The Hidden Motion Behind Kepler's Laws
Updated: Sep 1, 2021
It was Newton's good fortune to be able to solve the orbital motion of a planet about the sun using the only the velocity and acceleration. There was thus no need to go beyond second derivatives to get a complete description of the motion. It, therefore, remained a curiousity that Hamilton's finding in 1846 that as time varies, the velocity vector moves in a circle which lies in the plane containing the origin.
In fact, the eccentricity of the orbit is the ratio of the center of the velocity circle to the velocity of the circle. In the limit that the eccentricity tends to unity, the configuration space orbit degenerates into a parabola, and the velocity space circle goes through the origin. Newton found such behavior for the configuration space orbit, but that also remained a curiousity since it would correspond to a penetrating orbit that goes through its source.
There is a certain orthogonality between configuration space and velocity space orbits. At each point, the planet's radial vector is perpendicular to the radius vector in the velocity of the circle. The central force is directed radially inward in configuration space. We can say that in configuration space is parallel to the radius vector, while, in velocity space, the "kick", or jerk is tangent to the velocity circle, and, consequently, normal to the velocity vector. So what governs the dynamics in velocity space?
It is, indeed, odd that no one has gone to higher derivatives to analyze the dynamics of the nature of the "kicks" in velocity space. Just as the angular momentum, h, is conserved in configuration space, the velocity space angular momentum, L, is conserved by the velocity circle. We may say that the velocity vector sweeps out equal areas of velocity space in equal times, just as the configuration angular momentum does in configuration space.
The time rate of change of the vector acceleration is obtained by differentiating the acceleration vector with respect to time. The jerk vector is
in the tangential, t, and normal, n, directions, where p is the radius of curvature. As far back as 1845, Transon was to first to recognize that only the normal component of the jerk, J, depends on aberrancy, which is a measure of the local asymmetry of a curve: Just as the second derivative is to the radius of curvature, measured by an osculating circle, the third derivative is to the aberrancy, measured by an osculating parabola. If the normal component of the jerk vanishes, then, and only then, will the acceleration vector be parallel to the axis of the parabola.
[Parenthetically, it is only right we pay homage to Lazare Carnot (pere of Sadi) who introduced the notion of aberrancy, although it was Transon who obtained the explicit expression for it.]
Call L=v^3/p, the velocity space angular momentum. If we multiply numerator and denominator by the change in angle, and identify the product of the radius of curvature and the angular change as the change in the arc length, we get
where v is the ratio of the change in the arc length to the change in time, v=ds/dt. The tangential component of the jerk will give the time rate of change of the tangential force,
in direct analogy with its configuration counterpart. If we switch to inverse velocity, w=1/v,
we get Binet's equation in velocity space
where the prime denotes differentiation with respect the tangent angle, theta.
The radius of aberrancy is defined as
This rather formable looking expression can be simplified considerably by using the constancy of the velocity space angular momentum, so that it will reduce to
R=p/cos(delta) (w+w"), (**)
where delta is the angle of aberrancy:
Introducing (*) into (**) and rearranging gives
In comparison with the expression for the radius of curvature, where there is a cubic correction cos(theta) to the fact that the curve is not unit speed, there is only the first power correction when it comes to the aberrancy.
Now the eccentricity in configuration space is defined as the ratio of the distance the center,c, of the velocity circle is from the origin and the radius vector. When the two become equal, the orbit passes through the origin, and the corresponding orbit in configuration space is a parabola. If the direction theta=0 passes through the center of circle, the polar equation of the orbit in velocity space is
when this is plugged into Binet's equation, we get a fifth order time rate of change of the force,
dF/dt=8(L c)^2/v^5, (***)
but this time in velocity space!
The denominator of the radius of aberrancy depends on the fourth derivative of the curve, y=y(x), as well as the low-order derivatives. It is called the affine curvature in differential geometry. For a central conic, the affine curvature is a non-zero constant. However, for a parabola it vanishes, and consequently the parabola has its center of aberrancy at infinity. This occurs when the center of the velocity circle is equal to its radius. That this ratio of velocities is unity can be considered as the ultra-relativistic limit.
Just as the first derivative determines a unique tangent line to a curve making 2-point contact, the second derivative fixes a unique osculating circle that makes a 3-point contact. In the same way the third derivative determines a unique parabola making a 4-point contact.
The affine curvature is related to the Schwarzian derivative, which represents curvature.
The affine curvature,
is related to the Schwarzian derivative by the substitution y"=z'^(3/4), giving a third order derivative,
which for a fractional linear transform its constant.
The vanishing of the Schwarzian separates metrics of different constant curvature, k>0 and k<0, for elliptic and hyperbolic geometries, respectively. That it vanishes for the parabola, k=0, is related to the that the velocity circle passes through the origin velocity space. The orbit is described by an inverse-fifth law in the velocity, and it represents tangential "kicks" that are compensated in such a manner that at the end of the orbit, the particle returns to the same, initial, velocity.
The position of the velocity circle in velocity space determines the nature of the motion in configuration space. If the velocity circle encloses the origin, the motion will be elliptical in configuration space. On the other hand, if it doesn't enclose the origin, the motion will be hyperbolic. The separation occurs when the velocity orbit passes through the origin. This corresponds to a parabola. An osculating parabola replaces the osculating circle for a 4-point contact. If the normal component of the jerk vanishes, the acceleration vector is directed toward the center of the conic. However, if the conic is a parabola, the center recedes to infinity, and the acceleration vector is parallel to the axis of the parabola. This occurs when the circular orbit passes through the origin, and it is characterized by the vanishing of the Schwarzian.
Thus, the higher-order dynamics represented by the jerk is not superfluous to the Keplerian orbit, but offer a greater understanding of the nature of the motion and the forces at work in velocity space. For instance, an inverse-square law,
would produce an elliptic orbit in velocity space. The configuration and velocity space forces being related by the ratio L/h, the ratio of angular momenta in velocity and configuration space. But, this would be on the level of acceleration, and not jerks. In fact, equating the force (iv) with the centripetal force, v^2/p, results in v=GM/h, which is the constant radius of the velocity circle. So the inverse square law in velocity space (iv) produces a circular orbit.
As we go around an orbit, the distance D goes through an angle d(theta). The distance would be that in velocity space, Fdt, a velocity. the radial velocity would then be given by
Introducing (iv) results in
v=(L/h) (GM/v^2) (dt/d(theta)).
Using the conservation of L,
We come out with the well-known result that the magnitude of the velocity vector equals the gravitational parameter, GM, divided by the angular momentum, h. Since we used (iv), this also applies to an elliptical orbit in velocity space!