Differentiation Rules

Let \({\bf b}(t)\) and \({\bf c}(t)\) be differentiable paths in \({\mathbb R}^3\) and \(p(t)\) and \(q(t)\) be differentiable scalar functions: \[ \begin{array}{r@{\quad}l} {\it Sum Rule}\,\colon &{\displaystyle \frac{d}{{\it dt}}}[{\bf b}(t)+{\bf c}(t)] ={\bf b}'(t)+{\bf c}'(t)\\[15pt] {\it Scalar Multiplication Rule}\,\colon &{\displaystyle\frac{d}{{\it dt}}}[p(t){\bf c}(t)]=p'(t){\bf c} (t)+p(t){\bf c}'(t)\\[15pt] {\it Dot Product Rule}\,\colon & {\displaystyle\frac{d}{{\it dt}}}[{\bf b}(t)\,{\cdot}\, {\bf c}(t)]={\bf b}'(t)\,{\cdot}\, {\bf c}(t)+ {\bf b}(t)\,{\cdot}\,{\bf c}'(t)\\[15pt] {\it Cross Product Rule}\,\colon &{\displaystyle\frac{d}{{\it dt}}}[{\bf b}(t)\times {\bf c}(t)]={\bf b}'(t)\times {\bf c}(t)+{\bf b}(t)\times {\bf c}'(t) \\[15pt] {\it Chain Rule}\,\colon &{\displaystyle\frac{d}{{\it dt}}}[{\bf c}(q(t))]=q'(t){\bf c}'(q(t)) \end{array} \]

example 1

Show that if \({\bf c}(t)\) is a vector function such that \(\|{\bf c}(t)\|\) is constant, then \({\bf c}'(t)\) is perpendicular to \({\bf c}(t)\) for all \(t\).

solution Because \(\|{\bf c}(t)\|\) is constant, so is its square \(\|{\bf c}(t)\|^2 = {\bf c}(t)\,{\cdot}\, {\bf c}(t)\). The derivative of this constant is zero, so by the dot product rule, \[ 0=\frac{d}{{\it dt}}[{\bf c}(t)\,{\cdot}\, {\bf c}(t)]={\bf c}'(t)\,{\cdot}\, {\bf c}(t)+{\bf c}(t) \,{\cdot}\,{\bf c}'(t)=2{\bf c}(t)\,{\cdot}\, {\bf c}'(t); \] thus, \({\bf c}(t)\,{\cdot}\, {\bf c}'(t)=0\); that is, \({\bf c}'(t)\) is perpendicular to \({\bf c}(t)\).

example 2

A particle moves in such a way that its acceleration is constantly equal to \(-{\bf k}\). If the position when \(t=0\) is (0, 0, 1) and the velocity at \(t=0\) is \({\bf i}+{\bf j}\), when and where does the particle fall below the plane \(z=0\)? Describe the path traveled by the particle (assume \(t\geq 0\)).

solution Let \(\textbf{c}(t)=(x(t),y(t),z(t))=x(t){\bf i}+y(t){\bf j}+z(t){\bf k}\) be the path traced out by the particle, so that the velocity vector is \({\bf c}'(t)=x'(t){\bf i}+y'(t){\bf j}+z'(t){\bf k}\). The acceleration \({\bf c}''(t)\) is \(-{\bf k}\), so \(x''(t)=0,y''(t)=0\), and \(z''(t)=-1\). It follows that \(x'(t)\) and \(y'(t)\) are constant functions, and \(z'(t)\) is a linear function with slope \(-1\). Because \({\bf c}'(0)={\bf i}+{\bf j}\), we get \({\bf c}'(t)={\bf i}+{\bf j}-t{\bf k}\). Integrating again and using the initial position (0, 0, 1), we find that \((x(t),y(t),z(t))= (t,t,1-\frac{1}{2}t^2)\). The particle drops below the plane \(z=0\) when \(1-\frac{1}{2}t^2=0;\) that is, \(t=\sqrt{2}\) (because \(t\ge 0\)). At that instant, the position is \((\sqrt{2},\sqrt{2},0)\). The path traveled by the particle is a parabola in the plane \(y=x\) (see Figure 4.1), because in this plane the equation is described by \(z=1-\frac{1}{2}x^2\).

