MECHANICS

Mechanics, science concerned with the motion of bodies under the action of forces, including the special case in which a body remains at rest. Of first concern in the problem of motion are the forces that bodies exert on one another. This leads to the study of such topics as gravitation, electricity, and magnetism, according to the nature of the forces involved. Given the forces, one can seek the manner in which bodies move under the action of forces; this is the subject matter of mechanics proper.

The origins and foundations of mechanics

History

The discovery of classical mechanics was made necessary by the publication, in 1543, of the book De revolutionibus orbium coelestium libri VI (“Six Books Concerning the Revolutions of the Heavenly Orbs”) by the Polish astronomer Nicolaus Copernicus. The book was about revolutions, real ones in the heavens, and it sparked the metaphorically named scientific revolution that culminated in Newton’s Principiaabout 150 years later. The scientific revolution would change forever how people think about the universe.

In his book, Copernicus pointed out that the calculations needed to predict the positions of the planets in the night sky would be somewhat simplified if the Sun, rather than the Earth, were taken to be the centre of the universe (by which he meant what is now called the solar system). Among the many problems posed by Copernicus’s book was an important and legitimate scientific question: if the Earth is hurtling through space and spinning on its axis as Copernicus’s model prescribed, why is the motion not apparent?

To the casual observer, the Earth certainly seems to be solidly at rest. Scholarly thought about the universe in the centuries before Copernicus was largely dominated by the philosophy of Plato and Aristotle. According to Aristotelianscience, the Earth was the centre of the universe. The four elements—earth, water, air, and fire—were naturally disposed in concentric spheres, with earth at the centre, surrounded respectively by water, air, and fire. Outside these were the crystal spheres on which the heavenly bodies rotated. Heavy, earthy objects fell because they sought their natural place. Smoke would rise through air, and bubbles through water for the same reason. These were natural motions. All other kinds of motion were violent motion and required a proximate cause. For example, an oxcart would not move without the help of an ox.

When Copernicus displaced the Earth from the centre of the universe, he tore the heart out of Aristotelian mechanics, but he did not suggest how it might be replaced. Thus, for those who wished to promote Copernicus’s ideas, the question of why the motion of the Earth is not noticed took on a special urgency. Without suitable explanation, Copernicanism was a violation not only of Aristotelian philosophy but also of plain common sense.

The solution to the problem was discovered by the Italian mathematician and scientist Galileo Galilei. Inventing experimental physics as he went along, Galileostudied the motion of balls rolling on inclined planes. He noticed that, if a ball rolled down one plane and up another, it would seek to regain its initial height above the ground, regardless of the inclines of the two planes. That meant, he reasoned, that, if the second plane were not inclined at all but were horizontal instead, the ball, unable to regain its original height, would keep rolling forever. From this observation he deduced that bodies do not need a proximate cause to stay in motion. Instead, a body moving in the horizontal direction would tend to stay in motion unless something interfered with it. This is the reason that the Earth’s motion is not apparent; the surface of the Earth and everything on and around it are always in motion together and therefore only seem to be at rest.

This observation, which was improved upon by the French philosopher and scientistRené Descartes, who altered the concept to apply to motion in a straight line, would ultimately become Newton’s first law, or the law of inertia. However, Galileo’s experiments took him far beyond even this fundamental discovery. Timing the rate of descent of the balls (by means of precision water clocks and other ingenious contrivances) and imagining what would happen if experiments could be carried out in the absence of air resistance, he deduced that freely falling bodies would be uniformly accelerated at a rate independent of their mass. Moreover, he understood that the motion of any projectile was the consequence of simultaneous and independent inertial motion in the horizontal direction and falling motion in the vertical direction. In his book Dialogues Concerning the Two New Sciences (1638), Galileo wrote,

It has been observed that missiles and projectiles describe a curved path of some sort; however, no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in proving. …

Just as Galileo boasted, his studies would encompass many aspects of what is now known as classical mechanics, including not only discussions of the law of falling bodies and projectile motion but also an analysis of the pendulum, an example of harmonic motion. His studies fall into the branch of classical mechanics known askinematics, or the description of motion. Although Galileo and others tried to formulate explanations of the causes of motion, the focus of the field termeddynamics, none would succeed before Newton.

Galileo’s fame during his own lifetime rested not so much on his discoveries in mechanics as on his observations of the heavens, which he made with the newly invented telescope about 1610. What he saw there, particularly the moons of Jupiter, either prompted or confirmed his embrace of the Copernican system. At the time, Copernicus had few other followers in Europe. Among those few, however, was the brilliant German astronomer and mathematician Johannes Kepler.

Kepler devoted much of his scientific career to elucidating the Copernican system. Although Copernicus had put the Sun at the centre of the solar system, his astronomy was still rooted in the Platonic ideal of circular motion. Before Copernicus, astronomers had tried to account for the observed motions of heavenly bodies by imagining that they rotated on crystal spheres centred on the Earth. This picture worked well enough for the stars but not for the planets. To “save the appearances” (fit the observations) an elaborate system emerged of circular orbits, calledepicycles, on top of circular orbits. This system of astronomy culminated with theAlmagest of Ptolemy, who worked in Alexandria in the 2nd century ad. The Copernican innovation simplified the system somewhat, but Copernicus’s astronomical tables were still based on circular orbits and epicycles. Kepler set out to find further simplifications that would help to establish the validity of the Copernican system.

In the course of his investigations, Kepler discovered the three laws of planetary motion that are still named for him. Kepler’s first law says that the orbits of the planets are ellipses, with the Sun at one focus. This observation swept epicycles out of astronomy. His second law stated that, as the planet moved through its orbit, a line joining it to the Sun would sweep out equal areas in equal times. For Kepler, this law was merely a rule that helped him make precise calculations for his astronomical tables. Later, however, it would be understood to be a direct consequence of the law of conservation of angular momentum. Kepler’s third law stated that the period of a planet’s orbit depended only on its distance from the Sun. In particular, the square of the period is proportional to the cube of the semimajor axis of its elliptical orbit. This observation would suggest to Newton the inverse-square law of universal gravitational attraction.

By the middle of the 17th century, the work of Galileo, Kepler, Descartes, and others had set the stage for Newton’s grand synthesis. Newton is thought to have made many of his great discoveries at the age of 23, when in 1665–66 he retreated from the University of Cambridge to his Lincolnshire home to escape from the bubonic plague. However, he chose not to publish his results until the Principia emerged 20 years later. In the Principia, Newton set out his basic postulates concerning force, mass, and motion. In addition to these, he introduced the universal force of gravity, which, acting instantaneously through space, attracted every bit of matter in the universe to every other bit of matter, with a strength proportional to their masses and inversely proportional to the square of the distance between them. These principles, taken together, accounted not only for Kepler’s three laws and Galileo’s falling bodies and projectile motions but also for other phenomena, including the precession of the equinoxes, the oscillations of the pendulum, the speed of sound in air, and much more. The effect of Newton’s Principia was to replace the by-then discredited Aristotelian worldview with a new, coherent view of the universe and how it worked. The way it worked is what is now referred to as classical mechanics.