Section 22–2 The inverse operations

(Subtraction & division / Root & logarithm / Class of objects)

In this section, Feynman explains inverse operations from the perspectives of “subtraction & division,” “root & logarithm,” and the class of objects.

1. Subtraction & division:

“If a + b = c, b is defined as c − a, which is called subtraction. The operation called division is also clear: if ab = c, then b = c/a defines division (Feynman et al., 1963, section 22–2 The inverse operations).”

According to Feynman, the direct operations are addition, multiplication, and raising to a power. He defines two inverse operations as follows: 1. Subtraction: if a + b = c, then b is defined as c – a. 2. Division: if a ´ b = c, then b is defined as c/a. Alternatively, one may define a subtraction as an addition of an inverse, a − b = a + (−b). More important, we can define an inverse operation as an operation that undoes what was done by the previous operation. Thus, subtraction is an inverse operation of addition, and addition is an inverse operation of subtraction. Similarly, division is an inverse operation of multiplication, and multiplication is an inverse operation of division. To be precise, we should also define a, b, and c in terms of a class of objects.

To be comprehensive, one may discuss properties of subtraction and division with regard to commutativity and associativity. Specifically, subtraction does not have commutative property because 1 – 0 = 1 does not equal to 0 – 1 = –1; i.e., the order of numbers affects the outcome. Subtraction also does not have associative property because (1 – 1) – 1 = –1 does not equal to 1 – (1 – 1) = +1; i.e., the order of operations affects the outcome. Furthermore, division does not have commutative property because 1 ¸ 2 = 0.5 does not equal to 2 ¸ 1 = 2. Division also does not have associative property because (1 ¸ 2) ¸ 2 = 0.25 does not equal to 1 ¸ (2 ¸ 2) = 1. In summary, subtraction and division do not have commutative property and associative property.

2. Root & logarithm:

“Because b^a and a^b are not equal, there are two inverse problems associated with powers… (Feynman et al., 1963, section 22–2 The inverse operations).”

Feynman first poses an inverse problem: if b^a = c, what is b? In this case, b = _a√c and it is called the ath root of c. This is similar to the question: “What integer, raised to the third power, equals 8?” The other inverse problem is: if a^b = c, what is b? This case has a different answer: b = log_a c and it is similar to the question: “To what power must we raise 2 to get 8?” To distinguish the two problems, we may use the terms exponential function and power function. That is, the exponential function with base a (e.g., f(x) = a^x) and the logarithm function (e.g., g(x) = log_a x) with base a are inverse functions of each other. One may let students show that f[g(x)] = a^{log x} = g[f(x)] = log a^x = x. Similarly, the power function (e.g., u(x) = xⁿ) and the root function (e.g., v(x) = _n√x = x^1/n) are inverse functions of each other: u[v(x)] = (x^1/n)ⁿ = v[u(x)] = (xⁿ)^1/n = x.

Perhaps Feynman should have discussed more properties of power and logarithm operation. To illustrate how the power operation does not have commutative property and associative property, we can use the symbol ^ such that a ^ b = a^b. First, the power operation does not have commutative property because 1 ^ 2 = 1² = 1 does not equal to 2 ^ 1 = 2¹ = 2. Second, the power operation does not have associative property because (2 ^ 1) ^ 2 = 2² = 4 does not equal to 2 ^ (1 ^ 2) = 2¹ = 2. By using the similar method, we can prove that the root operation (Öx) and logarithm operation (log x) are not commutative and not associative. One may let students prove the following important properties: 1. Product property: log_b xy = log_b x + log_b y. 2. Quotient property: log_b x/y = log_b x – log_b y. 3. Power property: log_b xⁿ = n log_b x.

3. Class of objects:

“We are going to discuss whether or not we can broaden the class of objects which a, b, and c represent so that they will obey these same rules… (Feynman et al., 1963, section 22–2 The inverse operations).”

Feynman says that the relationships or rules are correct for integers since they follow from the definitions of addition, multiplication, and raising to a power. In division, the rules are not completely correct because mathematicians would emphasize that a division of a number by zero is undefined. On the other hand, the logarithm of zero is also undefined. Curiously, Feynman continues saying that we can broaden the class of objects…, although the processes for a + b, and so on, will not be definable. It is likely that he was not specifically referring to a + b, but log_a b and _aÖ b that cannot be defined without using irrational numbers or complex numbers.