Figure 4.1: The path of the particle with initial position (0, 0, 1), initial velocity \({\bf i} {+} {\bf j}\), and constant acceleration \({-}{\bf k}\) is a parabola in the plane \(y=x.\)

Figure 4.2: The image of the smooth path \({\bf c}\hbox{(}t\hbox{)} = \hbox{(}\cos^{3} t\hbox{,} \sin^{3} t\hbox{)}\), a hypocycloid, does not “look smooth.”

example 3

A particle moves along a hypocycloid according to the equations \[ x = \,\cos^3 t , \qquad y = \,\sin^3 t, \qquad a \le t \le b. \]

What are the velocity and speed of the particle?

solution The velocity vector of the particle is \[ {\bf v} = \frac{d x}{{\it dt}} {\bf i} + \frac{d y}{{\it dt}} {\bf j} = - ( 3 \,\sin t \,\cos^2 t) {\bf i} + ( 3 \,\cos t \,\sin^2 t) {\bf j}, \] and its speed is \[ s = \|{\bf v}\| = ( 9 \,\sin^2 t \,\cos^4 t + 9 \,\cos^2 t \,\sin^4 t)^{1/2} = 3\, |\sin t|\ \, |\!\cos t|. \]

Acceleration and Newton’s Second Law

The acceleration of a path \({\bf c}(t)\) is \[ {\bf a}(t)={\bf c}''(t). \]

If \({\bf F}\) is the force acting and \(m\) is the mass of the particle, then \[ {\bf F}=m{\bf a}. \]

Figure 4.3: A mass \(M\) attracts a mass \(m\) with a force \({\bf F}\) given by Newton’s law of gravitation: \({\bf F}={-}{\it GmM}{\bf r}/{r}^{3}\).

Figure 4.4: The position, velocity, and acceleration of a particle in circular motion.

Kepler’s Law

\[ T^2=r^3_0\frac{(2\pi)^2}{{\it GM}}. \] Thus, the square of the period is proportional to the cube of the radius.

example 4

Suppose that a satellite is to be in a circular orbit about the earth such that it stays fixed in the sky over one point on the equator. What is the radius of such a geosynchronous orbit? (The mass of the earth is \(5.98 \times 10^{24}\) kilograms and \(G=6.67 \times 10^{-11}\) in the meter-kilogram-second [kgs] system of units.)

solution The period of the satellite should be 1 day, so \(T=60\,\times\, 60 \times 24 ={86{,}400}\) seconds. From the formula \(T^2=r^3_0(2\pi)^2/{\it GM}\), we get \(r^3_0=T^2{\it GM}/(2\pi)^2\), and so \[ r^3_0 = \frac{T^2{\it GM}}{(2\pi)^2}=\frac{(86{,}400)^2\times (6.67\times 10^{-11})\times (5.98\times 10^{24})}{(2\pi)^2} \approx 7.54 \times 10^{22}\,{\rm m}^3. \]

Thus, \(r_0=4.23\times 10^7 \,{\rm m} ={42{,}300}\,{\rm km}\approx {26{,}200}\, {\rm mi.}\)

Historical Note

Kepler, Newton, Hamilton, Feynman, and Planck

As we discussed in the historical introduction, the law of planetary motion stating that the square of the period is proportional to the cube of the radius of an orbit is one of the three that Kepler observed before Newton formulated his laws of motion, known more generally as Newton’s mechanics. These mechanics enable us to compute the period of a satellite about the earth or a planet about the sun (when the radius of its orbit is given), and, as we will indicate shortly, trajectories of space missions. Kepler discovered and used results like this not only for circular orbits but more generally for elliptical orbits. Newton was able to derive Kepler's three celestial laws from his own law of gravitation. The neat mathematical order of the universe that these laws provided had a great impact on eighteenth-century thought.

Newton never wrote down his laws of mechanics as differential equations. This was first done by Euler around 1730. Newton made most of his deductions (at least those in published form) by geometric methods. Euler also showed how Newton’s equations followed from Maupertuis’ action principle. The clearest version of the action principle in mechanics, now known as Hamilton’s principle, is due to William Rowan Hamilton around 1830, who, as we all should now know, happens to also be the father of vector calculus. Hamilton’s version of Maupertuis’ principle was elegantly presented by Richard Feynman, as we discuss next.

FEYNMAN AND HAMILTON’S PRINCIPLE. In his legendary Caltech Lectures on Physics, Nobel Prize–winning physicist Richard Phillips Feynman (see Figure 4.5 and Figure 4.6) included what he called a “Special Lecture” on a topic clearly very close to his heart—one that he first heard about from his New York high school teacher, Mr. Bader. Mr. Bader told his (apparently bored) student Feynman how principles of maxima and minima apply to the trajectories of moving objects and in particular how the action principle of Maupertuis, Leibniz, and Hamilton (discussed in Section 3.3) applies to Newton’s mechanics, governed by \({\bf F}\,{=}\,{\it m{\bf a}}\).

Professor Feynman, at the end of his lecture, notes that “a physicist, a student of Mr. Bader, in 1942 showed how this action principle applied to quantum mechanics.” That student was Feynman himself, who received the Nobel Prize for his insights, which also included the discovery of Feynman integrals. The moral here is pay attention to your teachers—especially the best ones!

