When treating non-Euclidean theories of planetary motion, it is common to modify G in the metric
ds^2 = E dr^2 + G d\theta^2,
and leave E the identity. In flat space, G=r^2, in spherical space G=R^2 sin^2\theta, and in hyperbolic space, G= R^2 \sinh^2\theta, where R is a constant radial coordinate. Both cases are of constant curvature.
However, the more interesting case is to leave G "flat", and to concentrate on E. This will include forces such as the Weber force, and forces associated with surfaces of revolution such as the pseudosphere. This bugle surface was used by Beltrami as the prototype of hyperbolic geometry of constant curvature, and is known for its solitary wave properties through the sine-Gordon equation.
We consider those curves whose radii of curvature are proportional to their normals. If V' is the slope of the curve, the derivative of V with respect to the coordinate, r, then the radius of curvature is
R = (1+V'^2)^(3/2) /V".
The length of the normal to the curve would be
N= rV'(1+V'^2)^(1/2).
Assuming that N is n times R gives
rV'V" = +/- n (1+V'^2).
The ambiguity in sign is due to the two directions that the lengths can be measured, The factor (1+V'^2) measures deviations from a unit speed curve.
Separating variables and integrating lead to
1+V'^2 = (p/r)^(2n)
where p is the constant of integration, and we have chosen the negative sign by assuming that if the curve is concave with respect to the r axis then r and V" will be positve, while V' and N would be negative. N would be in the opposite direction to R.
Now, the problem is to determine V since E would measure the deviation from a unit speed curve,
E= (1+V'^2) =(p/r)^(2n).
As an illustration, consider n=1, the pseudosphere. The slope is then given by
V'= (p^2-r^2)^(1/2) /r.
Integration leads to
V= (p^2-r^2)^(1/2) - p sech^(-1) (r/p).
The curve is shown in the figure as two trumpets with respect to the r-axis.
The Lagrangian is proportional to the metric,
L= (p/r)^2 (dr/dt)^2 + G(d\theta/dt)^2.
The radial equation of motion is
d^2r/dt^2 - (1/r) (dr/dt)^2 - r^2 G'(d\theta/dt)^2 = 0
in the absence of an external force. We don't have to specify what G is because we are interested in the motional forces. They can be written as
(d/dt) (dr/dt/r),
and it is this term that Weber equated to zero to determine the coefficents in his force so that they would correspond to Ampere's law.
However, the radial equation is not Weber's law which results when n=1/2. The slope of the curve is now
V' = {(p-r)/r}^(1/2).
and this gives the curve
V={r(p-r)}^(1/2) -p tan^(-1)({r/(p-r)}^(1/2)).
This is shown in the following figure consisting of two inverted hemispheres.
This corresponds to a radial equation of motion
(d^r/dt^2)-(1/2r)(dr/dt)^2 -rG'(d\theta/dt)^2 =F
where F is Weber's force. The one-half in the equation of motion comes from Ampere's experiments measuring charges parallel and normal to each other. The curve, which is like two inverted catenarys is shown in the figure below.
For other values of n, we obtain soliton like curves shown in the first figure superimosed on the bugle surface. Solitons necessarily arise from the constrained motion of double well potentials that result when we specify G as r^2. The motional forces, however, are determined by the E coefficient in the metric. This will be further elaborated on in forcoming blogs. Here, we wish to emphasize the surface of revolution that are to associated with the motional forces and the particle aspect that arises therefrom.
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