This post is an English translation of a talk I gave in French during my research internship with Pavel Winternitz, almost a decade ago, lightly edited for reading. You can watch the original talk (in French) on YouTube.
What a theoretical physicist does
I’m a joint math and physics undergraduate, and I hope to become a theoretical physicist (well guess that didn’t happen, but I love my job as a software engineer at Stripe). So what does a theoretical physicist actually do? They start with an intuition about a particular physical problem. From that intuition, they develop a mathematical model that describes the system.
Then that model gets tested, both theoretically and experimentally. Theoretically, we derive every result the model is supposed to predict. Experimentally, we go out into nature and see whether the model is actually right. That could be at CERN, but the point is that it’s reality, and it’s the experimentalists’ job.
Take the greatest physicist in history, Newton. Like everyone else, he had noticed that objects fall to the ground. He had also read Kepler, who had stated three laws based on observation, one of which says the planets trace out ellipses around the Sun. Newton reasoned that there is something that makes objects fall to the ground, and something that makes the planets move around the Sun. His intuition was that these might be the same thing.
From there he built his model: his three laws of motion and the law of universal gravitation. He tested it theoretically by proving Kepler’s three laws and by describing the motion of objects on Earth. If I throw a ball, Newton’s equations and the law of gravitation describe its motion.
Then comes the experimental test. Newtonian mechanics works extremely well and is useful in every other branch of physics, which is why we learn it first. Today we know it’s an approximation that holds for speeds far below the speed of light and for macroscopic objects, which is equivalent to letting Planck’s constant go to zero.
So you might say: fine, Newton developed classical mechanics, we’re done, let’s go home and watch Game of Thrones. And that’s where I’d say no. Newton described the motion of classical objects mathematically, but then you have to solve the equations, and that is rarely possible. It only happens for systems we call integrable, where Newton’s equations can be solved analytically.
So the second part of a theoretical physicist’s job is to improve the theory: to make it richer with deeper physical concepts and more powerful mathematical tools. A simple example is energy. With it, we can solve problems we couldn’t solve with Newton’s laws alone, or could only solve with enormous difficulty.
Energy also gives us a better understanding of how nature works, through conservation laws: conservation of energy, of linear momentum, of angular momentum. To explain an object’s motion, we don’t just say it follows from Newton’s equations; we say energy or momentum is conserved, and that’s why the object follows the path it does.
One last reason to keep working on classical mechanics is to build new theories. It’s the Hamiltonian formulation of classical mechanics that lets us develop quantum mechanics. Without it, we couldn’t have.
Superintegrability is exactly this: developing classical mechanics even further. To talk about it, I first need to introduce the Hamiltonian. Physicists may find this part familiar, but bear with me.
The Hamiltonian
We start with generalized coordinates and generalized momenta. A coordinate system lets us position an object within a reference frame. I’ll call the coordinates q, so in Cartesian coordinates the qi are x, y and z.
The generalized momenta are the pi. In Cartesian coordinates they have a familiar physical meaning: p = mv, the mass times the time derivative of the position. In other coordinate systems the momentum is something else, but we don’t need that for now.
We can represent these quantities in phase space, which simply relates the qi and the pi. A one-dimensional system has a two-dimensional phase space; in general, an n-dimensional system has a 2n-dimensional phase space. It holds every possible configuration of coordinates and momenta.
The Hamiltonian is a function on phase space:
It could also depend on time, but never will in this post: we’re dealing with superintegrable systems, where the Hamiltonian must be a constant of the motion. (Writing vectors here is a slight abuse of notation; it saves me from writing q1 through qn and p1 through pn).
In its simplest form, the Hamiltonian is the total energy of the system: kinetic energy plus potential energy. The potential comes from the force, as in vector calculus: a conservative force can be derived from a potential. In Cartesian coordinates:
Why this form? Because p = mv, so the kinetic energy is
and this holds along each axis, giving px² + py² + pz², the squared norm of the momentum.
From the Hamiltonian we get the system’s equations of motion, which are equivalent to Newton’s laws. Newton gives n second-order equations for an n-dimensional system (F = ma). The Hamiltonian formulation gives 2n equations, coming in pairs, and from those pairs we recover exactly what Newton would give. These are Hamilton’s equations:
Used as is, this is just an equivalent formulation. Instead of finding the forces and writing Newton’s laws, I write the total energy of the system and use these equations to get the equations of motion. The real power comes next.
Poisson brackets and integrals of motion
The Poisson bracket is an operator on two functions on phase space, which depend on the generalized coordinates and momenta. By definition, the Poisson bracket of f and g is:
This may look strange, but there’s a reason for it: it’s what lets us find integrals of motion, functions on phase space that are conserved along the motion. The Hamiltonian is one: it’s conserved because total energy is conserved. Let’s prove why the bracket matters.
Take the Poisson bracket of the Hamiltonian H with another function on phase space, X:
Now plug in Hamilton’s equations: ∂H/∂pj is the time derivative of qj, and −∂H/∂qj is the time derivative of pj:
When I asked the room whether this rang a bell, someone in the audience recognized it right away: it’s the chain rule. This is just the time derivative of X:
So instead of computing the time derivative of X, I compute its Poisson bracket with the Hamiltonian, and I get exactly the same thing. If the bracket is zero, the time derivative is zero, and X is an integral of motion.