In What Do You Care What Other People Think?, Feynman (1988) writes: “I learned algebra, fortunately, not by going to school, but by finding my aunt’s old schoolbook in the attic, and understanding that the whole idea was to find out what x is - it doesn't make any difference how you do it. For me, there was no such thing as doing it ‘by arithmetic,’ or doing it ‘by algebra.’ ‘Doing it by algebra’ was a set of rules which, if you followed them blindly, could produce the answer: ‘subtract 7 from both sides; if you have a multiplier, divide both sides by the multiplier,’ and so on - a series of steps by which you could get the answer if you didn’t understand what you were trying to do. The rules had been invented so that the children who have to study algebra can all pass it. And that’s why my cousin was never able to do algebra (Feynman, 1988, p. 17).” It is possible that Feynman’s understanding of algebra may not be conventional.

Questions for discussion:

1. How would you define inverse operations?

2. How would you list the properties of subtraction, division, root, and logarithm?

3. Is the class of objects for addition the same as subtraction?

The moral of the lesson: subtraction, division, root, and logarithm are the inverse operations for addition, multiplication, power, and exponent respectively; in a sense, direct operations and inverse operations are opposite operations that undo each other.

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. Feynman, R. P. (1988). What Do You Care What Other People Think?. New York: W W Norton.

Section 22–1 Addition and multiplication

(Mathematical proof / Mathematical philosophy / Mathematical definitions)

In this section, we can understand addition and multiplication from the perspectives of mathematical proof, mathematical philosophy, and mathematical definitions.

1. Mathematical proof:

“we may find the Pythagorean theorem quite interesting … an interesting fact, a curiously simple thing, which may be appreciated without discussing the question of how to prove it… (Feynman et al., 1963, section 22–1 Addition and multiplication).”

Feynman explains that this chapter will not be developed from a mathematician’s point of view because mathematicians are mainly interested in how various mathematical facts are demonstrated. In addition, they are interested in how many assumptions are absolutely required and what is not required, but they are not interested in the result of what they prove. However, in the words of a mathematician, “[p]roofs are for the mathematician what experimental procedures are for the experimental scientist: in studying them one learns of new ideas, new concepts, new strategies... (Rav, 1999, p. 20).” Interestingly, Paul Erdös knew 37 different beautiful proofs of the Pythagorean theorem when he was seventeen. In The Pythagorean Proposition, Loomis (1986) compiled 367 proofs of the Pythagorean theorem.

Feynman says that we may find the Pythagorean theorem is an interesting fact, which may be appreciated without discussing the question of how to prove it. In his autobiography, Einstein (1949) says that “I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in ‘proving’ this theorem on the basis of the similarity of triangles; in doing so it seemed to me ‘evident’ that the relations of the sides of the right-angled triangles would have to be completely determined by one of the acute angles (p. 9).” In A Beautiful Question, Wilczek (2015) reconstructs Einstein’s proof by showing how two similar triangles can be added to form another similar triangle. Every physics student ought to have the pleasure of understanding this beautiful proof.

2. Mathematical philosophy:

“But we are not going in that direction, the direction of mathematical philosophy …   from the assumption that we know what integers are and we know how to count (Feynman et al., 1963, section 22–1 Addition and multiplication).”

Feynman prefers not to include mathematical philosophy from the assumption that we know what integers are and how to count. It is worthwhile to discuss mathematical philosophy that is related to the definition of number. For example, Frege argues that numbers are objects and defines numbers as extensions of concepts. In Gouvêa’s (2008) words, “it is no longer that easy to decide what counts as a ‘number.’ The objects from the original sequence of ‘integer, rational, real, and complex’ are certainly numbers, but so are the p-adics. The quaternions are rarely referred to as ‘numbers,’ on the other hand, though they can be used to coordinatize certain mathematical notions (p. 82).” Mathematicians may not agree with the definition of number, but we can simply define a number as a means for counting instead of a set of things that is more abstract.

It appears that we know what integers are, what zero is, and what it means to increase a number by one unit. As a suggestion, one may mention that a number divided by zero threatened the foundation of mathematics and it remains undefined. Furthermore, Seife (2000) writes that “[z]ero clashed with one of the central tenets of Western philosophy… (p. 25)” because the Greek universe rejects the concept of zero. On the other hand, one may discuss the origin of zero in early Hindu and Buddhist philosophical discourses about the concept of “emptiness” or “void.” Historically, the doctrine of “sunyata” (or void) is one of the profound contributions of philosophy from India. However, physicists may not agree with the definition of vacuum or void.

