Section 7–3 Development of dynamics

(Galileo’s principle of inertia / Newton’s force / Sun might be the angel)

In this section, the three interesting points are Galileo’s principle of inertia, Newton’s force, and the Sun might be the angel.

1. Galileo’s principle of inertia:

“Galileo discovered a very remarkable fact about motion, which was essential for understanding these laws. That is the principle of inertia (Feynman et al., 1963, section 7.3 Development of dynamics).”

It is controversial to say that Galileo discovered a principle of inertia in which a body may keep on coasting forever in a straight line. First, Galileo did not explicitly state a theorem or a general principle of inertia. Next, Galileo suggested a concept of circular inertia: “a ship, for instance, having once received some impetus through the tranquil sea, would move continually around our globe without ever stopping and placed at rest it would perpetually remain at rest, if in the first case all extrinsic impediments could be removed, and in the second case no external cause of motion were added (Galilei, 1613, pp. 113–114.)” In other words, a body may continue in its state of circular motion unless it is compelled to change that state by an external force.

In Principia, Newton (1687) states the First Law of motion (Motte's translation) as “Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon (p. 19).” In explaining his First Law, Newton writes that “A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.” In general, Newton suggested two kinds of uniform motion: progressive motion in a right line and circular motion (Westfall, 1971).

2. Newton’s force:

“Newton modified this idea, saying that the only way to change the motion of a body is to use force (Feynman et al., 1963, section 7.3 Development of dynamics).”

According to Feynman, Newton’s second law of motion means that the acceleration produced by a force is inversely proportional to a mass, or the force is proportional to the mass times the acceleration. Simply put, a force is needed to change the speed or the direction of motion of a body. In Principia, Newton (1687) states the Second Law as “[t]he alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed (p. 19).” The term motion means momentum and thus, Newton has specified a relationship between a motive force (F) and a change of momentum (Δmv). The equation F = ma is not stated in Newton’s second law.

Currently, physicists define force as an interaction between two objects rather than an innate property of an object (Brookes & Etkina, 2009). In Principia, Newton (1687) writes that “vis insita, or innate force of matter is a power of resisting, by which everybody, as much as in it lies, endeavors to preserve in its present state, whether it be of rest, or of moving uniformly forward in a right line (p. 9).” In short, Newton views inertia as an innate force of a body that is different from the motive (external) force. One may debate whether Newton’s innate force has a connotation of momentum or energy. It is not surprising that the law of conservation of force was developed subsequently (Nicolson, 1871).

3. Sun might be the angel:

“If there is a force toward the sun, the sun might be the angel, of course! (Feynman et al., 1963, section 7.3 Development of dynamics).”

Feynman explains that the brilliant idea in planetary (circular) motions is that no tangential force is needed to keep a planet coasting in its orbit. On the contrary, a presence of tangential force would increase the speed of the planet in a circular orbit. (The speed of a planet in an elliptical orbit can be changed by a central force because the path of the planet is not perpendicular to this force.) More important, the force needed to control the planetary motion is not a force around the sun but toward the sun. This force causes the actual motion of a body to deviate from the line of which the body would have gone. Interestingly, Feynman adds that if there is a force toward the sun, the sun might be the angel.

Kepler is sometimes cited for asserting planetary motion was driven by angels about the sun. Historically speaking, Kepler proposes the existence of a magnetic force between the planets and the Sun. Thus, it is not true that Kepler hypothesizes angels pushing the planets around the sun. In The Divine Comedy (c. 1308 to c. 1320), Dante writes “To those celestial lights, that towards us came, leaving the circuit of their joyous ring, conducted by the lofty seraphim… moves the third heaven… (Alighieri, 1909, p. 317).” Note that a seraphim is an angel that has six wings. Medieval scholars debated whether the angels were external or internal movers and whether the angels’ powers were independent of God (Tubbs, 2009).

Questions for discussion:

1. Should the principle of inertia be applicable to circular motions?

2. How is Newton’s original concept of force different from current physicists’ formulation of force?

3. Why did Feynman explain that the force needed to control the motion of a planet around the sun is a force toward the sun because of the principle of inertia?

The moral of the lesson: the force needed to control the motion of a planet around the sun is not a force around the sun but toward the sun.

References:

1. Alighieri, D. (1909/2009). The Divine Comedy (translated by Henry F. Cary). New York: Cosimo.

2. Brookes, D. & Etkina, E. (2009). Force, ontology, and language. Physical Review, Special Topics, Physics Education Research, 5, 010110.

