On Derivations of Kepler’s Laws

Given that a particle is continuously subjected to a force directed toward a fixed central point, and that the magnitude of the force is inversely proportional to the square of the particle’s distance from that central point, the usual way of determining the motion of the particle is by solving the appropriate differential equation. In this way it can easily be shown that the particle’s path will be a conic, i.e., an ellipse, hyperbola, or parabola, with the central point located at one focus, and that the line from the central point to the particle sweeps out equal areas in equal times. However, Newton’s original derivation in the Principia didn’t make use of differential equations. Instead, the path of the particle was determined by direct integration (or “quadrature”), after establishing the two constants of motion.

To review Newton’s derivation, we should begin by mentioning a fundamental proposition involving the composition of forces, which Newton presented as the first corollary of the basic laws of motion. Consider a particle initially stationary at point A as depicted in the figure below, and suppose that if an impulse I₁ alone is applied to it, the particle would arrive at point B after one unit of time.

Similarly suppose that if an impulse I₂ alone was applied to the particle, it would arrive at point C after one unit of time. Newton asserts, based on his three “laws” of motion, that if both impulses I₁ and I₂ are applied simultaneously to the particle at point A, it will arrive after one unit of time at the point D, which is the opposite point of the parallelogram with sides AB and AC. Of course, we now know that this assertion is actually false, as can be seen most simply by considering the case of two equal and parallel impulses. In such a case, according to Newton’s assertion, if particle A is subjected to both impulses it should travel exactly twice the distance it would travel if subjected to just one of them. This is equivalent to the claim that if either impulse alone would give the particle a speed v, then the composition of both impulses would result in the speed 2v. But we know from special relativity that this cannot be true, because it would imply that the composition of a finite number of such impulses would impart to the particle a speed exceeding the speed of light.

To examine this conflict with special relativity in more detail, let us represent an impulse by a constant force F applied over an incremental time interval Δt. Newton’s assertion signifies that a particle of mass m₀ would be accelerated from rest to the speed v = FΔt/m₀, and hence v is directly proportional to the total impulse FΔt, so doubling the impulse will double the resultant speed. However, according to special relativity (see the derivation in the note on the inertia of energy), the relation between force and acceleration (in the same direction) is given by

and therefore the final speed v_f imparted to a particle by a constant force F applied over an increment Δt is given by evaluating the integral

From this we find that the resultant speed v imparted to the particle is

which is somewhat less than what Newton’s first corollary predicts. Since Newton gives a proof of his corollary, claiming that it follows from the three basic laws of motion, and since those laws (suitably interpreted for contact interactions) are valid in special relativity just as in Newtonian dynamics, one might wonder if Newton’s proof was flawed. Actually his proof was valid, but of course it relied not just on the three laws of motion, but also on the definitions and common notions with which he preceded those laws. Specifically, Newton stipulated absolute space and time, and even gave (in the Scholium preceding the statement of laws) an explicit example of velocity composition, in which a sailor moves eastward with a speed of 1 unit on the deck of a ship that is moving westward with a speed of 10 units on the earth that is truly moving eastward with a speed of 10010 units. Newton says the sailor will be moving truly eastward with a speed of 10001 units, and relative to the earth he is moving westward with a speed of 9 units. Thus, Newton is working within a framework of Galilean spacetime, in which the composition of velocities is purely additive. In this framework, his proof of the corollary is perfectly valid. Inertial coordinate systems are not related by Galilean transformations, but by Lorentz transformations, so velocity compositions (for relatively moving systems of coordinates in terms of which inertia is homogeneous and isotropic) are not purely additive, and hence the parallelogram of forces is not strictly valid.

Still, for velocities that are small compared with the speed of light, Newton’s premises are quite viable, and the first corollary plays a key role in the marvelously successful science of dynamics that Newton presented in the Principia. Indeed the very first Proposition, in which the law of equal areas for orbits with central forces (i.e., the generalization of Kepler’s second law to arbitrary central forces) depends crucially on the parallelogram. The figure below shows how Newton proved that if a particle moves under the influence of a force of arbitrary magnitude directed toward some fixed central point, the line from the center to the particle sweeps out equal areas in equal times.

If the magnitude of the central is zero, a particle moving from A to B will continue moving along the straight line to C at constant speed, so the distance BC traveled in unit time equals the distance AB traveled in the same amount of time. In this case it’s clear that the line from the central point S to the particle sweeps out equal areas in equal times, because the triangles SAB and SBC have equal bases (AB and BC) and equal height AS, and hence have equal areas. Now, as Newton argues in Proposition 1 of the Principia, suppose when the particle reaches location B it is subjected to an impulse directed toward the central point S, such that a particle initially at rest at B would be driven to point E in one unit of time. We also know that the impulse of the particle along the line AB would drive the particle to point C after a unit of time. Therefore, invoking the parallelogram, Newton asserts that the combined effect of the inertia of the particle and the central impulse would drive the particle to point D after a unit of time. It is evident that the swept area during this time interval, represented by the triangle SBD, is equal to SBC, because both of these triangles are on the base SB, and both have the same height above this base line (since CD is parallel to SB). This same reasoning applies regardless of the magnitude of the impulse, provided only that it is directed toward the central point S. We can then imagine an impulse applied to the particle at D, again directed toward S, and again we find that the swept area in unit time is the same. Newton then asks us to consider increasing the number of triangles and reducing their widths, until we have a continuous curved path, and the particle is being subjected to a continuous central force.

