gravitational

Mysteries of the gravitational 2-body problem

John Baez

May 3, 2003

The inverse square force law was the first really exciting test of classical mechanics: it describes the motion of a planet around the sun in the solar system. It's called the Kepler problem, since Kepler was the one who guessed that planets moved in ellipses, based on tables of empirical data. Later Hooke told Newton to see if Newton's F = ma together with the inverse square force law would predict these elliptical orbits. Newton did just that, and went on to predict when Halley's comet would come back.

The inverse square force law was also the first really interesting test of quantum mechanics: it describes the motion of an electron around a proton in a hydrogen atom. A high school teacher named Balmer was the one who guessed the spectrum of hydrogen, based on tables of empirical data. Bohr came up with a rough-and-ready approach to quantum mechanics that explained Balmer's formula, but Schrödinger was the first to give a really detailed explanation.

In both cases the solution is "better than it needs to be". What I mean is this. In classical mechanics we can solve for the motion of a particle in any central force by doing an integral; if the force is attractive we'll get orbits that go round and round... but usually the orbits will precess. The magical thing about the inverse square force law

F = -k/r²

is that they don't precess: we get closed orbits! This is also true for the harmonic oscillator, where the force is described by Hooke's other law:

F = -kr

Even better, in both cases the motion is in an ellipse! It was an eerie stroke of luck for Newton that the Greeks - especially Apollonius - "just so happened" to have spent a lot of time studying conic sections just for their intrinsic beauty. That let Newton invent a proof using Euclidean geometry that the planets go around in ellipses, given classical mechanics and an inverse square force law. (He probably figured this out using calculus, but in the Principia he hid his tracks, since calculus wasn't rigorous, while Euclid's Elements was regarded as the pinnacle of rigor.)

In quantum mechanics we find that a hydrogen atom has (n+1)² bound states in the nth energy level, if we start counting at n = 0. Again this is "better than it needs to be". For a typical central force we expect bound states of different total angular momentum to have different energies; but for the inverse square force law something magical happens: there are states of different total angular momenta having the same energy! The nth energy level has:

2×0+1 = 1 state of angular momentum 0 - called s states in chemistry,
2×1+1 = 3 states of angular momentum 1 - called p states in chemistry,
2×2+1 = 5 states of angular momentum 2 - called d states in chemistry,
2×3+1 = 7 states of angular momentum 3 - called f states in chemistry,

and so on, up to

2n+1 states of angular momentum n,

for a total of

1 + 3 + 5 + ⋅⋅⋅ + 2n+1 = (n+1)²

states. Here I'm not talking about subtleties involving spin, which double the count of states and split some of these energy levels. I'm just talking about Schrödinger's original calculation.

The reason for both these "magical effects" is the Runge-Lenz vector: an extra conserved quantity besides energy and angular momentum, which is special to the inverse square force law! The formula for it is a bit funny:

      v x J     q
     ------- - --- 
        k      |q|

where q is the difference in positions of the two particles, |q| is the magnitude of q, v is the time derivative of q, J is the angular momentum, and k is the constant in the force law.

To understand the meaning of the Runge-Lenz vector, you need to know two things about it:

It points in the direction of the orbit's "perihelion" - in astronomy, the point where the planet comes closest to the sun.
Its magnitude equals the eccentricity of the orbit.

So, the fact that it's conserved means the orbit doesn't precess, and doesn't get more skinny or round as time passes.

For a proof of these facts, a proof that the Runge-Lenz vector is conserved, and an argument that uses it to deduce that the orbits in an inverse square force law are conic sections, try these homework problems of mine:

The Kepler Problem, in Postscript and PDF
- Answers by Toby Bartels, in Postscript and PDF
The Kepler Problem Revisited: the Runge-Lenz Vector, in Postscript and PDF
- Answers by Toby Bartels, in Postscript and PDF

Both classically and quantum mechanically, Noether's theorem relates conserved quantities and symmetries, so the fact that the Runge-Lenz vector is conserved means the inverse square force law has more symmetry than your average central force. But it's a rather sneaky "hidden symmetry", which changes the eccentricity of the orbits! My friend Michael Weiss says this book has a lovely diagram of this symmetry at work in the quantum case, turning an s-orbital into a p-orbital:

P. W. Atkins, Quanta: a Handbook of Concepts, Clarendon Press, Oxford, 1974.

In fact, the angular momentum and Runge-Lenz vector (rescaled by a function of the energy) give a total of 6 conserved quantities, which generate a 6-dimensional group of symmetries. What this group is depends on whether we are looking at solutions with negative energy (bound states, where the particle moves in an ellipse) or positive energy (scattering states, where it moves in a hyperbola).

In the case of bound states, the 6-dimensional symmetry group you get is SO(4), the group of rotations in 4d space!