3. 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.

4. Galilei, G. (1913). Letters on Sunspots (translated by S. Drake). In G. Galilei (1957). Discoveries and Opinions of Galileo. New York: Doubleday.

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

6. Nicolson, J. (1871). The Conservation of Force. Nature, 4, 47-48.

7. Tubbs, R. (2009). What Is a Number? : Mathematical Concepts and Their Origins. Baltimore: Johns Hopkins University Press.

8. Westfall, R. S. (1971). Force in Newton’s physics. London: Macdonald.

Section 7–2 Kepler's laws

(Kepler’s First law / Kepler’s Second law / Kepler’s Third law)

In this section, the three physical laws discussed are Kepler’s First law, Kepler’s Second law, Kepler’s Third law.

1. Kepler’s First Law (Law of orbits, 1609):

“Each planet moves around the sun in an ellipse, with the sun at one focus (Feynman et al., 1963, section 7.2 Kepler’s laws).”

Kepler found that each planet is orbiting around the Sun in an elliptical path with the sun at a focus of the ellipse. Feynman explains that an ellipse is not just an oval, but it is a curve that can be obtained by using a string and pencil method in which the sum of whose distances from two fixed points is a constant. In other words, Feynman defines an ellipse as a collection of points in a curve in which the sum of whose distances from the two foci is constant. In a “lost lecture” of Feynman on planetary motions, he provides another definition of ellipse: “if you wish, these two points are called the foci, and the focus means that light emitted from F will bounce to F’ from any point on the ellipse (Goodstein & Goodstein, 1996, p.150).”

In general, an ellipse can be mathematically represented by a polar coordinate equation: r(θ) = r₀/(1 + e cos θ) where r is the distance from a planet to the Sun, r₀ is the semi-latus rectum of the ellipse, e is the eccentricity of the ellipse, and θ is the angle with respect to the planet’s original position. (The term semi-latus rectum is a compound of the Latin: semi means half, latus means side, and rectum means straight.) Strictly speaking, the path of a planet is approximately elliptical because there are gravitational influences from other planets and celestial bodies such as comets and moons. Importantly, the first law provides a revolutionary idea based on astronomical observations that planetary orbits are elliptical instead of circular.

2. Kepler’s Second Law (Law of equal areas, 1609):

“The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time (Feynman et al., 1963, section 7.2 Kepler’s laws).”

Kepler’s second law refers to his observations that the Sun-planet line (an imaginary line connecting the Sun and a planet) is sweeping area at a constant rate; it can be mathematically represented as dA/dt = constant. This is based on Tycho’s data on the motion of Mars in which it moves faster when it is nearer to the Sun and slower when it is farther away from the Sun. This law provides another revolutionary idea: planets move around the sun at varying speeds. In essence, the orbital speed of a planet is related to how the Sun-planet line “sweeps out” equal areas in equal times.

Kepler initially ponders on whether a decrease in velocity of a planet is a cause of an increase in distance from the Sun, or vice versa. In his book titled The Cosmographic Mystery, Kepler (1596) conceived of a planet as being moved by a “motive soul.” Based on the astronomical data inherited, Kepler proposes that the planets are moving in their orbits due to a central force from the Sun. In addition, Kepler guesses that the central force is magnetic in nature because it was known that the Earth is a magnetic body. Note that the central force from the Sun does not provide a torque, and thus, the angular momentums of the planets remain constant.

Note: In Kepler’s laws, the Sun is assumed to be stationary. To be more accurate, the Sun and planets rotate about the center of mass of the solar system.

3. Kepler’s Third law (Law of periods, 1619):

“The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits (Feynman et al., 1963, section 7.2 Kepler’s laws).”

Feynman mentions that Kepler’s third law was discovered much later and this law is of a different category from the other two laws, because it does not focus on a single planet, but relates one planet to another. In essence, this law states that the square of the orbital period of a planet is proportional to the cube of the orbital size. Mathematically, it can be represented as T² µ d³ in which T is the time interval it takes a planet to move one round about its orbit and d is the major axis that can be expressed as the length of the greatest diameter of the elliptical orbit. Simply put, if the planets move in circles, as they nearly do, the time required to go around the circle is proportional to the 3/2 power of the orbital diameter (or radius).