3. Mathematical definitions:

“Now as a consequence of these definitions it can be easily shown that all of the following relationships are true… (Feynman et al., 1963, section 22–1 Addition and multiplication).”

According to Feynman, after we have defined addition, then we can start with nothing and add a to it, b times in succession, and call the result multiplication of integers (i.e., b ´ a). In addition, we can have a succession of multiplications: if we start with 1 and multiply by a, b times in succession (i.e., a^b). Mathematicians may not like Feynman’s approach and prefer to define the concept of group, ring, and field. They can be informally defined as follows: A group is a set in which we can perform addition or multiplication operation having some properties (i.e., closure, identity, inverse, & associativity). A ring is a group under addition and satisfies many properties of a group for multiplication. A field is a group under both addition and multiplication. In short, we can have different rules for different mathematical objects.

Feynman says that it is very hard to define properties of numbers such as continuity and ordering. Essentially, it is about the concept of the Dedekind cut and ordered continuum that provides a rigorous distinction between rational and irrational numbers. In 1872, Richard Dedekind proposes an arithmetic formulation of the idea of continuity. That is, the real numbers on a line (or number line) form an ordered continuum, such that any two numbers x and y must satisfy one and only one of the conditions: x < y, x = y, or x > y. He also postulates the concept of a cut that separates the continuum into two subsets, say X and Y, such that if x is any member of X and y is any member of Y, then x < y. This cut may correspond to a rational number or an irrational number.

Questions for discussion:

1. How would you prove the Pythagorean theorem using similar triangles?

2. How would you define the concept of numbers?

3. How would you explain the properties of numbers such as continuity and ordering?

The moral of the lesson: numbers have some nice properties such as a + 0 = a, and a.1 = a, but physicists can use them like a tool without the need of proving it.

References:

1. Einstein, A. (1949). Autographical notes (Translated by Schilpp). La Salle, Illinois: Open court.

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. Gouvêa, F. Q. (2008). From Numbers to Number Systems. In T. Gowers, J. Barrow-Green, I. Leader (Eds). The Princeton Companion to Mathematics. Princeton: Princeton University Press.

4. Loomis, E. S. (1968). The Pythagorean Proposition. Reston, VA: National Council of Teachers of Mathematics.

5. Rav, Y. (1999). Why do we prove theorems?. Philosophia Mathematica, 7(1), 5-41. 6. Seife, C. (2000). Zero: The Biography of a Dangerous Idea. New York: Viking.

7. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.

Section 21–5 Forced oscillations

(Steady-state response / Transient response / Resonance response)

In this section, Feynman discusses the steady-state response, transient response, and resonance response of an oscillator. However, it could be better reorganized as transient response, steady-state response, and resonance response.

1. Transient response:

“…we have found is the solution only if things are started just right, for otherwise there is a part which usually dies out after a while. This other part is called the transient response to F(t) (Feynman et al., 1963, section 21–5 Forced oscillations).”

Feynman discusses the forced harmonic oscillator that is expressed by the equation md²x/dt² = −kx + F(t). In this section, he focuses on the solution that is called a steady-state response instead of the transient response. Mathematically speaking, the general solution of the above equation is the sum of a particular solution (steady-state response) and the “homogeneous” solution (transient response). In chapter 24 on Transients, Feynman elaborates on the transient response with more details. In section 24-3 Electrical transients, he mentions that there are “dying exponentials,” in which one may have a much faster “dying rate” than the other. Thus, Feynman may agree with using the term “dying response” that could be better than “transient response.”

Feynman simply says that the other part which usually dies out after a while is called the transient response to F(t). However, this is potentially misleading because the transient response is a solution to F(t) = 0 and there should be resistive forces (e.g., air resistance such as bv). In essence, the transient response is a dying response because the driving force F(t) = 0 means it is not a periodic force that can continuously influence the oscillatory motion. Furthermore, the effects of the initial conditions will die out after some time because of the resistive forces. On the other hand, there are three types of dying responses to F(t) = 0: underdamped, overdamped, and critically damped. These different responses to F(t) = 0 are dependent on the relative magnitude of the resistive force and the natural frequency ω₀.

2. Steady-state response:

“Of course, the solution we have found is the solution only if things are started just right… while x = Ccos ωt and C = F₀/m(ω₀²−ω²) are called the steady-state response (Feynman et al., 1963, section 21–5 Forced oscillations).”

