Is anyone else around here participating in the Summer of Math Exposition?
Here's my entry: https://www.youtube.com/watch?v=QC3CjBZLHXs
People say "you don't fully understand something until you explain it" but in this case I didn't known that I *didn't* understand it until I *tried* to explain it. So there was a bit of a time crunch to learn what I needed to and get the video done within the deadline. I hope it's interesting and enjoyable despite being a bit rushed!
Basic knowledge of Hamiltonian and Lagrangian mechanics will be useful to follow the talk, though I will give a very short refresher of the essentials. If I do a good job, nothing more advanced should be required.
Tomorrow I will give an online lecture hosted by the Institute for Fundamental Study in Thailand. I will talk about my work in integrable systems and its context, accessible to students in physics and mathematics.
The lecture will start at 7 UTC, which is 8 AM here in the UK 🥐 ☕
A recording will appear on youtube afterwards.
Students sometimes wonder why we use quaternions, which are 4D, to represent 3D rotations, when we can use (2D) complex numbers to represent 2D rotations. But that's the wrong way to count dimensions. Complex numbers represent similarity transformations in 2D, which combine 2D rotations (1 degree of freedom) and uniform scalings (another 1 degree of freedom). But 3D rotations have 3 degrees of freedom!
This felt really familiar 😂
Tom Gauld is out of doubts one of my favourite illustrators. His latest new scientst’s strips are here: https://newscientist.com/author/tom-gauld/
The variational principle we find for the Toda hierarchy leads to a surprising observation. This set of equations, which are all semi-discrete, somehow contains another set of equations, which are continuous! We suspect that these continuous PDEs are also integrable, but understanding them will take a lot more work...
Or, maybe, it will just take someone read our paper, recognise those PDEs, and point us to where they have previously appeared.
Our new preprint is about the mixed situation: semi-discrete systems. It mainly considers systems consisting of particles on a 1-dimensional lattice, so with a discrete space, evolving in continuous time. We focus on a well-known system of this type (the "Toda lattice"). This equation, and its friends which make it integrable, (together: the "Toda hierarchy") are all semi-discrete.
Usually, variational principles produce a single ("Euler-Lagrange") equation, but the one we study produces a set of compatible equations. This is why it captures integrability.
So far I've talked about an integrable system as a differential equation. Both ODEs (eg planet orbiting a star) or PDEs (eg waves in a shallow canal) can be integrable. But also discrete systems, where time or space-time are a lattice instead of a continuum, can be integrable.
Lagrangian mechanics involves a "variational principle": solutions to the dynamical system minimise some abstract quantity, so finding solutions becomes an optimisation problem.
In many ways, Hamiltonian and Lagrangian mechanics are two sides of the same coin. But in integrable systems, this hasn't been the case. You won't see any Lagrangians in textbooks on the topic. Still, a Lagrangian theory of integrability does exist. In recent years I've contributed various bits and pieces to it
Actually, to claim "complete integrability" you need a slightly stronger compatibility condition expressed in terms of Hamiltonian mechanics and Poisson brackets, which implies nice things about the geometry of the orbits.
Don't worry if some of this terminology sounds unfamiliar. The only thing you need to know about mechanics to follow this story, is that almost everything you can do with "Hamiltonian" mechanics can also be done with "Lagrangian" mechanics.
"Compatible" can have different interpretations, but for now let's take it to mean "commuting". This means that it does not matter in which order you apply the equations.
For example, the planet orbiting the sun is integrable because it commutes with rotation: it does not matter whether you first apply the physical evolution and then rotate everything, or first rotate and then let physics do its thing. Both will get you to the same final state.
This work is about integrable systems of a certain kind. Integrable systems are in a sense the opposite of chaotic systems.
A simple example is a planet orbiting a star: it is very predictable, tracing out an ellipse over and over again. A system is integrable if it possesses some properties which make it well-behaved like that. One such property is that integrable systems have friends: an equation is integrable if it is part of a sufficiently large set of compatible equations.
Today, for the second time in as many weeks, I have a fresh new preprint on the arXiv:
(joint with Duncan Sleigh)
The last one got a thread on the birdsite, so let's give this one the same attention here.
(And no, I'm not *that* productive. Both papers were over a year in the making.)
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1st in the series. I've been collecting these for multiple years on https://blog.mathoffthegrid.com/p/collected-problems-2.html (and several followup pages)
My name Mats. I'm a postdoc at Loughborough University working in mathematical physics (integrable systems and variational principles), but I like to hear and talk about all kinds of maths and puzzles.
I'm also a long-distance runner, but I don't know if I'll be posting much about that.
I write in English, but if you post in Dutch, German, French or Spanish I'll probably understand what you're saying too ;-)
Mathematician by day and long distance runner by night (well, on the weekends mostly).
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