The Genius of Isaac Newton

A Modern Derivation of Newton’s Revolutionary Proof of the Inverse-Square Law

The Genius of Isaac Newton
Image by WikiImages from Pixabay

The genius of Isaac Newton has few parallels in the history of science. His masterpiece, the Philosophiæ Naturalis Principia Mathematica, often referred to as the Principia, is considered one of the most important scientific works in history (see this link). The renowned Indian physicist Subrahmanyan Chandrasekhar, one of the receivers of the 1983 Nobel Prize for Physics, wrote in his essayShakespeare, Newton and Beethoven or patterns of creativity:

“It is only when we observe the scale of Newton’s achievement that comparisons, which have sometimes been made with other men of science, appear altogether inappropriate both with respect to Newton and with respect to the others.”
Figure 1: Woolsthorpe Manor, the birthplace of Isaac Newton (source).

A Bird’s-Eye View of Kepler’s Laws

Between the years 1609 and 1619, the German astronomer and mathematician Johannes Kepler published his three laws of planetary motion that describe the orbits of the planets around the Sun.

Figure 2: A portrait of Johannes Kepler (1610) by an unknown artist (source).

The laws are:

  • The orbit of a planet is an ellipse with the Sun at one of the two foci.
Figure 3: According to Kepler’s first law, the orbit of the planets are ellipses with the Sun at one of the two foci (source).
  • A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. The animation below illustrates Kepler’s second law in action. In a fixed time period, the same blue area is swept out. The purple arrow (directed towards one of the foci of the ellipse) is the acceleration. The other arrow is the velocity (the components of the acceleration are also shown).
Figure 4: The second law in action. The meaning of the arrows is described in the text (source).
  • The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis (see Fig. 11) of its orbit.

Halley’s Visit to Newton

Christopher Wren, the acclaimed English architect, who was also an astronomer, mathematician and physicist, the English polymath Robert Hooke and the English astronomer, mathematician, and physicist Edmond Halley were discussing at a coffee house, after a meeting of the Royal Society, how to prove that the elliptic orbit of the planets was a consequence of the inverse-square force (according to which the gravitational force acting on the planets is inversely proportional to the square of their distance from the Sun). Halley then visited Cambridge to discuss the topic with Newton. The French mathematician Abraham de Moivre later described their meeting:

In 1684 Dr. Halley came to visit him at Cambridge. After they had been some time together, the Dr. asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sir Isaac replied immediately that it would be an ellipse. The Doctor, struck with joy and amazement, asked him how he knew it. Why, saith he, I have calculated it. Whereupon Dr Halley asked him for his calculation without any farther delay. Sir Isaac looked among his papers but could not find it, but he promised him to renew it and then to send it him…
Figure 5: From left to right: Isaac Newton (source), Christopher Wren (source), Edmond Halley (source), and Robert Hooke (source). Note that the identity of the man in the last portrait is still a matter of debate.

Newton’s biographer Gale Edward Christianson, in his book Isaac Newton, describes the events following Halley’s visit:

“[I]n November 1684… eleven months after Halley, Hooke, and Wren… discussion that had started the quest, a copy of De Motu arrived at Halley’s doorstep... [He] was astounded, for in his hands were the mathematical seeds of a general science of dynamics… At first, Newton… thought of De Motu as an end in itself. But once his creative powers were loosed, there was no checking their momentum. ‘Now that I am upon this subject’, he wrote Halley…, ‘I would gladly know the bottom of it before I publish my papers…’ De Motu would serve as the germ of his masterpiece, the greatest book of science ever written. Thus began eighteen months of the most intense labor in the history of science. In April 1686 Newton presented… the first third of his illustrious work. He titled it Philosophiae Naturalis Principia Mathematica…

The figure below (containing pages of the Principia) shows excerpts of Newton’s proofs of:

  • Kepler’s second law (see above).
  • The inverse-square law, according to which the attractive force acting on a planet that is orbiting in an elliptical trajectory around the Sun is proportional to 1/r² and points to one of the foci of the ellipse (this is, in fact, the inverse of the problem described by Halley to Newton).
Figure 6: On the left, the title page of Principia’s first edition (source). In the middle, an excerpt of Newton’s proof of Kepler’s second law of planetary motion. On the right, an excerpt of Newton’s proof of the inverse-square law (source).

To visualize the steps taken by Newton to prove Kepler’s second law of planetary motion, the following (slow) animation (Fig. 7) is useful.

Figure 7: Animation showing the steps taken by Newton to prove Kepler’s second law of planetary motion in the Principia (source).

A Modern Proof of The Inverse-Square Law

The following analysis follows the book by Symon. Consider the gravitational central force of the sun on a planet. A central force is a force F directed along the line joining an object and the fixed origin O of the force. The goal of this article is to determine the motion of a body acted on by such force (the inverse of the problem proposed by Halley, but which was also solved by Newton).

Figure 8: An attractive central force F acting on a body located at P. The force points towards a fixed point O. The variables r and θ are called polar coordinates

Some Simple Kinematics in a Plane

The polar coordinates (rθ) are related to the cartesian coordinates (xy) by:

Equation 1: Polar coordinates.

