Personal experience tells us that sudden changes in velocity and acceleration are disruptive, and cause erratic motions. Yet, we know there is a velocity analog to Kepler's II law whereby the radial vector sweeps out equal areas in equal times. This, in fact, is what gives rise to Newton's inverse-square law. For without it, we could only deduce that the acceleration is the ratio of the square of the velocity to the radial distance.
The third derivative of the displacement with respect to time brings in what is known as the "jerk". If it so happens that its normal component vanishes, the rate of change of the area is constant, or, in other words, the velocity vector sweeps out equal areas in velocity space in equal intervals of time. This, like the conservation of angular momentum in configuration space, implies a conservation of angular momentum in velocity space. It's this conservation that makes us suspect that there are closed orbits in velocity space that are produced by jerks, just like the inverse-square force applied to one of the foci creates elliptical motion.
It is well-known that to each elliptical orbit in configuration space, there is a hodograph, or velocity, diagram in configuration space. A hodograph is obtained by translating each velocity vector so that its tail is at the origin. Connecting the heads of the velocity vectors produces a hodograph. The hodograph was used by Hamilton to show that planetary orbits are conic sections if, and only, if the central force law is inverse-square. Yet, the same is true for Hooke's law as well.
It turns out that the hodograph of an orbit associated with a central force is the polar reciprocal of the orbit itself. The hodograph of a Keplerian orbit is obtained from its polar by rotating counterclockwise through a right angle and rescaling by a factor of the angular momentum (D Chakerian, "Central force laws, hodographs, and polar reciprocals").
The polar reciprocal depends on the pedal, or the perpendicular to the tangent line through any point, measured from the source. To each and very point where the curve is tangent to the line, there is an associated point on the polar inverse such that if p is the distance from the source to the perpendicular through the tangent line, the inverse 1/p is the distance from the source to the associated point on the polar reciprocal. The inversion theorem says that the polar reciprocal of the original curve is a unit circle centered at the source. In other words, circles under inversion are again circles. This establishes the fact that if the polar reciprocal is a conic section if and only if the original curve is a conic section.
The dual relationship between pedals and their tangent lines allows us to introduce two radii of curvatures, one for the original curve, rho, and the other for its dual, rho*. Rather, than complicating the situation, this leads to a great simplification for we search all those polar reciprocals for which the radius of curvature, rho*, is constant. This can be accredited to Newton himself who used it to establish his inverse square law.
The relation between the radii of curvature is
rho rho*= (r/p)^3. (1)
The ratio of the pedal to the radial coordinate is the sine of Newton's angle alpha which is made by the radial coordinate and the tangent to the curve at any given point. We may also write (1) as
rho rho* = (v/r d phi/dt )^3 (2)
and equating the two expressions lead to
p v = r^2 (d phi/dt) = h (3)
which is the conserved angular momentum per unit mass. Since the ratios always appear as those the total to their corresponding circular components, expressions (1) and (2) may easily be extended to higher motional phenomena, i.e.,
rho rho*= (r/p)^3 = (v/ r d phi/dt)^3 = (a/ v d phi/dt)^3 = (j/a d phi/dt)^3 = (s/j d phi/dt)^3. (4)
where a is the acceleration, j, the jerk and s, the snap. Equating pairs of terms in (4) lead to the definition of acceleration, jerk, and snap in terms of lower order terms:
a= v^2/r, j= a^2/v, and s= j^2/a. (5)
In this light, Newton's dynamical balance with centrifugal acceleration, in terms of the radius of curvature and acceleration is
v^2/rho= a sin alpha= a r d phi/dt/v (6)
gives the first relation in (5) when definition rho d phi/dt -v, has been introduced.
Newton used his expression for the radius of curvature
rho = r (1+z^2)^{3/2} / (1+z^2-z'), (7)
where z= r'/r, is his slope and the prime stands for differentiation with respect to phi. Rearranging (7) and introducing u=1/r, (7) reduces to
rho rho* =(v/r d phi/dt)^3, (8)
which is none other than (2), where
rho* = u" +u (9)
where
v =[(dr/dt)^2 +r^2(d phi/dt)^2] = [(dr/dt)^2+h^2/r^2] (10)
Moreover, (9) can be expressed as, using (6),
rho* = u"+u = a r^2/h^2. (11)
And if it is constant, Newton's inverse-square law must hold, a r^2= constant.
Here, the radius of curvature of the polar reciprocal is expressed in terms of the pedal dual, p*=1/r=u. And introducing (11) into (8), we obtain the first equality in (5) provided
v= rho (d phi/dt). (12)
So it is the conservation of angular momentum in (11) that requires a to be inverse-square.
