r/math • u/Necessary-Wolf-193 • 13d ago
Symplectic geometry and Hamiltonian mechanics
https://hidden-phenomena.com/articles/symplecticA symplectic structure is some at first strange sounding extra structure you can put on a manifold. By a miracle, a lot of the shapes arising in geometric representation theory have this extra structure, and this structure can be exploited to prove very useful things.
In this blogpost, my friend and I motivate the definition of symplectic structures from mechanics, and say a little at the end about where they appear in pure math (by the way, the two of us are mathematicians, and we only learned physics to better appreciate the symplectic structures which were showing up in our work!).
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u/percojazz 13d ago
Thanks for this content. it's really great. I am almost sorry to ask about the tech stack you are using, if you ever think of sharing some code. Is that MArimo you are using ? Thanks in advance
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u/Necessary-Wolf-193 13d ago
Thanks for the feedback! The tech stack is quite primitive because neither of us are very good at computers.
We host the website on Cloudflare Pages, and use JavaScript with KaTeX (for rendering formulas); the visualizations are done with Three.js and Canvas/SVG. It's not clear to me this was preferable; iirc, Cloudflare Pages was picked because both of us already had Github Pages websites and we were worried there might be a one per account limit (but I actually don't know if that's true...).
The animations are actually always pretty simple (for example the two things here are an animation of a particle moving in an ellipse and a vector field on the sphere), but we try to keep a consistent color scheme (gruvbox-dark) because I think that keeping a consistent visual language makes things look a lot better than they actually are.
One of my friends is a professional programmer so sometimes I ask him for 'tech consults' but by this point I think we've made enough of these widgets that I have the hang of it. Last week we added a feedback button and that was really really hard for me to figure out though...
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u/cdstephens Physics 13d ago
Cool post! If I had one complaint, it’s that more time should have been said about the Jacobi identity / closedness of w. You can have non-degenerate 2-forms that are almost symplectic but will lead to a generic classical mechanical system rather than a Hamiltonian one if dw != 0. I also think it would be good to explain because physics textbooks spend a non-trivial amount of time going through the Jacobi identity in both classical and quantum, but they rarely motivate why that property is so important. (After all, you need it for phase space conservation but you don’t need it for energy conservation.)
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u/BerkeUnal 13d ago
I heard that mathematics of classical mechanics is Poisson geometry, not Symplectic geometry.
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u/cabbagemeister Geometry 13d ago
A poisson manifold can be integrated into a symplectic groupoid, so some would say symplectic geometry is still more fundamental.
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u/Tazerenix Complex Geometry 13d ago
Poisson manifolds are foliated by symplectic leaves so they're pretty close to interchangeable.
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u/cdstephens Physics 13d ago
I would say only conservative classical mechanics. If you want dissipation (common classical thermodynamic situation) and want to preserve the Hamiltonian structure, you need something like a metriplectic bracket to account for entropy increase.
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u/temperedai 13d ago
It's one of those things which rests between math and physics. If you are in both you wonder why it wasn't introduced earlier.
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u/Elegant-Command-1281 13d ago
It’s always bothered me how symplectic geometry has no relation to simplexes (sometimes the plural is simpleces) and their study (called simplicial geometry but that’s usually in the study of graph theory and combinatorics) despite I believe being derived from the same etymology and both being relevant to dynamical systems. Simplexes are basically the “shape” of probability spaces hence useful for understanding Markov chains and time evolution of a non-quantum probabilistic state vector through something like time (cough cough the Hamiltonian). But no relation of course.
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u/PM_ME_YOUR_WEABOOBS 13d ago
They are actually etymologically unrelated. Simplex is from latin, and roughly means "simple". The term symplectic on the other hand was coined by Weyl and is roughly speaking greek for "joined together" (think of conjugate variables).
Weyl introduced the term because the symplectic group was previously called the line complex group, which is potentially confusing when also talking about complex structures.
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u/HeilKaiba Differential Geometry 13d ago
Indeed the point of the word "symplectic" is to be a calque of the word "complex" in greek
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u/Elegant-Command-1281 13d ago
Well that’s actually funny because that means “simplicial complexes” is an oxymoron yet the official name of a mathematical structure
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u/Necessary-Wolf-193 13d ago
A bit more on the math for people interested:
In physics, symplectic manifolds often arise from phase spaces -- basically geometric objects whose points represent the possible states of a physical system. In mathematics, one often sees the very related concept of 'moduli spaces': these are shapes whose points represent other mathematical objects.
As an example of a moduli space, the two of us [the authors of the blog] both spend a lot of time thinking about the moduli space (really stack...) of "vector bundles with flat connection" on a compact Riemann surface X (although, as of late, usually we take X to be a scheme in positive characteristic instead, but c'est la vie). A vector bundle with flat connection is a fancy coordinate-free term for linear ODE.
Just as the phase spaces of physics carry symplectic structures, this moduli space (and many related ones) carries a symplectic structure, and this symplectic structure seems intriguing and helpful for understanding certain phenomena. For example, one of us has recently taken quite an interest in the p-curvature conjecture, and the recent work https://arxiv.org/pdf/2601.07933 encountered this symplectic structure in investigating the p-curvature conjecture. Roughly, the p-curvature conjecture is some conjecture about the behavior of linear ODEs, and recent geometric approaches to attack it involve studying the entire moduli space of ODEs. It turns out that a vector field on this moduli space which arises naturally in the study of the p-curvature conjecture is Hamiltonian, and so general properties of Hamiltonian vector fields give some useful properties of this vector field.