The inverse square force law gives stable, closed, elliptical orbits in any dimension.

You seem to be thinking about this other face: if the force is proportional to , the bound orbits will always be closed only in two special cases: the inverse square law () and the harmonic oscillator (). This is Bertrand’s theorem.

Furthermore, a potential obeying will, for a density concentrated in a small ball in dimensions, give a force proportional to where . So we get an inverse square force law in 3 dimensions, an inverse cube force law in 4 dimensions, and so on.

But my article was not about this stuff! It was about how a harmonic oscillator in 4 dimensions can simulate the motion in an inverse square force law in 3 dimensions if we change the time variable, change the center of the orbit, and project from a circular orbit in 4 dimensions down to an elliptic orbit in 3 dimensions!

]]>• Planets in the fourth dimension, *Azimuth*.

There are many ways to think about this, and apparently the idea in some form goes all the way back to Newton! It involves a sneaky way to take a particle in a potential

and think of it as moving around in the complex plane. Then if you *square* its position—thought of as a complex number—and cleverly *reparametrize time*, you get a particle moving in a potential

This amazing trick can be generalized! A particle in a potential

can transformed to a particle in a potential

if

]]>It will fulfill Hamiltons equations with respect to the new scaled time variable. E.g. the scaled time variable is related to the original one by . Do the Poincaré time transformation from to . If fulfills Hamiltons equations wrt to time , so will the new Hamiltonian wrt the scaled time .

Quote: “I now think that in the Hamiltonian formalism, there should be two functions on the same phase space: the ‘ordinary Hamiltonian’ that generates evolution in ordinary time, and the ‘Moser-Göransson’ Hamiltonian that generates evolution in Moser-Göransson time. I guess one is just the other times r. ”

So would be the Hamiltonian that generates the time evolution in the scaled/’Poincaré’ time (or what you called ‘Moser-Göransson’ time). The whole procedure is like some kind of canonical transform (but for the time) which preserves the form of Hamiltons equations.

]]>Sorry, I don’t know what’s the mathematical content of saying “ is a valid Hamiltonian”.

]]>I missed your comment originally (my post was sort of hit-and-run, sorry!)

The GIF almost certainly doesn’t obey Kepler’s 2nd law; I just decreased the frame duration linearly as the satellite approaches the closest point in its orbit. That said, you’re more than welcome to it- I don’t claim any rights to something I basically ripped off from your post!

]]>OK, if we switch to Moser-Göransson time, what changes? We still have:

But now, using a prime to denote the derivative with respect to the new time, we have:

with constant magnitude:

For the second derivate we have, as we would hope:

and the Hamiltonian for Moser-Göransson time is simply:

If we look at the invariants, they have the same form, but simplify slightly due to the fixed magnitude :

If we want to invert the map, we get a reasonably simple expression for the position vector:

But to obtain the velocity , I can’t see any easier way than to compute the scalar as the magnitude of the position vector, and divide by that and a constant factor to extract from the imaginary part of .

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