Like information invisibly encoded in strands of DNA, engineered works — bridges, dams, aircraft, disk drives, motors, and so on — are characteristically imprinted with math. With few exceptions, math is the foundation as well as the structure for all technological achievement; and on the wings of computers, it may carry us farther than anyone ever imagined. Take a journey along the evolutionary path of mathematical analysis, and you see not only the past but the future.

From maps to Mars

Mathematical modeling, in one form or another, is behind virtually all technological progress since the beginning of scientific discovery.

Follow the trail of Euclidean geometry, for example. Starting in ancient times and extending through the Renaissance, it has lead to such technological advancements as surveying, mapping, navigation, and astronomy. Granted the tools and methods were primitive by today’s standards, but they allowed man for the first time to explore the extent of his own universe.

The next big leap was the development of tools to model time-variant events. In the 17th century, mathematicians worked out the theory to describe real-world (dynamic) phenomena — motion. The breakthrough came from Newton and Leibnitz, who independently developed differential calculus.

One feat in particular demonstrated the power of the new math. With a relatively simple model, Newton not only reproduced known planetary orbits, he predicted future positions. The accuracy of his celestial mechanics is impressive by any measure and it had broad repercussions, challenging man’s view of the universe and his role in it.

Rather than attempting to describe a planetary orbit as a static curve, Newton’s revolutionary invention of a dynamic model for motion expressed how changes in velocity are related to position. The model, as Leibnitz said it must be, was a set of differential equations expressing the relationship between position and its derivatives.

A stone’s throw

Dynamic equations, while facilitating analysis of complicated systems, presented their own set of challenges. Suppose you want to model the trajectory of a rock thrown through the air. Using Newton’s Law of Motion, and assuming gravity is constant near the earth’s surface, the differential equation describing the stone’s height u as a function of time is:

where t = time, m = mass, and F is the constant force (gravity) acting on the stone. The function

is the second derivative of position with respect to time, or acceleration.

The solution to this equation is typically expressed as a polynomial in t

where the constants a and b must be determined from a known position and velocity corresponding to a given time. In the 18th century, this was about as far as one could take it.

Few dynamic problems are as simple as projectile motion, however. Consider the motion of planets in our solar system. The main influencing force, as Newton pointed out, is interplanetary gravitational pull. For just two planets, the force of gravity is proportional to each planet’s mass and inversely proportional to the square of the separation distance. When you consider the entire solar system, the force on each planet is the sum of the forces imposed by every other planet. Using Newton’s Law of Motion, the position of Earth, Mercury, Jupiter, and every other planet in our solar system is:

where u_i indicates the position of planet i, t = time, m_i = mass of planet i, and C = Newton’s constant of gravity. Note that u_i is a vector with three components, the planet’s x, y, z coordinates.

This equation, although it involves a more complicated force, is somewhat similar to that of the tossed stone. But there are differences that make the solution orders of magnitude more difficult. For instance, you can’t solve for the position of a single planet by itself; you need the position of all other planets to determine the force acting on the one of interest.

More perplexing is the fact that there doesn’t seem to be an analytical solution. Newton showed that if only two planets were in space, they would travel in elliptical, parabolic, or hyperbolic orbits around the center of gravity. Since then, despite enormous research efforts, most mathematical issues related to systems of three or more planets remain unresolved; to this day, no one has found an analytical solution to the planetary equations of motion.

Partial victory

We’ve defined just two problems so far, both on the basis of a single independent variable. Many physical processes, however, involve multiple independent variables. Analyzing phenomena of this sort requires a more advanced form of math known as partial differential equations.

One of the most well-known partial differential equations of all time is the Poisson equation. Developed in the early 19th century, it applies to many things observed in physics including the gravitational field of a planetary system. In the previous example we assumed the masses of the planets to be located at infinitely small points. A more realistic analysis accounts for distributed mass, which is exactly what Poisson’s equation does:

The function u is the gravity potential, g is the gravitational constant, and ρ is the mass density as a function of independent variables x, y, z.

Poisson’s equation also applies to electromagnetics, geophysics, and chemical engineering. Other partial differential equations of note are the heat-transfer equation

which describes heat flow in solid objects, and the wave equation

which expresses wave propagation in acoustics and electromagnetics. In both cases, the function u depends on time t as well as position (x, y, z). The remaining term, f, is a source function that varies with time, position, and the unknown function u.

These and other partial differential equations — including the Schrödinger equation in quantum mechanics, the Navier equation in structural mechanics, and the Navier-Stokes and Euler equations in fluid dynamics — are fundamental to all sciences and have greatly aided the development of modern day technology.

Nature’s curve

Partial differential equations are typically easier to solve if all terms involving the function of interest and its derivatives can be reduced to a linear combination with independent coefficients. If that’s the case, the equation is said to be linear; otherwise it’s nonlinear.

The distinction between linear and nonlinear partial differential equations (PDEs) is crucial when it comes time to solve them. Many linear PDEs — including Poisson’s equation, the heat equation, the wave equation, and the equation describing the tossed stone — reduce to an exact form by proper manipulation. Some of the techniques include separation of variables, superposition, Fourier series, and Laplace transforms.

In contrast, nonlinear PDEs — such as the Navier-Stokes equation and those that express planetary motion — are seldom solvable in analytical form. Instead you must rely on numerical solutions and approximations. Many algorithms for this purpose exist today and are quite accurate even when compared to exact mathematical solutions.

Three centuries ago Leibnitz envisioned as much, predicting the eventual development of a general method that could be used to solve any differential equation.

It was in pursuit of such dreams as this, and the more pressing needs of navigators and astronomers, that early mathematicians looked for easier calculation methods. They devoted centuries to creating tables, notably of logarithms and trigonometric functions, that allowed virtually anyone to quickly and accurately calculate answers to a variety of problems.

The tables, however, were only a stepping stone. Even in his time Leibnitz realized the benefits of automating math with machines. “Knowing the algorithm of this calculus,” he said, “all differential equations that can be solved by a common method, not only addition and subtraction but also multiplication and division, could be accomplished by a suitably arranged machine.”

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