The classical Kepler problem - the simplest example of a planetary system - has a super simple solution: planets make ellipses around the sun. In more technical terms, it has the property of maximal super-integrability. In this post, we’ll find out what that means, what the consequences are, and how it manifests itself in the classical and relativistic setting.
Integrability
Let’s first define the notion of integrability. In our case, we’ll be working with Hamiltonian systems, and the definition of integrability will be Liouville integrability. An important notion in Hamiltonian mechanics is that of Poisson brackets. These brackets take two functions f and g and give derivatives of these functions with respect to the coordinates of the phase space, positions and momenta:
Liouville integrability then says (barring some technical requirements) that if in a Hamiltonian system with n degrees of freedom, there are n integrals of motion (time-independent constant functions), it is solvable in quadratures. Solvable in quadratures means that we can in principle write down integrals that solve the equations of motion of the system. Whether we can actually integrate them depends on some particulars.
Liouville-Arnold Theorem
Of course it is good to know the system can, in principle, be solved, but there is more information in the fact that a system is integrable than just that. One of the most important consequences of integrability is that the phase space has a particular shape. This is stipulated by the Liouville-Arnold theorem, which says that for an n-degree-of-freedom integrable system, the compact energy level sets are diffeomorphic to n-tori. Essentially, the phase space, for the right energies, amounts to a stack of donuts labeled by the values of conserved quantities.
The Classical Kepler Problem
This is the problem of a particle subject to a force dropping off as the square of the distance from some central point, having the familiar Hamiltonian
To quite a high accuracy, this is exactly what’s going on in the system Sun-Earth or considering a wave function instead of a point-particle, the Hydrogen atom with its electron orbiting a proton.
We now know it’s sensible to look for the integrals of motion of the system to help us figure out whether it can be solved and what the phase space will look like. It’s immediately clear that energy E as well as the three components of angular momentum L are conserved, because of the invariance under time translations and rotations. Having these four integrals of motion in a three dimensional system already gives us something more constrained than just integrability: super-integrability.
But it doesn’t stop here. The Kepler problem has an additionally conserved vector, the so-called Laplace-Runge-Lenz vector
in principle adding another three integrals of motion. Now, only one of these components cannot be built from the energy and angular momentum, but still, we’ve got a 3-dimensional system with 5-independent integrals. The upshot of all these additional integrals is that the phase space is just one-dimensional, causing the motion (for negative energies) to be purely periodic. This notion is called maximal super-integrability.
Enter Relativity
In relativistic theories, energy and angular momentum are still conserved, but the LRL vector is lost as integral of motion. This causes a particle travelling around the centre, that closed the ellipse for the classical case, to just miss the point it left in the previous round, creating a flower-like pattern.
Now, interestingly, the way the ellipse precesses around the central point, is dependent on the type of relativistic theory we use. If we take the first order correction from Einstein’s General Relativity to the gravitational problem (spin-2 field), we find that the precession rate is six times the one in electromagnetism (spin-1). Taking a scalar field (spin-0), we even find the precession rate has the opposite sign of the one in GR! This, in the 1910s, was a clear argument for Einstein’s theory in favour of the scalar theory proposed by Nordström, when comparing the theories to data on the anomalous precession of Mercury.
Final Remarks
Making systems relativistic generally comes with significant complications, prohibiting the exact calculation of parameters relevant for studying for example gravitational wave data. The two body problem in full GR already doesn’t have an analytic solution. This demands approximate calculations to get all possible information from the conducted experiments. Using the lens of integrability, conserved quantities and loss of those in the context of a certain order of relativistic corrections might help us to improve the tractability of these calculations.
This blog post was written subsequent to a presentation in the Basic Notions seminar. For more information about this seminar, go to the seminar's webpage.