Velocity Space "Limit" Cycles for Systems with Conservation of Angular Velocity

Since the time of Hamilton it has been known that velocity space circles correspond to configuration space ellipses under the condition of conservation of angular momentum. What are the corresponding geometrical forms under the conservation of angular velocity?

Recall that Hamilton's equation is


where m is the gravitational parameter. To transform to velocity space, the arclength, ds, is replaced by the "fictitious" time parameter ds=dt/r. There then results


where the Hamiltonian can now be introduced,




where K=-2E, the constant curvature with H=E, the conserved energy. The Riemann metric is obtained by introducing the inverse velocity, q=1/v. [Milnor, "On the geometry of the Kepler problem"]

Now, in the case of constant angular velocity, Hamilton's equation reads


Introducing the fact that now the angular velocity, w, is constant, results in

dv/d theta=mr/w.

Using the conservation of energy,



d theta=2dv/(v^2-2E). (1)

In non-Euclidean geometries, distance is measure by the logarithm of the cross-ratio, and since the logarithm is related to the arc cosine,

log X =2 i arccos[(X+1)/2 X^(1/2)],

the arclength is related to the angle, theta. Equation (1) can be written as

d theta= (dv/k){1/(v-k) -1/(v+k)},

where k=(2E)^1/2. The integral can be simply performed,

theta= (1/k)ln{(v-k)/(v+k)=-(1/k)coth(v/k), (2)

for k real, or

theta=-i (1/|k|)cot(v|k|),

for k imaginary.

To look for circular orbits in velocity space, we do not have to go beyond Eq. (2),

v=k coth theta.

The form of the curve is a "limit" cycle coming in from negative infinity and wrapping around the center with a radius k=2, as shown in the diagram.

For v=2 tanh theta, the system originates at the origin and wraps around a circle, again with radius 2, as shown in the following diagram.

We have referred to these as limit cycles since they are obtained for increasing values of theta.

In reference to the Kepler problem in (inverse) velocity space, imaginary values of k would correspond to elliptic geometry of constant positive curvature involving trigonometric functions theta=arccos(v/|k|). For constant angular velocity, the hyperbolic functions correspond to periodic orbits, as the diagrams show.

For positive values of the energy we get open trajectories---which are well-known since the time of G. van Gutschoven around 1662, and later studies by Newton and Johann Bernoulli. They are referred to as kappa curves since the curves take that form as shown in the following diagram.

The kappa is the radial curve of the tractrix. The radial curve is the locus of the end of the radius of curvature vector attached to a fixed point. Beltrami's model of a non-Euclidean geometry consists of a surface of constant negative curvature, which is a "pseudo-sphere". The pseudo-sphere is obtained by revolving the tractrix about its axis of symmetry. So what is a trigonometric solution for constant angular velocity, w, is related to a hyperbolic function in geometries of negative constant curvature under the condition that the angular momentum is conserved.

Although the kappa curves are very well-known, and exhaustive search of the literature did not turn up the limit cycle curves. This appears very odd.

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