It’s noteworthy that the very first Proposition in the Principia makes an argument based on taking a limit, which immediately distinguishes it from all the Euclidean works that it so closely resembles superficially. The transition between a polygonal path, subject to a finite sequence of discrete impulses, and a continuous curved path, subject to a continuously applied force, is fairly intuitive, although from the modern standpoint of mathematical rigor we might expect the transition between discrete and continuous action to be given a more explicit justification. The so-called limit paradox illustrates that the properties of the continuous limit of a sequence of polygonal paths need not be the limit of the corresponding properties of the polygonal paths. However, in Newton’s first Proposition, it isn’t hard to show that the requisite properties do indeed carry over, so the proof could be made rigorous – marred only by the fact that the assumption of Galilean spacetime which underlies the parallelogram assertion is empirically false. Moreover, it’s conceivable that gravitational action actually is quantized at some (presently imperceptible) level, so the fact that Newton’s demonstration covers the discrete as well as the continuous cases could even be seen as a strength.

Using vectors and the (now) well-known properties of vector operations, including differentiation, the proof of the law of equal areas (in the context of Galilean spacetime) is fairly immediate. Letting r denote the position vector of the particle relative to the central point of attraction, we simply note that the area continuously swept out by this vector per unit time is given by the magnitude of the cross product, and the rate of change of this is, by the chain rule of differentiation

The cross product of parallel vectors is zero, so the second term on the right side is identically zero, and the first term is zero if the acceleration vector is parallel to the position vector, which is to say, if the force (which is proportional to the acceleration) is directed toward the origin. Thus for any central force the rate of change of the area sweeping rate is zero, so equal areas are always swept out in equal times. Letting h denote the vector , we recognize this as the angular momentum, and its constancy simply expresses the conservation of angular momentum in the absence of any tangential forces. The magnitude of this conserved vector is

where we’ve let v denote the magnitude of  (which of course is not the same as ), and we’ve made use of the identity , which follows immediately by differentiating both sides of the relation . The squared speed v² equals the sum of the squares of the radial and tangential components of the velocity, which are rω and  respectively, where ω is the angular speed. Substituting for v² into the above expression gives the constant of motion h = r²ω for a particle subject to a central force.

Newton also discovered (based on his Galilean premises) another conserved quantity for orbital motion subject to a purely central force, namely, what we would call the energy of the particle. This is slightly ironic, since kinetic energy came to be regarded as an invention of Leibniz, who called it “the living force” (vis viva), as distinct from “the dead force” (vis mortua) of momentum, the latter being regarded by the Newtonians as the more fundamental. Nevertheless, the conservation of the sum of vis viva and potential energy of a particle (the latter corresponding to the work done on the particle by the force of gravity) is clearly demonstrated by Newton in Propositions 39 and 40 of the Principia, and used in Proposition 41 to give an explicit construction for the orbit of a particle subject to any central force.

Newton begins by showing that the change in the speed of a particle as it moves from one “height” to another in a central force field is independent of the path traveled. This can be seen by considering two particles, initially at the same location and with the same speeds, but one moving directly toward the center of force, and the other moving at some angle, as illustrated below.

We are considering an arbitrarily small region, so the central force (which acts in the “downward” direction in this figure) is essentially constant between the initial height r₁ and the subsequent height r₂. The particles both begin at point A, moving with speed v, and both are subject to the same downward acceleration α. One of the particles is moving directly downward toward point B, and the entire downward acceleration contributes to increasing its speed. The time required for this particle to descend from r₁ to r₂ approaches (r₁ – r₂)/v, since the variation in speed over the interval can be made arbitrarily small by considering a smaller interval. Therefore, the speed of the particle when it reaches point B is v + α(r₁ – r₂)/v. Taking this to the limit for arbitrarily small intervals, the differential change in speed for a differential change in radial position is given by dv = αdr/v. On the other hand, the time required for the particle moving with speed v at an angle θ from the downward direction to descend from r₁ to r₂ is greater by a factor of 1/cos(θ), because the distance it must travel is greater by this factor. Also, note that the component of the acceleration parallel to AC will contribute to increasing the speed of the particle, whereas the perpendicular component of the acceleration merely acts to change the direction. The component of acceleration parallel to AC is a factor of cos(θ) smaller than the full acceleration (again relying on the parallelogram assertion). Thus the rate of acceleration is smaller but the duration of the acceleration is greater, and the product of the acceleration and the duration is the same, so the two particles continue to have the same speed in their respective directions.

