Surfaces of constant, negative, curvature like the pseudosphere and the hourglass have geodesics which are independent of the angular coordinate. However, in order to get a surface of constant curvature, the angular coordinate must be taken into consideration because gaussian curvature needs two planes, one in the radial and the other in the angular directions. The Euler-Lagranged equations are distinct from the radial geodesics because they encode how the motion occurs on the surface.
The geodesics are meridians: on the pseudosphere they are tractrix curves which have been rotated about their axis of symmetry. They are the shortest, north-south, curves at constant angle. The pseudosphere can be derived from the cylindrical metric
\dot{s}^2=\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2,
where \theta is the azimuthal angle. Assuming z=V(r) the metric becomes
\dot{s}^2=(1+V'^2{r))\dot{r}^2+r^2\dot{\theta}^2. (1)
A surface of constant negative curvature results when
1+V'^2=(a/r)^2, (*) (8
where a is a constant. Integration leads to
V(r)=\pm\{\sqrt{a^2-r^2}-\arcsech(r/a}\},
which is a pseudosphere, two trumpets where horns are pasted together.
If we treat (1) as a Lagrangian, the Euler-Lagrange equations are
\ddot{r}-\dot{r}^2/r-(r^3/a^2)\dot{\theta^2}=0. (2)
The geodesics are are determined by
\frac{d}{dt}(\frac{\dot{r}}{r}=0 (3)
at \dot{theta}=0. The integral is
r=r_0 e^{Ct} (4)
which are exponentially increasing radial lines. This is balanced by a 'flat' angular part which form circles at constant radius. Taken together they produce a surface of constant negative curvature \kappa=-1/a^2.
The surface of revolution depends solely on the radial coordinate, while the angular coordinate is necessary to establish its cuvature. For other choices of the angular curvature, the surface will not be constant. So the radial part determines the trumpet shape while the angular coordinate tells us how to navigate on the surface, and whether or not the curvature will be constant and negative.
If we consider (2) with an external force, F,
\ddot{r}-\frac{\dot{r}^2}{r}-\frac{r^3}{a^2}\dot{\theta}^2=F (5)
we have Weber's equation. The azimuthal Euler-Lagrange equation
\frac{d}{dt}(r^2\dot{\theta}=0. (6)
indicates that angular momentum, h, is conserved so that (5) becomes
\ddot{r}-\frac{\dot{r}^2}{r}-(h/a)^2\frac{1}{r}=F. (7)
Weber, in his 1845 article used (3) to determine the ratio of parallel to transverse charge flows, and found it to be 1/2. Yet, it was incorrect of him to use (3) and then disregard it in determining his equation. Eqn (5) does not suffer from that defect, yet, instead of Coulomb's potential we have an inverse radial law. This occurs in electrodynamics in the radiative zone; the inverse square law becomes inverse radial. Here, it is necessary to establish the proper balance between exponential radial geodesics and circles formed at constant radial distances. These circles are known as horocycles and are equidistance from points at infinity that occur at the ends of the trumpet.
If, instead of angular momentum, we were to consider constant angular velocity, \omega=r\dot{\theta}, Weber's equation,
\ddot{r}-\frac{\dot{r}^2}{r}-\frac{\omega^2}{a^2}r=F,
showing that a harmonic force is involved. But this condition can only occur at r=const, according the the azimuthal Euler-Lagrange equation.
It is strange, to say the least, that Weber didn't realize that (3) produces exponentially increasing geodesics so that his equation does not really belong to the Euclidean plane. Weber offered a synthesis of Ampere's motional forces and the static Coulomb law. But there is nothing sacred about Coulomb's law for if it is replaced by the inverse radial force, then Weber's equation belongs to the a surface of constant, negative curvature!
The angular curvature need not be flat. For if we consider the inverse of (*)
1+V'^2=(\frac{r^2}{a^2}) (**)
we get the hourglass surface,
V(r)=\pm{r\sqrt{r^2-a^2}-a^2\arccosh(r/a)} (8)
where the coefficient, a, determines the throat of the hourglass , shown in the figure.
Now, in order to get a surface of constant curvature, we must consider the angular part of the metric to be exponential,
ds^2=(r/a)^2dr^2+a^2 \exp{(r/a)^2}d\theta^2 . (9)
So here we see a pattern being formed: one component of the metric must be exponential diverging while the other part 'flat.' The geodeisc equation is not
\ddot{r}+\frac{\dot{r}^2}{r}\frac{d}{dt}(r\dot{r}=0, (10)
or r\dot{r}=D, where D is the diffusion coefficient. Another integration gives r^2=2Dt+C,
where if C=0, we get brownian motion! This is surprising since in Weber's view, the kinetic energy term is not retarding charge flow, but it is in the same direction as the acceleration. Yet we get mere diffusion instead of exponential diverging geodesics.
Angular momentum consideration,
e^{(r/a)^2}\dot{\theta}=h,
now leads to the Euler-Lagrange equation.
\ddot{r}+\frac{\dot{r}^2}{r}-\frac{h^2}{a^2}\frac{e^{-(r/a)^2}}{r}=F (11)
In addition to the kinetic energy assisting the flow, instead of retarding it, the force, which is again inverse radial, is modified by a gaussian diffusion about the origin. This is in stark contrast to a Debye screening potential which is exponential decaying. Here, there is symmetric diffusion about the origin which is a singular point. Brownian motion diffusion corresponds to the horocycles of the pseudosphere while the gaussian law of diffusion replaces the expoential growth of the radial geodesics. At least one component of the metric must be exponentially diverging in order to get a surface of constant, negative, curvature.
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