Some physicists prefer to state the Kepler’s third law as a relationship of the semi-major axis of the orbit to its sidereal period. Interestingly, Kepler (1619) writes that “it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances, that is, of the actual spheres (p. 411).” Thus, one may write “mean distances” to the Sun instead of “semi-major axis.” More important, the orbital period of a planet is also dependent on its semi-minor axis. Furthermore, physicists should realize that this third law is based on Kepler’s calculation (or manipulation) of Tycho’s data (Donahue, 1988). However, Tycho’s data are not as superb as some scholars claimed because they were observed by naked eyes without the use of a telescope.

Questions for discussion:

1. What is a good definition of an ellipse for Kepler’s first law?

2. Could you explain how the durations of summer and winter are slightly different by using Kepler’s second law?

3. Should Kepler’s third law be stated in terms of mean distances or semi-major axis?

The moral of the lesson: the motions of planets are in elliptical orbits and do not have constant speeds; their orbital periods are dependent on the size of orbits.

References:

1. Donahue, W. H. (1988). Kepler’s fabricated figures: Covering up the mess in the New Astronomy. Journal for the History of Astronomy, 19(4), 217-237.

2. 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.

3. Goodstein, D. L., & Goodstein, J. R. (1996). Feynman’s Lost Lecture: the motion of planets around the sun. New York: W. W. Norton & Company.

4. Kepler, J. (1596). Mysterium cosmographicum (The Secret of the Universe). Tübingen: Georgius Gruppenbachius.

5. Kepler, J. (1619). Harmonice Mundi (The Harmony of the World). translated by E. Aiton, A. Duncan & J. Field. Philadelphia: American Philosophical Society.

Section 7–1 Planetary motions

(Law of gravitation / Nicolaus Copernicus / Tycho Brahe)

It may seem strange that the lecture on Newton’s law of universal gravitation is delivered before Newton’s three laws of motion. However, this chapter on the law of gravitation is further improved in the first Feynman’s Messenger lecture delivered at Cornell University. In this section, the three interesting points discussed are the law of gravitation as well as the contributions of Nicolaus Copernicus and Tycho Brahe.

1. Law of gravitation:

“…every object in the universe attracts every other object with a force which for any two bodies is proportional to the mass of each and varies inversely as the square of the distance between them (Feynman et al., 1963, section 7.1 Planetary motions).”

Feynman states the law of gravitation as every object in the universe attracts every other object with a force which for any two bodies is proportional to the mass of each and varies inversely as the square of the distance between them. Mathematically, the law of gravitation can be expressed by the equation, F = Gmm′/r². Importantly, Newton (1687) writes that the force “will be reciprocally proportional to the square of the distance of the centers (p. 158).” Thus, some physicists prefer the term distance to be more precisely stated as the “distance between centers of mass of the two objects.” Alternatively, Newton’s law of universal gravitation can be stated as “[e]very particle attracts any other particle with a gravitational force of magnitude F = GMm/r² (Halliday, 2005, p. 331).” Essentially, this law is applicable to particles.

Historically, Ismaël Bullialdus (1605-1694) published a book titled Astronomia Philolaica in 1645 and suggested an inverse square force law before Hooke and Newton. Bullialdus argued that Kepler’s planetary force if it existed, would diminish according to the inverse square of the distance just like Kepler’s law of light propagation. That is, Bullialdus did not agree with the existence of this force because of his metaphysical tenet that a moving object must contain in itself a principle of its motion (Jammer 1999). Importantly, Newton (1687) gave a mathematical justification for applying the inverse square law to large spherical objects as if they are particles.

2. Nicolaus Copernicus:

“…The story begins with the ancients observing the motions of planets among the stars, and finally deducing that they went around the sun, a fact that was rediscovered later by Copernicus (Feynman et al., 1963, section 7.1 Planetary motions).”

Feynman explains that the ancients observing the planetary motions among the stars, and finally deducing that they orbited around the sun. In fact, Nicolaus Copernicus was aware that Aristarchus of Samos (c. 310 – c. 230 BC) had already proposed the heliocentric theory (Sun-centered universe) earlier. Although Copernicus rediscovered this theory later, there were still debates as to whether the planets really orbited around the sun in the fifteenth century. More important, the “exact” planetary motions around the sun were deduced by having more astronomical observations. They are elaborated in the next section as Kepler’s three laws of planetary motions.