Result 1. {H, X} = 0 implies that X is an integral of motion.
Integrable and superintegrable systems
Now we can define integrable systems properly. An integrable system is a Hamiltonian system that admits n integrals of motion in involution. If P1 is the Hamiltonian and P2 through Pn are the other integrals, they must satisfy:
Why the extra conditions? In principle I could have infinitely many integrals of motion: from any n of them, I can build others that are functionally dependent on them. I want integrals that are functionally independent of one another, so they characterize the system precisely. Otherwise it’s pointless: I could take an integral of motion, multiply it by 2, multiply it by 3, and there’d be no physics left.
So what’s a superintegrable system? They’re systems that admit more than n integrals of motion, up to 2n − 1.
Why 2n − 1? Phase space has 2n dimensions. With 2n integrals of motion, all that’s left is isolated points: no motion, nothing physically interesting. Just below that, at 2n − 1, we get curves, and those curves are the trajectories the system can follow. The definition has the same form as above, but with m integrals of motion, where
and the additional integrals commute with H but not necessarily with each other.
Why they matter
Superintegrable systems have symmetries that form algebras of quadratic, cubic and higher order, which is interesting in its own right for mathematicians. If you’ve studied physics, you’ve almost certainly met some of them without knowing they were superintegrable:
The classical harmonic oscillator, the physics of a spring.
Kepler systems, which govern planetary orbits. Their superintegrability underlies the Hohmann transfer, used to move satellites between orbits and position them.
Quantum Coulomb systems, which give the energy levels of the hydrogen atom. Starting from that system, perturbation theory lets us build up the structure of the whole periodic table.
The quantum isotropic harmonic oscillator, the one that is the same in every direction.
Superintegrable systems admit the maximum number of symmetries, and that forces analytic and algebraic solvability. That’s the key point: I don’t need differential and integral calculus to solve the system. I can do it purely algebraically, with no differential equations at all.
That brings better understanding and better intuition, which is what a theoretical physicist is after. We could just do math, but we also like to understand the physics behind it. Whenever a new superintegrable system is found, it tends to end up in a model, because it can be solved explicitly. Current research focuses on discovering, classifying and solving these systems, and on elucidating their algebraic structure.
Fewer dimensions, closed orbits
One property was especially important for my own research: superintegrability shrinks the region of phase space where a trajectory can live.
Each integral of motion imposes a constraint, much like constraints in an optimization problem. Integrability already imposes n integrals, taking us from the 2n dimensions of phase space down to n. Each additional integral removes another dimension, so with k extra integrals we reach n − k, where k runs from 1 to n − 1.
At the maximum, 2n − 1 integrals, we have a maximally superintegrable system, and every trajectory lives on a one-dimensional curve. So every bounded trajectory must be closed, and every bounded orbit is periodic. That’s what makes it possible to find trajectories algebraically, and it’s what I worked on in my research project. But first, a concrete example.
Kepler’s laws, the algebraic way
In a classical mechanics course, the Kepler problem is solved analytically, with Newton’s laws or, more often, the Lagrangian formulation, from which the Hamiltonian formulation follows. It’s long and tedious. Here I’ll do it the cool way: purely algebraically. No differential equations solved, no integrals computed. Just superintegrability.
A reminder of Kepler’s laws:
The planets follow elliptical orbits around the Sun. Strictly, the Sun sits at one focus; here we’ll simply put it at the origin. For very different masses, like the Earth and the Sun, that’s a reasonable approximation.
In equal times, a planet sweeps out equal areas. In other words, the rate at which area is swept out is constant.
The square of the orbital period T is proportional to the cube of the ellipse’s semi-major axis a: T² is proportional to a³.
I’ll prove the first two laws. The third follows from them.
The Hamiltonian
The Hamiltonian is kinetic energy plus potential energy. I’ll set the mass to 1, as we often do to avoid carrying m around:
The gravitational force is proportional to 1/r², so the potential is proportional to 1/r, with r = √(q1² + q2²). We take k > 0 so the force is attractive. If it were repulsive, the Earth would fly away.
Angular momentum
Hamilton’s equations would give us:
Solving these is exactly what we want to avoid. Instead, we look for integrals of motion. The first is H itself. For the second, take the angular momentum and compute its Poisson bracket with H:
So by Result 1, L is an integral of motion. The classical mechanics course proves angular momentum conservation differently, but this way is faster.
The Laplace-Runge-Lenz vector
Since we’re in two dimensions, a maximally superintegrable system needs 2n − 1 = 3 integrals. The third is harder to find. And yet Kepler is the easiest superintegrable system there is, which tells you why there’s still research going on in this area.