We include the first part of Feynman’s lecture here and more of it in the Internet supplement; see Volume II, Lecture 19, of the Feynman Lectures on Physics for the entire lecture.

The Principle of Least Action, by Richard Feynman

When I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, “You look bored; I want to tell you something interesting.”

Then he told me something which I found absolutely fascinating, and have, since then, always found fascinating. Every time the subject comes up, I work on it. In fact, when I began to prepare this lecture I found myself making more analyses on the thing. Instead of worrying about the lecture, I got involved in a new problem. The subject is this—the principle of least action.

Mr. Bader told me the following: Suppose you have a particle (in a gravitational field, for instance) which starts somewhere and moves to some other point by free motion—you throw it, and it goes up and comes down [see Figure 4.7].

It goes from the original place to the final place in a certain amount of time. Now, you try a different motion. Suppose that to get from here to there, it went like this [see Figure 4.8], but got there in just the same amount of time.

Then he said this: “If you calculate the kinetic energy at every moment on the path, take away the potential energy, and integrate it over the time during the whole path, you’ll find that the number you’ll get is bigger than that for the actual motion.”

In other words, the laws of Newton could be stated not in the form \(F\,{=}\,{\it ma}\) but in the form: The average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another.

Let me illustrate a little better what this means. If you take the case of the gravitational field, then if the particle has the path \(x\hbox{(}t\hbox{)}\) (let’s just take one dimension for a moment; we take a trajectory that goes up and down and not sideways), where \(x\) is the height above the ground, the kinetic energy is \(\textstyle\frac{1}{2}m\,\hbox{(}{\it dx}\)/\({\it dt}\,\hbox{)}^{2}\), and the potential energy at any time is mgx. Now I take the kinetic energy minus the potential energy at every moment along the path and integrate that with respect to time from the initial time to the final time. Let’s suppose that at the original time \(t_{1}\) we started at some height and at the end of the time \(t_{2}\) we are definitely ending at some other place [see Figure 4.9].

Then the integral is \[ \int^{t_2}_{t_1}\Bigg[\frac{1}{2}m\, \bigg(\frac{{\it dx}}{{\it dt}}\bigg)^2 - {\it mgx}\Bigg]\,{\it dt}. \]

The actual motion is some kind of curve—it’s a parabola if we plot against the time—and gives a certain value for the integral. But we could imagine some other motion that went very high and came up and down in some peculiar way [see Figure 4.10].

We can calculate the kinetic energy minus the potential energy and integrate for such a path . . . or for any other path we want. The miracle is that the true path is the one for which that integral is least.

MAX PLANCK AND THE PRINCIPLE OF LEAST ACTION. Max Planck (see Figure 4.11), one of the greatest scientists of the modern era and the discoverer of the “quantization” of nature, was also a profound believer in the mathematical design of the universe, and in particular the principle of least action. He argued that the universality of this principle was evidence of a divine creator, and the structure of the universe a result of divine reason. On June 29, 1922, on “Leibniz Day” in Berlin, Germany, just a few years after World War I, with all its terrible carnage, Planck delivered an address honoring this great scholar and his discovery of the principle of least action.

In the following paragraphs we excerpt some of Planck’s remarks.

Modern science, in particular under the influence of the development of the notion of causality, has moved far away from Leibniz’s teleological point of view. Science has abandoned the assumption of a special, anticipating reason, and it considers each event in the natural and spiritual world, at least in principle, as reducible to prior states. But still we notice a fact, particularly in the most exact science, which, at least in this context, is most surprising. Present-day physics, as far as it is theoretically organized, is completely governed by a system of space–time differential equations which state that each process in nature is totally determined by the events which occur in its immediate temporal and spatial neighborhood.

This entire rich system of differential equations, though they differ in detail, since they refer to mechanical, electric, magnetic, and thermal processes, is now completely contained in a single dictum—the principle of least action. This, in short, states that, of all possible processes, the only ones that actually occur are those that involve minimum expenditure of action. As we can see, only a short step is required to recognize in the preference for the smallest quantity of action the ruling of divine reason, and thus to discover a part of Leibniz’s teleological ordering of the universe.footnote #

In present-day physics the principle of least action plays a relatively minor role. It does not quite fit into the framework of present theories. Of course, admittedly it is a correct statement; yet usually it serves not as the foundation of the theory, but as a true but dispensable appendix, because present theoretical physics is entirely tailored to the principle of infinitesimal local effects, and sees extensions to larger spaces and times as an unnecessary and uneconomical complication of the method of treatment. Hence, physics is inclined to view the principle of least action more as a formal and accidental curiosity than as a pillar of physical knowledge.

Figure 4.12: Trajectory of the Genesis spacecraft from the earth to a periodic orbit about a million and a half kilometers from earth and the interesting return trajectory to earth.