Kepler's Second Law in Velocity Space: Conservation of Angular Momentum
Updated: Jul 22, 2021
As far back as 1846, Hamilton tried to explain orbital motion as time varies, the velocity vector moves in a circle which lies in the same plane containing the origin. This he presented to the Royal Academy under the title "The hodograph, or a new method of expressing in symbolical language the Newtonian law of attraction".
Hamilton contended that if the force of gravity varies inversely with the square of the distance, as shown by Newton, the tips of the arrows trace out a circular curve. Thus, “the angular motion of a body in its orbit is exactly represented, with all its variations, by the circular motion on the hodograph”.
To the Greeks, the universe should be the epitome of symmetry, and a circle is more symmetrical than any other conic. On this notion Ptolemy created his circular orbits of the motion of the planets, and when that failed he added epicycles, or circles upon circles to make up for the errors incurred. Undoubtedly, Kepler's first attempt was to explain the motion of Mars as a circular orbit.
Analogously, Kepler made use of the first derivative in his Stereometria doliorum (1615) to design cylindrical barrels to hold the maximum amount of wine under given conditions. Kepler was prompted to write such a book because of the unsatisfactory barrel of wine he bought for his wedding.
The first derivative needs two points to specify a line; the second needs three and treats the curvature of a circle. Third order derivatives are needed to define affine curvatures for which conics are the only curves of constant affine curvature. The concept of aberrancy goes back to Carnot (pere! Lazare not Sadi) in his treatise on synthetic geometry. He recognized the radius of curvature and arclength as two such tools to measure curvature; the third one being "the angle that is formed, at the point describing the curve, by the tangent and the line which bisects the infinitesimally small chords drawn through the curve parallel to the tangent." In the limit as these chords approach the parallel tangent, an invariant angle is formed: this is the angle of aberrancy.
Now what does this have to do with Keplerian orbits? The crux of the matter lies in the conservation of momentum that allows one to eliminate the frequency between the conservation of momentum and the radial component of acceleration. (The perpendicular component vanishes, that's what gives you the conservation of angular momentum. Setting the radial component of acceleration equal to an inverse square law gives a Keplerian ellipse upon integration.
So, it would appear that one does not have to go beyond acceleration, or, equivalently second derivatives to describe the motion. That the radius vector sweeps out equal areas in equal times is a statement of Kepler's second law. But, what about the nearly perfect orbits that Hamilton claimed existed in velocity space? For that one has to beyond second derivatives! Yet, alas, we will find that it doesn't apply to central forces, as Hamilton had envisioned.
Consider a curve, y=y(x). The curvature,
is given in terms of the first and second derivatives, the prime denoting derivative. Differentiating k, and taking its inverse give the derivative of the radius of curvature
tan A =y'-(1+y'^2)y'''/3y"^2,/2) tan A (*)
where A is known as the angle of aberrancy,
tan A =y'-(1+y'^2)y'''/3y"^2,
a formidable beast involving the third derivative.
Now consider the acceleration vector
a=v* t +v^2/p n,
decomposed in terms of the tangential, t, and normal, n, components. The asterisk will be used to denote time derivatives. The normal component contains the centripetal acceleration, v^2/p. The time rate of change of the acceleration is (affectionately referred to as the "jerk". See S H Shot, "Jerk: The time rate of change of acceleration")
where the time rates of the transverse, t*=(v/p)n and n*=-(v/p)t have been used. The normal component can be expressed as a total time derivative (v^3/p)*. This has the form of the time rate of change of an area--but not an area in configuration space! Its conservation was known to Shot, who failed to realize its physical significance.
The area of a sector
dA=(1/2) v vd\theta=(1/2)v^2/p ds,
where we introduced the arclength ds=p d\theta. Dividing through by dt, and noting that v=ds/dt, gives the rate of change of the area as
Now, if the velocity vector sweeps out equal areas in equal times will A* be a constant. In configuration space this would be proportional to the angular momentum, h, A*=h/2; but not so in velocity space. Takin the time derivative of A* shows that the normal component of the jerk j vanishes. The condition is
3v^2 v*=p*=dp/ds v=3 v tan A
so that the tangential component of the acceleration is
v*=(v^2/p) tan A,
which says that the aberrancy is the ratio of the tangential acceleration to the centripetal acceleration. The condition that this be so is that the angular momentum, per unit mass, L=v^3/p be conserved in velocity space. That the aberrancy is related to the tangential component of acceleration means that no central force, directed along the radius, can have an effect upon it. There is a theorem, called Sciacci's theorem, that allows the acceleration to be evaluated into radial and tangential components. The vanishing of the tangential component requires the conservation of angular momentum in space, which is an additional condition to the one we have employed.
Actually, the conservation of angular momentum in configuration space can be exchanged for the conservation of total energy. In switching from tangential-normal to tangential-radial forms, the tangential component picks up another term,
v*=v(dv/ds) -> v(dv/ds)+(v^2/p cos\theta)(dr/ds).
Now, following Newton, we set the coefficient of the second term equal to F, the force directed to one of the foci of the ellipse. And if F is the negative gradient of a scalar potential,
the above terms give
which has to vanish. This gives v^2/2-V=const., which is the conservation of energy. No mention of angular momentum, or its conservation, is required. Likewise, the conservation of L should also lead to a conservation law.
The conservation of angular momentum can now be seen as the vanishing of the third derivative. The acceleration vector in the transverse-normal form is
a=(v^2/p)[(p/v^2)v* e_t+ e_n].
The normal component can be eliminated with (in the plane)
which when introduced into the acceleration gives
The coefficient of the radial component,
the force directed to the center of force. The vanishing of the tangential component should result in a conservation law:
Since x=r\cos\theta, this is the conservation of angular momentum, h=xv. (Recall that Newton used the complementary angle \alpha, which changes cosine into sine.)
Now if L is conserved,
so that the vanishing of the tangential component of the acceleration implies
\tan A+\tan\theta=0. (**)
Consequently, the third derivative in the expression for the aberrancy vanishes in (*).
In general, the axis of aberrancy and the slope of the curve will not be aligned. We must then expect a tangential component of the acceleration in addition to a radial component of the acceleration. It is important to note that (**) does not involve the conservation of momentum in configuration space, h=xv, but only that in velocity space L=v^3/p.
Finally, under conservation of L, if we introduce the radius of aberrancy as
R=p cos A/(1-dA/d/theta),
j= - (pL/R cos A) t,
directed tangentially, and opposite to the tangential component of the acceleration, v*t.
The conservation of angular momentum in velocity space has the following consequence: The vanishing of the normal component of the jerk, indicating the conservation of L, implies that the acceleration vector is directed to the center of the conic. If the conic is an ellipse, this means that the acceleration vector is directed to the center of the ellipse, and we are back to Hooke's model where the Sun occupies the center of the ellipse.
So when we consider velocity space, we must introduce aberrancy which measures the asymmetry of the curve with respect to the normal drawn through it at any point. Another measure that needs introduction is the affine curvature which involves 4th derivatives. Yet, when we upgrade it a notch instead of considering the affine curvature as [(y")^(-2/3)]"=0, which is proportional to the discriminant of the conic, we use [(z')^(-1/4)]", we come out with the Schwarzian derivative (a 3rd-order invariant), which is a non-zero constant for an ellipse and a hyperbola, and vanishes for a parabola. But that's another story.....