It should be readily apparent from (4) that we are in store for a whole host of generalizations involving higher order components of the motion,
The time rate of change of the vector acceleration is the vector jerk. In terms of the tangential and normal components, it is given by (e. g., Schot, "The time rate of change of acceleration")
j= (d^2v/dt^2- v^3/rho) t + (1/v) d/dt(v^3/rho) n (13)
Introducing (12) into (13), leads to the vanishing of the normal component of the jerk, if
h v^2 (d phi/dt) =const. (14)
Introducing the polar dual, w=1/v, into the tangential component, and using (14) results in
rho* = w" + w = j/h^2 w^2. (15)
Expression (15) has exactly the same identical structure as the radius of curvature of the support function---except now in velocity space! The condition that the rhs of (15) be constant is that the jerk must decay as the inverse-square of the velocity. Rather, if (3) is imposed instead of (14), the constancy of (15) requires v r^2=const., which is familiar from subsonic mass flow where the mass density is constant: the flow velocity must decrease as the inverse of the surface area if the mass flow is to be constant.
If we introduce (15) into the fourth equality in (4), we come out with
rho =a / d phi/dt, (16)
and not (12). The rate of change of the arc length is the acceleration in velocity space just as the velocity is the rate of change of the arc length in configuration space. This implies Newton's dynamic equilibrium condition (6) becomes
a^2/ rho =j sin(alpha)
where sin(alpha)=a d phi/dt/j.
The solution to (15) is a conic section
w = A + B cos(phi) (17)
where A=j v^2/h^2, analogous to GM/h^2 in configuration space, GM being the gravitational parameter. In principle, there is nothing to prevent closed elliptic orbits, B<A. in velocity space due to the action of a jerk so long as it decays as the inverse-square of the velocity. And they need not be circular orbits either.
The proof of the inverse-square law, whether it be radial distance or velocity, depends on the conservation of angular momentum, i.e., there can be no component out of the plane of rotation.
The relation
rho rho*= (r/p)^3=1/sin^3(alpha) (18)
lead to the condition rho*= constant, when Newton showed that rho sin^3(alpha) was constant. The two forms of the expression for the conservation of angular momentum, r^2(d phi/dt) and pv lead to
rho =(r^2/p) d phi/dphi*. (19)
The polar reciproal radius of curvature will obey an analogous relation
rho* =(r*2/p*)d phi*/d phi. (20)
Multiply the two together gives
rho rho* = r^2 r*2/p p* (21)
rAnd because r=1/p* and r*=1/p, we easily get back the original relation (18). Using v=h r,
and v*=h r*, (i.w., h = p*v=pv*) expression (18) can be written as
rho rho* =v^2v*^2/(r d phi/dt))(r*d phi*/dt) x 1/h^2 (22)
which differs from the original formula by a mere scaling factor, h^2.
We can split (22) into
rho = v^2/r d phi/dt h and rho* = v*2/r* d phi*/dt h. (23)
Introducing rho=v/d phi/dt into the first gives r h=v, or h =p*v. In the second, multiplying numerator and denominator by r*^2 gives
rho =a* r*2/h^2 (24)
which, if constant, is the inverse square law. Note that the accelerations a=v^2/r and
a*=v*^2/r* belong to the original curve, and its polar reciprocal. This requires the angular momentum to belong to both curves in terms of the pedal variables, i.e., r h=v or h=p*v. This is contrary to Chakerian's claim. The acceleration is always directed inwarded toward the source. This is true if a=v^2/r; contrarily, if a*=v*2/r, v* is the radial velocity tangent to the polar curve. But, its distance to the source is r* and not r.
The argument used by Chakerian is that the angular momentum is twice the area or half of the product of the pedal and velocity parallel to the tangent line to the curve. This says that h=pv, or v=hr* is a constant multiple of r*. Not only is
v= h r* or v*=h r (25)
but the acceleration, a, satisfies
a= h v* or a*- h v. (26)
Since a=v^2/r and a*=v*^2/r* these could imply
v^2= h v*r, and v*2=hv r*,
which contradict (25). In order to satisfy (25), one would need to define the accelerations as
a = v^2/r* and a= v*^2/r.
But since r* is not the distance from the given point on the tangent to the curve connecting it to the source, in the first case, and r is, again, not the distance from the tangent to the polar curve connecting it with the source, in the second. One could try to write the acceleration as the square of the geometric mean of velocities, v and v*, but there is no compelling reason to do so since the curve and its polar can be described independently of one another. It is only in the case of angular momentum that the pedal coordinate brings in the reciprocal of the radial coordinate.
Moreover, the invariancy of the angular momentum would require the products vr and v*r* to be constant and equal. Equivalently, h=v/r*=v*/r.
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