Section 9–5 Meaning of the dynamical equations

(Initial conditions / Good approximation / Dynamical equations)

In this section, the three interesting concepts are initial conditions of a simple harmonic motion, a good approximation of a differential equation, and the meaning of dynamical equations.

1. Initial conditions:

“…for then we can start with the given condition and compute how it changes for the first instant, the next instant, the next instant, and so on… (Feynman et al., 1963, section 9–5 Meaning of the dynamical equations).”

Feynman states the “given condition” of a simple harmonic motion as time t = 0, x = 1, and v_x = 0 that allows us to compute the subsequent motion of an object. He explains that the object continues to move because there is a force on it when it is at any position except at the equilibrium position x = 0. However, the object does not start to move at x = 0, but it would move if it is slightly disturbed from a stable equilibrium state. As an alternative, physics teachers can explain that the object would continue to move at the equilibrium position because it possesses an amount of kinetic energy or momentum. Specifically, the presence of a restoring force in simple harmonic motion may reduce the kinetic energy of the object.

Some physicists may prefer the term “initial conditions” instead of “given condition.” In classical mechanics, “initial conditions” may include initial positions and velocities of all particles that determine a system’s future. On the other hand, there are two “necessary (mathematical) conditions” of simple harmonic motion: (1) the direction of the force is towards the equilibrium position, and (2) the magnitude of the force is proportional to the displacement from the equilibrium position and it can be expressed as F = -kx where k is a constant. Moreover, physics teachers should remember that simple harmonic motion is an idealization. One may include two “ideal (physical) conditions”: (1) there is no air resistance or frictional force, and (2) the spring is perfectly elastic if the extension or compression of the spring is relatively short.

2. Good approximation:

“…if ϵ is very small, we may express the position at time t + ϵ in terms of the position at time t and the velocity at time t to a very good approximation as x(t+ϵ) = x(t) + ϵv_x(t) (Feynman et al., 1963, section 9–5 Meaning of the dynamical equations).”

Feynman expresses the position of an object at time t + ϵ in terms of its position at time t and its velocity at time t to a good approximation as x(t+ϵ) = x(t) + ϵv_x(t). He clarifies that this expression is more accurate if the time interval ϵ is shorter. In addition, the acceleration can be determined by Newton’s second law and the velocity at a short time interval later can be approximately expressed as v_x(t+ϵ) = v_x(t) + ϵa_x(t) = v_x(t) − ϵx(t). Physics students that do not like numerical methods should realize that Feynman enjoys calculations that involve approximations. In Surely You’re Joking, Mr. Feynman, he says that “I had a lot of fun trying to do arithmetic fast, by tricks, with Hans. It was very rare that I’d see something he didn’t see and beat him to the answer, and he’d laugh his hearty laugh when I’d get one... (Feynman, 1997, p. 194).”

Motions of objects can be modeled by using ordinary differential equations that may not be solved exactly. Thus, we can design numerical algorithms for differential equations and use a computer to simulate the motions of objects and form an approximate view of the motions. In general, the use of a numerical method may achieve an accurate approximate solution to a differential equation. Currently, there are many programs (or software packages) that help to solve differential equations. With today’s computers, an accurate solution can be obtained within seconds. More important, the simple harmonic motion is based on Hooke’s law that is a first-order linear approximation. To be more accurate, a real spring could be modeled by using F(x) = kx + γx² for some constant γ as a second-order approximation.

3. Dynamical equations:

“…Eq. (9.15) is dynamics, because it relates the acceleration to the force; it says that at this particular time for this particular problem, you can replace the acceleration by −x(t) (Feynman et al., 1963, section 9–5 Meaning of the dynamical equations).”

According to Feynman, the equation v_x(t+ϵ) = v_x(t) + ϵa_x(t) is merely kinematics because it determines how the velocity of an object changes depending on the magnitude of acceleration. On the other hand, the equation v_x(t+ϵ) = v_x(t) − ϵx(t) is dynamics because it is related to a force that is in terms of −x(t). To be precise, the acceleration could be expressed as –(k/m)x(t) instead of −x(t). The motion of the object that is attached to a gadget is dependent on the spring constant (k) of the gadget and the mass of the object (m) in motion. To avoid possible confusions, Feynman could set k/m equal to 1 later.

Kinematics refers to the nature and characteristic of the motions. Simply phrased, it focuses on the motions of objects without concern with the forces that cause motions. On the contrary, dynamics is concerned with the motions of objects due to the influence of forces. As a comparison, velocity is a kinematical quantity, whereas momentum is a dynamical quantity. Interestingly, the dynamical property of simple harmonic motion (SHM) can be related to circular motion. For example, French (1971) explains that “[i]n order to display the dynamical identity of this component motion with SHM, we can take the expressions for F_x and a_x separately, introducing the angular velocity ω and putting v = ωA (p. 234).”

Questions for discussion:

1. What are the conditions that should be specified in the simple harmonic motion?

2. What are the approximations that are necessary for the simple harmonic motion?

3. What is the meaning of the dynamical equations?

The moral of the lesson: if we know the position and velocity of an object that is attached to a spring at a given time, we would know its acceleration, which tells us the new velocity, and we would know a new position approximately — this is how the machinery works.