The unit vectors in the directions xyr, and θ are shown in the figure below.

Figure 9: The unit vectors in cartesian and polar coordinates.

The relations between them are given by:

Equation 2: Relation between unit vectors.

The position vector is equal to:

Equation 3: The position vector r.

Differentiating Eq. 3 twice with respect to time, we obtain, after some simple algebra the acceleration vector expressed in polar coordinates:

Equation 4: The acceleration vector in polar coordinates.

Time Evolution of the Orbiting Body

It is straightforward to show that the angular momentum (the rotational equivalent of linear momentum) of a body acted on by a central force is constant:

Equation 5: The angular momentum of a body acted on by a central force is constant.

Since is constant, the position and velocity vectors, x and v, always remain in a fixed plane (orthogonal to L), substantially reducing the problem’s complexity. Eq. 4, together with Newton’s second law of motion F = ma, give us the equations of motion we want to solve:

Equation 6: Equations of motion for a body acted on by a central force.

Now, if we write the angular momentum as

Equation 7: The plane angular momentum.

differentiate it with respect to t, and use Eq. 5 we obtain the second line of Eq. 6. Integrating Eq. 5 we get:

Equation 8: The constant angular momentum (which depends on the initial conditions).

where the constant L depends on initial conditions. Integrating the first line of Eq. 6 and using Eq. 7 or Eq. 8, we obtain:

Equation 9: Integrating the first line of Eq. 6 we obtain the energy, another constant that depends on the initial conditions.

The total energy E is another constant that depends on initial conditions. Solving Eqs. 8 and 9 for /dt and dr/dt respectively and then integrating, we obtain equations for r(t) and θ(t):

Equation 10: Solving Eqs. 8 and 9 for /dt for dr/dt and then integrating, we obtain equations for r(t) and θ(t).

Note that in Eq.10 there are four constants that depend on initial conditions, namely, namely, ELr₀, θ₀.

Finding the Trajectory

Finding the exact solutions r(t) and θ(t) of the two equations in Eq. 10 is usually quite tricky. What is more straightforward is calculating the trajectory of the orbiting body, namely r(θ).

First, consider Eq. 6 and Eq. 8 and transpose the term containing to the right-hand side of Eq. 6. We obtain:

Equation 11: Definition of the effective potential.

where we defined an effective potential. The two-dimensional problem becomes therefore a one-dimensional problem for r, since /dt was eliminated by Eq. 8.

Defining a new, auxiliary variable u=1/and substituting it into Eq. 11, we obtain, after a little algebra:

Equation 12: Differential equation for the auxiliary variable u.

Since we are interested specifically in the gravitational force acting on an orbiting body, F becomes:

Equation 13: Newton’s gravitational force.

Substituting Eq. 13 into Eq. 12 and solving it, we obtain the equation of a conic section, namely:

Equation 14: Solution of Eq. 12 for the gravitational central force Eq. 13 recalling that u=1/r.

(recalling that u=1/r). To go from Eq. 12 to Eq. 14, we solved the homogeneous equation, found a particular solution, and then put both together.

Figure 10: Types of conic sections: 1. Parabola 2. Circle and ellipse and 3. Hyperbola (source).

In Eq. 14:

  • r=0 is the focus of the conic section
  • θ₀ is the orientation of the orbit in the plane containing the curve.
  • is a constant. Since θ₀ is arbitrary we can choose A to be positive.

We are particularly interested here in the elliptical trajectory. It has two turning points corresponding to θ₀=0 and θ₀=π in Eq. 14. From Eq. 9 and Eq. 11, we see that the turning points occur when the effective potential equals the total energy E. Therefore, solving Eq. 11 for r setting the effective potential to be equal to Ewe obtain two equations, one for each turning point, expressed in terms of E and L. Comparing these equations with Eq. 14 for θ₀=0 and θ₀=π we obtain the value of A:

Equation 15: Value of A in Eq. 14.

From analytic geometry, we know that the equation of the ellipse in polar coordinates is:

Equation 16: The equation of the ellipse in polar coordinates.

where ε is the eccentricity (the value ε of an ellipse is greater than zero but smaller than 1).

Figure 11: The geometrical parameters of the ellipse (based on source).

Eq. 14 for the ellipse can be re-written as:

Equation 17: Eq. 14 for the ellipse re-written in terms of the parameters shown in Fig. 11.

Two important relations obeyed by an ellipse are:

Equation 18: ε and a, written in terms of A and B.

Comparing Eq. 14, and 17 we obtain in terms of the angular momentum L:

Equation 19: Value of B in terms of the angular momentum L.

Using ε=A/we get:

Equation 20: The eccentricity of the ellipse expressed in terms of the constants of motion.

We finally arrive at the equation of the ellipse followed by a body orbiting another body located at one of the foci of an ellipse. It is given by:

Equation 21: Trajectory of a body orbiting another body located at one of the foci of an ellipse.

Thanks for reading and see you soon! As always, constructive criticism and feedback are always welcome!

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