Copernicus (1473 – 1543) proposed a heliocentric theory that positioned the Sun at the center of the universe in which the Earth and the other planets rotating around it in circular paths at uniform speeds. In essence, Copernican theory is different from Ptolemy’s geocentric theory (Earth-centered universe) in placing the Earth at the center of the universe. A weakness of geocentric theory is the need of using epicycles (or cycles within cycles) to explain the “retrograde motions” of planets. By using astronomical instruments such as a triquetrum, Copernicus achieved more accurate observations and concluded there are “epicycles on epicycles” in the geocentric theory. However, the heliocentric theory is not quite correct because the Sun is also not stationary.

3. Tycho Brahe:

“Tycho Brahe had an idea … that these debates about the nature of the motions of the planets would best be resolved if the actual positions of the planets in the sky were measured sufficiently accurately (Feynman et al., 1963, section 7.1 Planetary motions).”

Feynman elaborates that Tycho Brahe (1546 – 1601) had an idea that was different from anything proposed by the ancient philosophers. Brahe’s main idea was that the debates about the nature of the orbital motions of the planets would be resolved if the actual positions of the planets in the sky were measured more accurately. In other words, Brahe proposed an empirical viewpoint in which the planetary motions should be determined by what we measure instead of what we think. This was a revolutionary idea because ancient philosophers prefer using deep philosophical arguments instead of performing careful experiments.

Tycho Brahe built large astronomical instruments, such as a triangular sextant and a revolving steel quadrant, that help to achieve accurate planetary observations. Tycho’s naked eye measurements of planetary parallax were accurate to the arcminute, or about 1/30 width of the full moon. However, he disagreed with Copernicus and proposed a “geo-heliocentric” model (or Tychonic system) in which the Sun orbiting the Earth, while the other planets orbiting the Sun. Curiously, Tycho (as an empiricist) maintained his belief of perfectly circular orbits for all celestial bodies despite his advocate of accurate astronomical observations.

Although Tycho opined that the planetary motion should be based on accurate astronomical observations, he believed that the Earth was just too heavy to be continuously in motion. In Tycho’s words, “This innovation expertly and completely circumvents all that is superfluous or discordant in the system of Ptolemy. On no point does it offend the principle of mathematics. Yet it ascribes to the Earth, that hulking, lazy body, unfit for motion, a motion as quick as that of the aethereal torches, and a triple motion at that (Gingerich, 1993, p. 181).”

Questions for discussion:

1. How would you state Newton’s law of universal gravitation?

2. How is the Copernican model different from Ptolemy’s geocentric model?

3. Was Tycho’s belief of circular orbits based on accurate astronomical observations?

The moral of the lesson: every object attracts every other object with a gravitational force that is proportional to the mass of each and varies inversely as the square of the distance between centers of mass of the two objects.

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. Gingerich, O. (1993). The eye of heaven: Ptolemy, Copernicus, Kepler. New York: American Institute of Physics.

3. Halliday, D., Resnick, R., & Walker, J. (2005). Fundamentals of Physics (7th ed.). New York: Wiley.

4. Jammer, M. (1999). Concepts of Force. New York: Dover Publications.

5. Newton, I. (1687/1995). The Principia. Translated by Andrew Motte. New York: Prometheus.

Section 6–5 The uncertainty principle

(Probability density / Einstein’s worry / Electron cloud)

The uncertainty principle is discussed in chapter 2, chapter 5, chapter 6, chapter 37, and chapter 38 of The Feynman Lectures. In this section, the three interesting points discussed are a probability density, Einstein’s worry, and electron cloud.

1. Probability density:

“…We can give a probability density p₁(x), such that p₁(x)Δx is the probability that the particle will be found between x and x+Δx (Feynman et al., 1963, section 6.5 The uncertainty principle).”

Dr. Sands explains the uncertainty principle from a perspective of probability. He elaborates that the concepts of probability are useful in describing the behavior of 10²² or more molecules in a gas because it is clearly impractical to write down the position and velocity of all these molecules. Furthermore, the use of probability is not just a matter of convenience for very complex situations, but it is essential to a description of atomic phenomena. Based on Heisenberg uncertainty principle, there are always uncertainties in the specifications of positions and velocities. Thus, physicists specify the probability of a particle as p₁(x)Δx will have a position between x and x+Δx in which p₁(x) is the probability density that the particle will be found.  

