isolated piece of mathematics.
Thanks, Zoran.
Re my last remark in #5: I see now how to solve it: of course while a Lagrangian submanifold inclusion is not formally étale, the inclusion of the formal neighbourhood is. (That’s obvious of course.)
So, hm, given a symplectic manifold with $\omega \colon X \to \mathcal{O}(\flat_{dR} \mathbf{B}^2 U(1))$ its homotopy fiber is something like the union of formal neighbourgoods of all Lagrangian submanifolds of $X$: a map from another manifold $Y$ into this fiber is a map that factors through some Lagrangian submanifold.
Hmm..
Added some expository accounts with historical review:
Agamenon R. E. Oliveira, History of Two Fundamental Principles of Physics: Least Action and Conservation of Energy, Advances in Historical Studies 3 2 (2014) [doi:10.4236/ahs.2014.32008]
Walter Dittrich, The Development of the Action Principle – A Didactic History from Euler-Lagrange to Schwinger, SpringerBriefs in Physics, Springer (2021) [doi:10.1007/978-3-030-69105-9]
Douglas Cline, Variational Principles in Classical Mechanics, University of Rochester (2021) [pdf, online version]
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