Updated: Sep 4, 2021
In our last blog, we appreciated that Newton could avoid higher than second order time derivatives because the conservation of angular momentum lead to a closed set of equations, once the angular velocity was eliminated.
We want to show here that the corresponding elliptical trajectory in configuration space is, in general, non-circular, but-rather a carthoid. To obtain a circular trajectory in velocity space we are forced to treat the tangential component of the jerk. The normal component vanishes giving the conservation of angular momentum in velocity space: the velocity radius vector traces out equal areas in equal time intervals. The limiting circular hodograph, for an eccentricity equal to 1, occurs for an inverse-fifth power law in the velocity. There are no other integral power laws for arbitrary values of the eccentricity.
In other words, the velocity hodograph must reflect the fact that the tangential velocity in the elliptical orbit is not constant except when the foci coincide. The circular hodograph with a constant velocity vector equal to GM/h can only distinguish between closed orbits with velocity c<GM/h, and open ones c>GM/h. It does not reflect the nature of the elliptical, or hyperbolic, orbits. Independent of the value of c/GM/h, velocity orbits are cardioids.
In 1846 Hamilton introduced the velocity hodograph which, as Feynman appreciated, is a powerful geometric tool. Newton's laws together with the inverse square law always produces a circular velocity diagram...Or does it?
The shape of the orbit depends on where the origin of the velocity diagram is located. If the origin happens to coincide with center, then the distance between the two foci shrinks to zero and the planet has the same speed in all parts of the orbit. If the point is anywhere between the center and the circumference of the diagram, the orbit is an ellipse. The closer the point is to the circumference, the more elongated the elliptical orbit becomes until it becomes a parabola when the point intersects the circumference of the circle....But is it what is commonly believed to be a circle?
We found that when that happens, the eccentricity, or the relative velocity, becomes unity and a parabola results. But the equation we used to derived it was the tangential component of the jerk on the condition that the normal component vanishes. That gave us the condition that equal areas of swept out of velocity space occurs in equal time intervals. This is the analog of the conservation of momentum in configuration space. Now, here's the rub.
The tangential force that gave rise to the limiting position of the point intersecting the circumference of the velocity space orbit was an inverse-fifth force--in the velocity! For any other position, the velocity is no longer a circle intercepting the origin. This is a little strange.
But are we dealing with a circular orbit for the velocity? If x=(x,y,z) represents a system of cartesian coordinates, the angular momentum points in the z direction x X v =h=(0,0,h)
where the velocity vector is tangential
v= (v_x,v_y,0)= R(-sin(theta),c+cos(theta),0)
where R=GM/h, the constant velocity vector and c=(c_1,c_2,0) is a constant vector obtained by integrating Newton's second law of motion. While it is true that
is formally the equation of a circle, it just is a mere identity, R=R. What we are actually after is the equation of the speed,
v=R(1+e^2+2e cos(theta))^(1/2), (*)
where e=|c|/R, the relativity velocity, aka the eccentricity. The angle (theta) is the same as the true anomaly in the equation of the ellipse
r=(h/R)/(1+e cos(theta)), (**)
which is obtained from the velocity vector by applying the definition of the angular momentum h=x X v, and solving for r.
Analogous to the orbit in configuration space (**), the speed (*) determines the orbit in velocity space, not the identity R=R. To conclude that the velocity vector does not traverse the entire hodograph during the motion is inaccurate. That "Eq. (*) shows that the true anomaly is limited; it may just move on a circular arc." The particle in orbit does not move on a circular arc. For small values of the eccentricity, (*) is nearly circular, but as e->1, the orbit is a cardioid, and the angle can go a full 2pi, regardless of the value of e. This is shown in the following diagram.
Also shown is the elongated elliptical orbit in configuration space, corresponding to the cardioid.
In short, you cannot fit a cardioid to a circle, except at extremely small values of e. But the reason for having to do so is physically meaningless.
According to Osipov and Belbruno, velocity space possesses one and only one Riemannian metric ds^2 with the arclength ds=dt/|x|, which is commonly referred to Levi-Civita's regularization procedure. This metric is smooth and has constant positive curvature for total energies less than zero. The geodesics are precisely the circles (or lines) are associated with Keplerian orbits.
However, ds/dt=v, and not 1/|x|. Levi-Civita's regularization is concocted to reduce the order in Newton's equation, |dv/dt|=-GM/r^2 by one, |dv/ds|=GM/r=v^2/2-E, where E is the total energy. It is then necessary to go to the inverse velocity in order to get the Poincare' metric of the disc. It is known that half-circles, or straight lines through the center are the geodesics for constant curvature. However, there is no physics introducing a dimensionality errata formula in order to get a preconceived result. Although it slows down the motion in regions where v is high, it is not physically justifiable.
So why not start directly from the equation
dv/d(theta)= - R(cos(theta),sin(theta),0),
obtained by introducing the conserved angular momentum h=r^2d(theta)/dt?
Since d(theta)/ds=p, the radius of curvature, we obtain upon squaring
(dv/ds)^2p^2=R^2, or ds^2=dv^2,
and the non-Euclidean(ness) of the space has disappeared!
Rather, it is the jerk that governs the orbits in velocity space. The tangential component of the jerk is
where a is the acceleration. In the case the rate of change of the force is given by
where L=v^3/p=v^2 d(theta)/dt, is the conserved angular momentum in velocity space, obtained by requiring that the normal component of the jerk vanish, and c is a constant velocity. The equation of the orbit is simply
where we used v=p d(theta)/dt, and the primes stand for differentiation with respect to (theta).
Introducing the inverse speed, w=1/v, Binet's equation becomes
Since the right hand side equals dF/dt/(Lw)^2, it confirms we are dealing with an inverse-fifth force. Multiplying both sides by w', integrating and setting the integration constant equal to zero give
Reverting to the velocity, we see that the solution to
is, in fact, v=c/2\cos(theta), a circle through the origin. This is the velocity hodograph.
Introducing the arc length through p d(theta)=ds, the metric can be written as
ds^2 = p^2(dv)^2/(c/2)^2(1-2v^2/c^2), (***)
where the square of the radius of curvature plays the role of a conformal factor. This does not have the form of a Riemann metric, except in the case where v is normal to dv, in which case it becomes a Lobachevsky metric in velocity space.
If we can to cast (***) as a bona fide Riemann metric of constant curvature, we need to introduce the transform
2 v/c = z/(1+z^2/k^2),
where k=(2)^1/2 c, Eqn (***) becomes the Riemann metric
with constant positive curvature,k, proportional to the constant amplitude velocity, c. The radius of curvature, p, plays the role of a conformal factor.
But, this is valid only in the limit e=1. For values e<1, the inverse-fifth power law is no longer valid, and the hodograph is no longer circular. In the hyperbolic case, a loop in the cardioid can appear.
Binet's equation can be written in general for power laws as
The only two exponents that lead to an integral power law and a closed orbit are n=0 and n=3. We have discussed the latter in a previous blog.
The n=0 case corresponds to an inverse-square law, which we know leads to an elliptical orbit. The energy equation is
where E is a constant of integration. To obtain an elliptical orbit, E>0 since
where the eccentricity is e=(1-E/c^2)^1/2.