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: 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. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

Section 9–4 What is the force?

(A program for forces / Gravitational force / Restoring force)

1. A program for forces:

“…we have to have some formula for the force; these laws say pay attention to the forces. Our program for the future of dynamics must be to find the laws for the force (Feynman et al., 1963, 9–4 What is the force?)”

The title of this section “What is the force?” is potentially misleading. Instead of explaining the concept of a force, it is about some formulae for the force to deduce the motion of objects. More important, Feynman mentions that it is about a program for the future of dynamics to determine the laws for the force. In essence, the section is an introduction to the use of numerical methods to deduce the motion of objects. In general, numerical methods may involve the use of algorithms to solve mathematical problems and it is built on the foundation of approximation theory. Furthermore, numerical analysis is revolutionized by the invention of computers as well as the development of mathematical theories and algorithms.

Physics teachers may elaborate the program for forces by using Wilczek (2005) words, “…the law F = ma, which is sometimes presented as the epitome of an algorithm describing nature, is actually not an algorithm that can be applied mechanically (pun intended). It is more like a language in which we can easily express important facts about the world. That’s not to imply it is without content. The content is supplied, first of all, by some powerful general statements in that language such as the zeroth law, the momentum conservation laws, the gravitational force law, the necessary association of forces with nearby sources and then by the way in which phenomenological observations, including many (though not all) of the laws of material science can be expressed in it easily… (p. 11).” According to Wilczek, the zeroth law may refer to the law of conservation of mass.

2. Gravitational force:

“Thus the law of gravity tells us that weight is proportional to mass; the force is in the vertical direction and is the mass times g (Feynman et al., 1963, 9–4 What is the force?).”

The gravitational (radial) force near the earth’s surface is proportional to the mass of the object and is nearly independent of height for heights that are small compared with the earth’s radius R. According to Feynman, Newton’s law of gravitation tells us that weight is proportional to mass and is equal to the mass times g: W = mg = GmM/R², in which g = GM/R² and it is known as the acceleration of gravity. Essentially, the free fall of an object under gravity, which leads to the equations v_x = v₀+gt and x = x₀+v₀t + ½gt², is related to a theoretical definition of weight, W = mg. However, there is no consensus on the definition of weight. For example, one may prefer an operational definition of weight that is based on a weighing scale.

The debate on the definition of weight started during the man-in-space project in the 1960s (King, 1962). Some physicists explain that an astronaut would feel weightless and “float in the air” in a satellite coasting around the Earth. Thus, physics educators review the definition of weight and debate whether “weight is a gravitational force” (gravitational weight) or “weight is measured by a weighing scale” should be adopted for science education. The issue is not simply between the “theoretical definition” of weight as a gravitational force and the “operational definition” of weight. The controversy is also about whether weight should be theoretically defined as a “gravitational force” or “force on the support” (alternatively, a support force).

3. Restoring force:

“…the force is greater, the more we pull it up, in exact proportion to the displacement from the balanced condition, and the force upward is similarly proportional to how far we pull down (Feynman et al., 1963, 9–4 What is the force?).”

Feynman gives an example of a gadget (or a spring) which exerts a force proportional to the distance and the force is directed in an opposite direction. Based on Newton’s second law, we can describe the motion using the equation m(dv_x/dt) = –kx. In short, it means that the velocity of an object (connected to the gadget) changes at a rate proportional to x and it is in the x-direction. For reasons of simplicity, we can choose k/m = 1 and solve the equation as dv_x/dt = −x. Feynman needs not say that nothing will be gained by retaining numerous constants or imagine there is an accident in the units. We can interpret this as a sense of humor and explain that physicists have the freedom to arbitrarily (instead of accidentally) define the units involved.

What seems missing here is an explanation of the equation involving Hooke’s law (F = –kx). This equation is explained later in chapter 12: “[t]his principle is known as Hooke’s law, or the law of elasticity, which says that the force in a body which tries to restore the body to its original condition when it is distorted is proportional to the distortion. This law, of course, holds true only if the distortion is relatively small (Feynman et al., 1963, section 12–3 Molecular forces).” In De potentia restitutiva, Robert Hooke (1678) succinctly phrases the law of elastic bodies as “Ut tensio, sic vis,” which means, “the extension is proportional to the force.” Interestingly, Hooke claims his discovery of the law by stating a Latin anagram “ceiiinosssttuu” (French, 1971) without clarifying the “elastic bodies.”

Questions for discussion:

1. Why do physicists need a program of forces?

2. Do you agree with Feynman’s definition of weight?

3. What does Feynman mean when he says that “nothing will be gained by retaining numerous constants” and “imagine there is an accident in the units”?

The moral of the lesson: physicists need a program for the future of dynamics to determine the laws for the force that involves the use of numerical methods.

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. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

3. Hooke, R. (1678). De potentia restitutiva. In R. T. Gunther (1931). Early Science in Oxford, Vol. VIII: The Cutler Lectures of Robert Hooke. Oxford: Oxford University Press.

4. King, A. L. (1962). Weight and Weightlessness. American Journal of Physics, 30(4), 387.

5. Wilczek, F. (2005). Whence the force of F= ma? III: Cultural diversity. Physics Today, 58(7), 10-11.