In the case of scattering states, the 6-dimensional symmmetry group you get is SO₀(3,1), the connected component of the Lorentz group! This group is famous in special relativity... who'd have thought it was lurking in Newtonian gravity?

For each n, SO(4) has an irreducible representation of dimension (n+1)², which explains why the hydrogen atom has this many states in its nth energy level, discounting spin.

Now, you've probably seen those cute pictures where people draw an atom like a tiny little solar system, with electrons racing around in elliptical orbits. This is silly because it neglects the big difference between classical and quantum mechanics. The uncertainty principle means that electrons don't have well-defined orbits!

But, the analogy between the atom and the solar system becomes rather deep if relate the two using geometric quantization. If you've got just one planet orbiting the sun in an ellipse, or just one electron orbiting your nucleus, you've got a system where the Runge-Lenz vector, you've got a system with SO(4) symmetry! - and quantizing the first gives the second.

If you want more details, read this:

Victor Guillemin and Shlomo Sternberg, Variations on a Theme by Kepler, Providence, R.I., American Mathematical Society, 1990.

This is required reading for anyone interested in the Kepler problem, the Runge-Lenz vector, and the SO(4) or SO(3,1) symmetry it gives rise to!

Unfortunately, I still don't feel I know the "real reason" why the Kepler problem has a hidden SO(4) or SO(3,1) symmetry. Guillemin and Sternberg's book shows that if we only consider the bound states of the Kepler problem, we get a physics problem that is secretly the same as the motion of a free point particle on the unit sphere in 4d space! A bit more precisely, they're the same via a "generalized canonical transformation",where we reparametrize time as well as changing the other variables. This is very beautiful, because mathematically this means we're looking at geodesic motion on S³, which is the same as SU(2), the double cover of the rotation group.

However, Guillemin and Sternberg need a few yucky calculations to reach this conclusion, so I don't feel the subject has been completely demystified. Perhaps this book will make it clearer:

Bruno Cordani, The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Pertubation, Birkhäuser, Boston, 2002.

It sounds good, but I haven't read it yet.

I also regard it as mysterious that an object moving in an inverse square force law traces out a conic section. There are lots of ways to prove it, of course. Newton did it using Euclidean geometry. My homework problems above give two other ways. The one using the Runge-Lenz vector is pretty... but I'm still looking for the truly beautiful way, where you leave the room saying: "Inverse square force law... conic sections... of course! Now the connection is obvious!"

If you want to think more about why the inverse square law leads to elliptical orbits, try the argument in Feynman's "lost lecture":

David L. Goodstein and Judith R. Goodstein, Feynman's Lost Lecture: the Motion of Planets Around the Sun, New York, Norton, 1996.

He gives a fascinating argument, but it's not what I'd call truly beautiful... and James Dolan told me he read a review where someone claims it's not even a watertight argument. Does anyone know about that review? I'm curious if the reviewer had a real objection, or was just too constipated to follow Feynman's relaxed way of presenting arguments.

In any event, the book is definitely worth reading. I didn't listen to the CD that came with it, but I'm sure that's fun too.

Once Minhyong Kim lent me this book by the famous Russian mathematical physicist Arnol'd:

Vladimir I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals, trans. Eric J. F. Primrose, Birkhauser, 1990.

It has an appendix containing some really weird ways to show that motion in an inverse square law traces out conic sections. Some of them involve taking the problem and redefining time. This should be related to the "generalized canonical transformation" that I mentioned above. Unfortunately, I haven't put the puzzle pieces together yet.

So, if you have profound insights on these issues, let me know!

Here are some more references which may help you find these insights:

Myron Bander and Claude Itzykson, Group theory and the hydrogen atom, Rev. Mod. Phys. 38 (1966), 330-345 (part I), 346-358 (part II).
V. A. Dulock and H. V. McIntosh, On the degeneracy of the Kepler problem, Pacific Jour. Math. 19 (1966), 39-55.
M. J. Englefield, Group Theory and the Coulomb Problem, Wiley-Interscience, New York, 1972.
Herbert Goldstein, Prehistory of the Runge-Lenz vector, Am. Jour. Phys. 43 (1975), 735-738.
Herbert Goldstein, More on the prehistory of the Runge-Lenz vector, Am. Jour. Phys. 44 (1976), 1123-1124.
H. V. McIntosh, Symmetry and degeneracy, in Group Theory and its Applications vol. 2, ed. Ernest Loebl, Academic Press, New York, 1968-75, p.75.
John Milnor, The geometry of the Kepler problem, AMS Notices 90 (June-July 1983), 353-365.
G. E. Prince and C. J. Eliezer, On the Lie symmetries of the classical Kepler problem, Jour. Phys. A, Math. Gen. 14 (1981), 587-596.

Is not this, perhaps, the secret of every true and great mystery, that it is simple? - C. Kerenyi