Feynman shows that the particular solution (steady-state response) of md²x/dt² = −kx + F(t) is x = Ccos ωt in which C = F₀/m(ω₀²−ω²). That is, an object with mass m oscillates at the driving frequency (ω), but its amplitude depends on the magnitude of the force (F₀) and the difference between the natural frequency (ω₀) of the oscillator and the driving frequency of F(t). In chapter 25, Feynman renames the steady-state response as the “forced” solution that is dependent on the “forcing frequency” (ω) of the driving force F(t). The steady-state response is sinusoidal in shape that remains unchanged while the ‘transient response” dies out. We should notice that there are no arbitrary constants in the “forced” solution that are related to the initial position or initial velocity.

According to Feynman, if ω is very small compared with ω₀, the displacement and the force are in the same direction. One may clarify that the equation md²x/dt² = −kx + F(t) can be reduced to kx = F(t) for very low frequency when the object moves slowly in the same direction as the force (md²x/dt² ® 0). If ω is very high, Feynman uses C = F₀/m(ω₀²−ω²) to explain that C is negative and the denominator becomes very large, and thus, the amplitude is small. Alternatively, one may reduce the equation md²x/dt² = −kx + F(t) to md²x/dt² = F(t) because the term kx is smaller relative to md²x/dt² as higher frequencies implies larger acceleration. By integrating the equation twice, we have x = -(F₀/mω²) sin ωt that has a negative sign and it means that the displacement is “out of phase” to the driving force.

3. Resonance response:

“If we happen to get the right timing, then the swing goes very high, but if we have the wrong timing, then sometimes we may be pushing when we should be pulling, and so on, and it does not work (Feynman et al., 1963, section 21–5 Forced oscillations).”

Theoretically, the swing should oscillate at an infinite amplitude if ω is exactly equal to ω₀. Feynman explains that this is impossible because the equation is wrong by excluding some frictional terms, and other forces that occur in the real world. Some may prefer a more realistic model that includes resistive forces that could be represented as kv. Essentially, this resonance response is an idealization that is applicable to an unrealistic undamped forced oscillator that has no resistive forces. However, this ideal model may be used to explain the response of a bound electron to an electromagnetic field such as the scattering of light in the classical theory (Kleppner & Kolenkow, 1973).

Feynman provides another reason why the amplitude of oscillation does not reach infinity. In short, it is likely that the spring breaks! This ending may seem abrupt but one may emphasize that the spring does not definitely obey Hooke’s law within the elastic limit. As a suggestion, we may use Feynman’s (1988) investigation of the Space Shuttle Challenger disaster to conclude the chapter. In general, the elastic constant of an object is dependent on its temperature. Importantly, Feynman demonstrated that the O-ring made of rubber doesn’t spring back quickly at the temperature of ice water. The O-ring of the space shuttle Challenger, having lost its ability to seal, caused the disaster.

Questions for discussion:

1. Does an undamped forced oscillator have a transient response?

2. How would you explain the steady-state response of a forced oscillator if the driving frequency is very low or very high?

3. What should be the response of an oscillator when the driving frequency is equal to the natural frequency?

The moral of the lesson: an idealized undamped forced oscillator that has no resistive forces should have a steady-state response or resonance response.

References:

1. Feynman, R. P. (1988). What Do You Care What Other People Think? New York: W W Norton.

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. Kleppner, D., & Kolenkow, R. (1973). An Introduction to Mechanics. Singapore: McGraw-Hill.

Section 21–4 Initial conditions

(Define initial conditions / Determine arbitrary constants / Relate total energy)

The three important points of this section are how to define initial conditions, determine arbitrary constants, and relate the total energy of an oscillator to the amplitude of the oscillations.

1. Define initial conditions:

“The constants A and B, or a and Δ, or any other way of putting it, are determined, of course, by the way the motion started, not by any other features of the situation. These are called the initial conditions (Feynman et al., 1963, section 21–4 Initial conditions).”

Feynman says that the arbitrary constants A and B, or a and Δ, are determined by the initial conditions. He adds that the initial conditions are dependent on the way the motion was started instead of any other features of the situation. As a suggestion, one may define the initial conditions as the kinematical conditions of an object’s initial motion such as initial displacement and initial velocity that determine the subsequent motion of the object. In other words, initial conditions are dependent on the time we set to zero when we start to observe the object’s motion and how we start the motion of the object. Mathematically speaking, infinite possible solutions of a differential equation can be reduced to a unique solution when we know the initial conditions such as the initial position, initial velocity, or initial acceleration.

