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.