For space dimensions, consider orbits in fields produced by elementary particles, particles that are monopole sources of those fields. Sources like mass for gravity and electric charge for electromagnetism.
The force law for such fields is 1/r
D-1 for distance r and D space dimensions. This is the same as for geometric dilution of radiation from a point source.
So we must consider a particle in the Newtonian limit that travels in a central force.
Bertrand's theorem states that only two power laws give closed orbits: r and 1/r
2. Related to that is what power laws give stable orbits. One starts with a particle in a circular orbit and one then perturbs that particle's motion. Will it do a stable oscillation or will it either fall in or fly away?
Equations of motion in polar coordinates for a central force f(r) for r the distance from the center and (theta) the position angle in the plane of the orbit, and ignoring the particle's mass,
\( \displaystyle{ \frac{d^2 r}{d t^2} - r \left( \frac{d \theta}{d t} \right)^2 = f(r) } \\ \displaystyle{ \frac{d^2 \theta}{d t^2} + 2 \frac{d r}{d t} \frac{d \theta}{d t} = 0 } \)
The second equation gives us conservation of angular momentum:
\( \displaystyle{ \frac{d^2 r}{d t^2} - \frac{h^2}{r^3} = f(r) } \\ \displaystyle{ r^2 \frac{d \theta}{d t} = h } \)
and making the position angle the independent variable and using the reciprocal of r: u = 1/r:
\( \displaystyle{ \frac{d^2 u}{d \theta^2} + u = - \frac{ f(1/u) }{h^2 u^2} } \)
Substituting a power law f(r) = - K r
-p gives us
\( \displaystyle{ \frac{d^2 u}{d \theta^2} + u = \frac{K}{h^2} u^{p-2} } \)
Let's now expand around a circular orbit, expanding the radius in a series in the position angle:
\( u = u_0 (1 + e \cos (g \theta) + \cdots ) \)
For a circular orbit,
\( \displaystyle{ \frac{K}{h^2} (u_0)^{3-p} = 1 } \)
\( g^2 = 3 - p \)
Since p = D - 1, for D space dimensions,
\( g^2 = 4 - D \)
Circular orbits will be stable if g
2 > 0, unstable if < 0, borderline stable if = 0. So circular orbits are stable for D <= 3, borderline stable for D = 4, and unstable for D >= 5. For D = 4, the possible angular momentum for a circular orbit is fixed by the force-law constant.
Extending the analysis to quantum mechanics, one finds that the solutions are well-behaved only for D <= 3. For D = 4, some solutions are concentrated at the origin, and for D >= 5, all solutions are.