Start from Newton’s second law with the gravitational force:
Now cross it with the angular momentum. With unit mass, L = r × p = r × ṙ, so:
That’s a vector triple product, so we use the identity we all love:
The product r · r is just r², and by the product rule:
(For a circle, this shows the position and velocity vectors are perpendicular, since ṙ = 0 when the radius never changes). Factoring out −r³, the bracket becomes a total time derivative:
Since angular momentum is constant, its time derivative vanishes, so:
Putting everything on one side:
So this vector is conserved along the motion. It’s the Laplace-Runge-Lenz vector:
Its dot product with L is zero, so A is perpendicular to L. Since L is perpendicular to the plane of the motion, A lies in that plane.
Three integrals, one conic section
The norm of A works out to:
and its components are:
Both components are conserved, since the vector keeps the same orientation and norm. But that gives four integrals of motion, H, L, A1 and A2, when there should be three. So two of them must be functionally dependent. Their Poisson brackets are:
To reduce to three, choose coordinates whose first axis points along A. Then A’s first component is its norm and its second component is zero:
Setting A2 = 0 and solving the component equations for the momenta gives:
Now substitute these into L = q1p2 − q2p1:
Multiply both sides by L, subtract q1A1, divide by k, and square:
Expanding:
and collecting terms:
Look at what’s left. A1 is fixed by the integrals of motion and the coupling constant k. Everything here is either an integral of motion, a constant of the system, or a coordinate. Nothing else.
When I asked the room what this was the equation of, the answer came back: a conic section. And which conic sections are closed? Ellipses.
Kepler’s first law: proven.
The second law
Take a small slice of area swept out along the ellipse. It has sides r and r + dr, and an arc of length r dφ. As dφ and dr go to zero, the slice becomes a triangle with height r and base r dφ, so:
(For a more rigorous version, you can start from the area integral below; I prefer the triangle because it’s more intuitive.)
Now write the angular momentum, with p1 = q̇1 and p2 = q̇2, in polar coordinates q1 = r cos φ, q2 = r sin φ:
The terms in ṙ cancel, leaving:
You can see the resemblance:
The right-hand side is exactly the rate at which area is swept out. And L is an integral of motion, so it’s constant:
Kepler’s second law: proven. Still easy, right?
Polynomial superintegrability
When we look for new superintegrable systems, it helps to narrow the search. Current research looks for integrals of motion that are polynomial in the generalized momenta: functions multiplied by combinations of pr, pθ, px, py, pz.
Polynomial integrals are useful for quantization, since we use the foundations of classical mechanics to work on quantum mechanics. They’re also simply easier to work with than other functions. The problem is hard enough already, so we keep the difficulty to a minimum.
Until the 2000s, people believed that second-order systems, with second-degree polynomial integrals, were essentially the only ones you could find, using the technique of separation of variables. That’s the best-understood part of the field, and many thought it was where the story would end.
Then, in 2009, a paper by Tremblay, Turbiner and Winternitz opened the way to higher-order polynomial integrals. It relaunched the whole field of superintegrability. And Tremblay was my CEGEP teacher, which was a nice surprise when I found out.
My research project
My project (page 18 on this link) was to find the trajectories of fourth-order superintegrable systems, meaning systems whose integrals of motion are fourth-degree polynomials. I was given the integrals, and my job was to find an equation describing the orbit algebraically, which superintegrability makes possible.
The Hamiltonian is a bit different from Kepler’s. In polar coordinates it has the kinetic energy, a function of θ divided by r² (which is what allows separation of variables), and a function of r that keeps the orbits bounded: either the Kepler potential we just worked through, or the harmonic oscillator:
That’s the first integral of motion. For Kepler, the second was the angular momentum, because there was no force in θ. Here there is one, so instead I group the angular terms and factor out the r², which leaves:
I checked that its Poisson bracket with H really is zero. The third integral, for the Kepler and harmonic oscillator cases, depends on further functions tied to particular potentials that a postdoc had found earlier. Written compactly it fits on a slide; written out in full it’s really long and painful.
As with the Kepler problem, the goal was to combine everything into functions that depend only on the coordinates and the integrals of motion. I got two such functions, but they’re too complicated to read trajectories off directly. All I can do is test whether a given point satisfies them.
So I find trajectories numerically: I choose initial conditions, compute the values of the integrals of motion they correspond to, and trace out the resulting orbits, along with the constraints on r and θ and the potential governing each one.
Some of my plots were judged unsatisfactory by Pavel, because it wasn’t clear from them that the orbits close. Someone in the audience pointed out that they looked like Lissajous curves, and they do, informally. With Lissajous curves you can count how many times the curve touches the edges of its bounding box; mine touched them so many times that you couldn’t tell whether the orbit was really closed. So I’ll have to find better examples. They’re still pretty, though.
Questions from the audience
Are there superintegrable systems in quantum mechanics? Yes. There you work with operators, and with commutation and anticommutation relations. I didn’t cover it because I ran right up to the time limit, but it’s another interesting topic, and there are real connections to make between the quantum and classical sides.
What’s the physical meaning of the Laplace-Runge-Lenz vector? Its norm is the mass times the coupling constant times the eccentricity of the orbit, and it points along the major axis. There’s a lot more to say about it, but that would need a talk of its own.