In the previous chapter, Dr. Sands gives a slightly different explanation of the uncertainty principle: the uncertainty in position is related to the error in our knowledge of the momentum of the object whose position we are measuring. In addition, he mentions that the uncertainty in position measurements is related to the wave nature of particles. Nevertheless, Heisenberg distinguishes two kinds of uncertainty: objective and subjective. In his own words, “[t]hese uncertainties may be called objective in so far as they are simply a consequence of the description in the terms of classical physics and do not depend on any observer. They may be called subjective in so far as they refer to our incomplete knowledge of the world (Heisenberg, 1958, p. 28).”

2. Einstein’s worry:

“…Einstein was quite worried about this problem. He used to shake his head and say, ‘But, surely God does not throw dice in determining how electrons should go!’ (Feynman et al., 1963, section 6.5 The uncertainty principle).”

Dr. Sands emphasizes that the most precise description of nature must be stated in terms of probabilities. On the contrary, some physicists have the opinion that they could know the speed and position of a particle simultaneously. For example, Einstein argues that God does not throw dice in determining how electrons should go. Moreover, Einstein, Podolsky, and Rosen (1935) pose the question whether a quantum mechanical description of physical reality can be considered complete. Interestingly, Einstein’s worry is related to a “spooky action at a distance” and it has led to the concept of quantum entanglement.

Dr. Sands mentions that only one or two physicists were working on the problem of describing the physical world in a different way such that uncertainties can be removed. Historically, many notable physicists attempted this problem which results in important contributions in quantum mechanics. For example, Bohm and Aharonov (1957) publish a new version of the Einstein–Podolsky–Rosen (EPR) paradox by reformulating the original argument in terms of spin. Subsequently, Bell (1964) proposes a theorem that could be used to test the EPR paradox. Most important, Einstein’s worry has led to active research by the physics community on quantum entanglement and useful applications such as quantum cryptography.

Note: In a speech titled Simulating physics with computers, Feynman (1982) discusses EPR paradox without mentioning Bell’s theorem.

3. Electron cloud:

“…our best “picture” of a hydrogen atom is a nucleus surrounded by an “electron cloud” (although we really mean a “probability cloud”) (Feynman et al., 1963, section 6.5 The uncertainty principle).”

Dr. Sands elaborates that the uncertainty in the position of an electron in a hydrogen atom is as large as the atom itself. Based on quantum mechanics, physicists do not describe the electron as moving in an “orbit” around the hydrogen atom or a proton. That is, we speak of a probability, p(r)ΔV, of observing the electron in a volume ΔV at a distance r from the proton. We may visualize the hydrogen atom as having an “electron cloud” that is surrounding the proton. The density of the electron cloud is proportional to the probability density of the electron where it will be found. Essentially, we can describe the electron as having a probability somewhere at a location because nature permits us to know only the chance of locating it.

In the earlier editions of The Feynman Lectures, the probability density of a hydrogen atom is described by the expression p(r) = A exp(−r²/a²) in which the constant a is the “typical” radius of the atom. Thus, there is a small chance of finding the electron at distances from the nucleus significantly greater than a which is about 10⁻¹⁰ meter. In the New Millennium edition of The Feynman Lectures, the probability density for an undisturbed hydrogen atom is revised as p(r) = A exp(−2r/a). This expression can be derived by using a three dimensional (time-independent) Schrödinger equation in spherical coordinates. The mistake of Dr. Sands in using p(r) = A exp(−r²/a²) could be due to his guess that the probability density of the hydrogen atom has a normal distribution.

Questions for discussion:

1. Should a precise description of an atom be described only in terms of probabilities?

2. Is Einstein’s worry about the quantum mechanical description of nature justified?  

3. How is the probability density of an undisturbed hydrogen atom derived?

The moral of the lesson: the uncertainty principle describes an inherent fuzziness that exists in any attempt to describe nature.

References:

1. Bell, J. S. (1964). On the Einstein Podolsky Rosen Paradox. Physics, 1(3), 195-200.

2. Bohm, D., & Aharonov, Y. (1957). Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky. Physical Review, 108(4), 1070-1076.

3. Einstein, A. Podolsky, B. & Rosen, N. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47(10), 777-780.

4. Feynman, R. P. (1982). Simulating physics with computers. International journal of theoretical physics, 21(6), 467-488.

5. 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.

6. Heisenberg, W. (2007/1958). Physics and Philosophy: The Revolution in Modern Science. New York: HarperCollins.