It follows that we can determine how the speed varies with “height” (i.e., radial distance from the center of force) for a particle moving purely radially, and this same relationship will be applicable to particles following any other paths. Recalling that dv = αdr/v, we have vdv = αdr, so if we know the radial acceleration α as a function of r, we can directly integrate this relation. In particular, suppose α(r) = –M/r² for some constant M. This represents an inverse-square law of attraction. In this case we write

Evaluating the integrals, we get

Re-arranging the terms, it follows that

and therefore the quantity

is constant. Now, as we noted previously, the squared speed v is the sum of the squares of the radial and tangential components of the velocity, so we have

Recalling that h = ωr² is a constant, we can substitute for ω = h/r² in this expression to give the relation

Substituting this expression for v² into the equation for the constant energy E, and solving for dr/dθ, we get

Thus the relation between q and r is given by the integral

Evaluating the integrals, we get

Thus we can put

and then solve for r to give the expression

which is the equation of a conic section. Since E and h are constants, we can evaluate them at any convenient point, such as at a point of nearest approach (perigee), where the radial distance from the center is r₀ and the speed is purely tangential with magnitude v₀. In this case we have v₀ = ω₀r₀, and so h = r₀v₀. We also have E = v₀²/2 – M/r₀. Making these substitutions into the above equation, we arrive at the equivalent expression for the conic path

It’s interesting that, aside from the fairly trivial integrations giving the expressions for the kinetic and potential energies (each being the inverse of the classical form d(xⁿ)/dx = nx^n–1), we can begin with the premise of a central inverse-square force and reach the conclusion that the path must be a conic section entirely by purely algebraic steps, except for the integration of the form

and conversely to prove that conical paths with a central force imply an inverse-square force law we need only perform one non-trivial differentiation. As discussed in another note, Newton presented a full demonstration of this latter proposition in the Principia (in Propositions 11 to 13 of Book 1), but in the first edition omitted a corresponding explicit demonstration of the inverse proposition. Instead he merely asserted that the inverse proposition “follows”. In later editions he amplified on this a bit, and ultimately (by the third edition) gave a fairly complete sketch of a proof, but it was still of a non-constructive nature, compared with the demonstration of the direct proposition. Only in Proposition 41 did he give a constructive demonstration, but only “granting the quadratures of curvilinear figures”. In other words, Newton didn’t actually present the integration, he merely indicated how to determine the path in terms of the specified integral.

It is known that Newton came to his version of the calculus by way of differentiation first, and only subsequently dealing with integration, and identifying these two operations as the inverses of each other. This is in contrast with Leibniz, who began with integration and then linked it with differentiation. In the first edition, Newton labored to avoid introducing calculus in algebraic form, and it’s conceivable that he found it difficult to handle the quadratures of complicated functions in the synthetic geometric style of the Principia. This might explain why he gave a full demonstration that conic orbits imply inverse square force (which can be done purely with differentiation), but only a sketch of a proof for the inverse proposition (which requires integration). Ironically, the link between the two propositions relating the force to the shape of orbits consists of the assertion that integration and differentiation are the inverses of each other, which is nothing other than the fundamental theorem of calculus. It’s clear that Newton regarded knowledge of this inverse relationship as his own private possession, which he was not willing to share with others, because he encrypted a statement of this in the form of an anagram in a letter to Leibniz. Perhaps his reticence about presenting a complete proof of the inverse problem of orbits was due to his desire to keep that knowledge secret. The alternative explanation – that he failed to recognize the connection between the pairs of inverse relationships – seems improbable.

Incidentally, the integration involved in Newton’s derivation for inverse square forces is one with which he could well have been familiar from his purely mathematical studies. Since sin(π/2 – θ) = cos(θ), we know that asin[cos(θ)] = π/2 – θ, and therefore

If we then put x = cos(θ) we have dx = –sin(θ)dθ, and making these substitutions into the above expression we get

Hence we have the integral

which is precisely the kind of relation that was studied by Newton in his investigations of infinite series derived from the binomial theorem. The series expansions for several inverse trigonometric functions were published at about the same time by James Gregory. Given the derivative for arcsin(x), it’s simple to employ the chain rule to find the derivatives of nested function, such as

The right hand expression is precisely the expression to be integrated in the orbital derivation, provided we set

which of course furnishes the integration shown previously. It’s interesting that one the integrals he had studied in his younger days from a purely mathematical perspective was also the very operation needed to derive conical orbits from the inverse-square central force law. Nevertheless, Newton resisted what must have been a great temptation to present this explicit derivation in the Principia.

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