Feynman explains that we cannot specify the acceleration with which it started because that is determined by the spring, once we specify x₀. We may rephrase this explanation as the initial acceleration can be deduced using ma = -kx₀ and thus, it is redundant. One should clarify that it is not necessary to use the initial position and initial velocity to determine the unique solution of a second-order differential equation. Instead of using the initial position and initial velocity, we may deduce the subsequent motion of an object using the initial velocity and initial acceleration. Alternatively, we can use independent information such as the position of the object at two different times instead of the initial position and initial acceleration that are not independent.

2. Determine arbitrary constants:

“Now let us consider what determines the constants A and B, or a and Δ. Of course, these are determined by how we start the motion (Feynman et al., 1963, section 21–4 Initial conditions).”

To determine the constants A and B, we can first use the formula involving the initial position x = Acos ω₀t + Bsin ω₀t to find that x₀ = A.1 + B.0 = A. Next, we can use the formula involving the initial velocity v = −ω₀Asin ω₀t + ω₀Bcos ω₀t to find that v₀ = −ω₀A.0 + ω₀B.1 = ω₀B. Solving these two equations, we find that A = x₀ and B = v₀/ω₀. However, it is important to clarify the meaning of x₀ because some students may consider it to be the amplitude of the oscillation. Furthermore, some students may also relate v₀/ω₀ to x₀ because of the formula v = ωr. Thus, there should be a derivation of the general formula relating the velocity and displacement, v² = ω²(x_max² – x²). Using this formula, we can deduce that v₀ = ωÖ(x_max² – x₀²).

To determine the constants a and Δ, we can use the formulas involving the initial position x = a cos (ω₀t + Δ) and initial velocity v = −ω₀a sin (ω₀t + Δ). It is useful to determine a and Δ because a is the amplitude of the oscillation and it is related to the elastic potential energy (½kx²) and kinetic energy (½mv²) of the oscillator. However, Feynman did not determine the constants a and Δ of the simple harmonic motion. By specifying x = x₀ and v = v₀ at the initial instant t = 0, we can find that x₀ = a cos Δ and v₀ = −ω₀a sin Δ. Using the identity cos² Δ + sin² Δ = 1 and solving the equations, we can find that a = √(x₀² + v₀²/ω₀²) and tan Δ = −v₀/ω₀x₀. Therefore, the maximum potential energy of the oscillator can be expressed as ½ka², whereas its maximum kinetic energy can be expressed as ½ mω₀²a².

3. Relate total energy:

“The energy is dependent on the square of the amplitude; if we have twice the amplitude, we get an oscillation which has four times the energy (Feynman et al., 1963, section 21–4 Initial conditions).”

Feynman mentions that the potential energy is not constant and the potential never becomes negative. He elaborates that there is always some energy in the spring, but the amount of energy fluctuates with x. In a sense, it is sloppy to simply use the term potential energy instead of elastic potential energy. More important, one may explain that the elastic potential energy of the spring never becomes negative just like the kinetic energy because of the term x². In addition, it is not true that there is always some elastic potential energy when the spring is not extended or compressed. Based on the expression of ½kx², the elastic potential energy of a spring is usually positive when the spring is stretched or compressed, but there is no elastic potential energy if the spring is in its natural length. 

Feynman explains that the energy is dependent on the square of the amplitude of oscillation has four times the energy if the amplitude is doubled. He adds that the average elastic potential energy of the oscillator is half the maximum elastic potential energy or half the total energy. Similarly, the average kinetic energy of the oscillator is also half the total energy. Feynman did not offer an explanation of the factor of ½ using symmetrical considerations. We can determine the average kinetic energy by integrating ò½kx²dt over a period of oscillation ΔT and then dividing the integral sum by the period ΔT. As an alternative, we can explain that the graph of elastic potential energy and kinetic energy of an oscillator are both sinusoidal in shape and have the same amplitude, and thus, they must have the same average value.

Questions for discussion:

1. How would you define the initial conditions of an oscillating object?

2. How would you determine the arbitrary constants A and B, or a and Δ?

3. How would you relate the average elastic potential energy of an oscillator to its amplitude of oscillation?

The moral of the lesson: the total energy of a simple oscillator can be calculated if its initial conditions are specified.

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.