Section 10–1 Newton’s Third Law

(Mathematical analysis / Three-body problem / Action equals reaction)

In this section, the three interesting points discussed are mathematical analysis, three-body problem, and action equals reaction.

1. Mathematical analysis:

“First, there are quite simple cases of motion which can be analyzed not only by numerical methods, but also by direct mathematical analysis (Feynman et al., 1963, section 10–1 Newton’s Third Law).”

It seems that Feynman tries to fill a gap of the previous lecture although he is supposed to introduce Newton’s third law of motion. Firstly, he mentions that there are simple cases of motion which can be analyzed not only by numerical methods, but also by mathematical analysis. As an example, it is possible to use mathematical analysis to show a planet’s orbit is elliptical instead of using numerical methods as discussed towards the end of the previous lecture. Feynman is referring to the use of a central force that is inversely proportional to the square of the distance (between the Sun and a planet). This two-body problem is sometimes known as the Kepler problem that can be solved by integration using polar coordinates (r and θ).

Instead of discussing Newton’s second law of motion in the previous lecture, there could be some discussions on how Newton developed his third law. In one of his scholium of the Principia, he acknowledges the contributions of Sir Christopher Wren, Dr. Wallis, and Huygen. In his own words, Newton (1687) writes that “Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject (p. 26).” In addition, Newton discusses the experiment related to collisions of hanging pendulums.

2. Three-body problem:

“That is the famous three-body problem, which so long challenged human powers of analysis… (Feynman et al., 1963, section 10–1 Newton’s Third Law).”

Feynman possibly fills another gap of the previous lecture by discussing the three-body problem, which challenged human powers of analysis for a long time. For example, if there are two bodies going around the sun, there is no simple formula for the motions of the three bodies. Furthermore, there are situations such as the motion of the stars in a globular cluster where both methods fail. Although we can use direct mathematical analysis or numerical methods to solve simpler problems, it is still challenging (if not impossible) to achieve exact solutions for very complicated problems involving many bodies. Interestingly, Feynman bridges the difficulties of solving Newton’s second law to the need of principle of conservation of linear momentum that is the foundation of Newton’s third law.

The three-body problem can be cited as a classic example of a chaotic system. This problem dates back to the 1680s when Newton shows that his law of gravitation could predict the orbit of two bodies held together by gravity—such as the Sun and a planet. In 1889, King Oscar II of Sweden offered a prize to anyone who could solve the three-body problem. Henri Poincaré won the King’s prize without solving the general problem. Poincaré simplified the problem by considering two enormous bodies and a third that was very small: the small particles would move in a complicated (or chaotic) manner around the two large astronomical bodies. Recently, Šuvakov and Dmitrašinović (2013) identify 13 new families of solutions by tweaking an existing solution on a computer simulation.

3. Action equals reaction:

“This principle is that action equals reaction (Feynman et al., 1963, section 10–1 Newton’s Third Law).”

According to Feynman, the principle of Newton’s third law is mainly action equals reaction. In other words, if particle A exerts a force (F_BA) on particle B, then particle B will push on particle A with an equal force (F_AB) and in the opposite direction. This pair of forces effectively acts in the same line. Alternatively, one may explain that there are five features of Newton’s third law: (1) forces occur in pairs; (2) the nature of the two forces are the same (e.g., gravitational or electromagnetic); (3) both forces have the same magnitude; (4) both forces acts along the same line but in opposite directions; (5) both forces act on a different object. In a sense, the principle of Newton’s third law is not simply action equals reaction.

In this chapter, Feynman assumes that action equals reaction is a truth. (In Volume 2, Section 26.2, he discusses the failure of the law of action and reaction when two charged particles are moving on orthogonal trajectories.) A special case of action equals reaction is to consider a third particle that is not on the same line as the other two particles. Importantly, the “net” forces on the first two particles are neither equal nor opposite. Mathematically, the forces on the three particles, say A, B, and C, can be resolved into parts such that F_AB = -F_BA, F_BC = -F_CB, and F_AC = -F_CA. If there are four particles, then we expect ⁴C₂ = 6 combinations of particles that have corresponding components of mutual interaction that are equal in magnitude and opposite in direction.

Questions for discussion:

1. What are the simple cases of motion which can be analyzed by numerical methods and direct mathematical analysis?

2. What are the difficulties in solving the three-body problem?

3. How would you state the principle of Newton’s third law of motion?

The moral of the lesson: if the first particle exerts a force on the second particle and pushing it with a certain force, then the second particle will push on the first with an equal force and in the opposite direction; these two forces act in the same line.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Newton, I. (1687/1995). The Principia (translated by A. Motte). New York: Prometheus.

3. Šuvakov, M., & Dmitrašinović, V. (2013). Three classes of Newtonian three-body planar periodic orbits. Physical review letters, 110(11), 114301.