Chapter 12.

Newtonian Science

12.1. Isaac Newton (1642-1727)

• Born on Christmas day, 1642, in small village of Woolsthorpe in Lincolnshire, England
  • 12 years after Kepler's death
  • Same year as Galileo's death
• Not a particularly noteworthy childhood
  • Father died a few months before Newton's birth
  • Mother later remarried, after which Newton raised by grandmother on family farm
  • Was apparently fond of tinkering and building things as child
  • Nothing in his school work or undergraduate studies reveal magnitude of his intellect
• All his life he gives impression of being an unloved man
  • Never marries
  • Never has a loving relation with either a woman or man as far as is known
  • Always fearful that people would steal his ideas
  • Never works with any other scientist
  • Newton's achievements were solitary
• Educated at Trinity College, Cambridge University
  • Entered in 1661, studied mathematics as applied to astrology
  • Isaac Barrow was his tutor, formerly Professor of Greek, then Professor of Mathematics, one of recognized scholars at Cambridge
  • Years of 1665 and 1666 were Plague years with university closed; Newton spent most of time on family farm, mother again a widow; these years prove to be golden for Newton
    • mathematics: binomial theorem, differential calculus (fluxions)
    • optics: theory of colors, theory of light
    • mechanics: gravity as inverse square law, first two laws of motion
• Cambridge years
  • 1667 returned to Cambridge
  • 1669, Barrow resigns his chair in favor of Newton who at age 27 became Lucasian Professor of Mathematics
  • After 1669, becomes involved primarily in work in chemistry and theology until he leaves Cambridge in 1699
  • Although Newton does some publishing in his early years after 1669, he is early on involved in dispute with Robert Hooke, then President of the Royal Society of London; consequence of which he does not publish to any appreciable extent
  • 1684, Edmund Halley encouraged Newton to publish his research on mechanics and later even paid for publication
  • 1687, less than 2 years later, Mathematical Principles of Natural Philosophy is published; it is an instant success and reveals Newton's genius for original thinking
  • After 1687, nervous breakdown, lasted about 5 years
• London years
  • 1699, appointed Master of Mint, lead currency reform
  • Served in Parliament in 1689 and 1701
  • 1704, published his Opticks
  • 1705, knighted under reign of William and Mary; became President of Royal Society and remained so until death
  • 1727, died, buried in Westminster Abbey

12.2. Newton's Principia

• Probably greatest single book in scientific discourse
• Although parts of book published between 1667 and 1687, bulk of work including translating many proofs from calculus to geometrical done in two years from 1684 to 1686
• No more encompassing treatise has ever been published; lays out the mathematical formulation of theory and applies it to many long-standing problems: planetary motion, lunar motion, tides, etc.
• Provides a standard for doing scientific investigations (along with Opticks in 1704)
• Establishes for all times mathematics not only as language of physics (and all science) but as a means of knowing
  • "Philosophy is written in that great book which ever lies before our eyes--I mean the Universe--but we cannot understand it if we do not learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures without whose help it is impossible to comprehend a single word of it, without which one wanders in vain through a dark labyrinth." Galileo Galilei

12.3. Newtonian Concepts of Motion

• Scalar quantity
  • Physical quantity that requires only a magnitude to fully specify
  • Examples: mass and energy
• Vector quantity
  • Physical quantity that requires both a direction and a magnitude to fully specify
  • Examples: force, velocity, momentum, and acceleration
• Frame of reference
  • Reference frame is place from which we observe and measure motion
  • Origin: a point in reference frame from which measurements begin
  • Fundamental reference direction: a direction in space defining orientation of reference frame
  • Secondary reference directions: in combination with fundamental direction they define dimensionality and nature of space
• Mass
  • Measure of a body's inertia, i.e., its resistance to change in its state of motion; fundamental measure of amount of matter in body
• Force
  • Pushes or pulls that change body's state of motion
• Velocity
  • Time rate of change of position (speed) in particular direction
  • Instantaneous velocity (at a single instance of time)
• Acceleration
  • Acceleration is the time rate of change of velocity in a particular direction
  • Instantaneous acceleration
• Momentum
  • Fundamental measure of quantity of motion, both magnitude and direction, a body possesses
  • Large-mass bodies able to transfer greater quantity of motion when bodies interact (collisions)
  • Large-velocity bodies able to transfer greater quantity of motion when bodies interact
  • Thus momentum must equal mass times velocity, i.e.,
  • P = mv
  • State of motion is momentum a body possesses in some frame of reference
• Conservation of momentum: René Descartes, 1644
  • Total momentum in the Universe is a constant, although interaction of various bodies with each other continuously redistributes total momentum among individual bodies of the Universe
    • Fundamental principle which tells us how world may behave
    • Bodies may exchange quantities of motion, but may neither create nor destroy motion
  • Total momentum of Universe is finite constant, albeit a very large constant; suggests Universe has finite nature
  • Conservation principle also exists for angular momentum, quantity of rotational motion
  • Finite quantity of rotational motion in Universe which may neither be created nor destroyed only exchanged between interacting bodies

12.4. Newton's Theory of Motion

• 1st Law of Motion
  • A body remains in its state of motion unless acted on by an external force (Galileo's principle of inertia defining "natural motion")
  • momentum = constant
• 2nd Law of Motion
  • A body acted on by an external force will change its momentum in direction of force, such that the greater the force the greater the change in momentum
  • force = time rate of change of momentum
  • f = d(P)/dt
• 3rd Law of Motion
  • Forces always occur in pairs, i.e., for every action there is an equal and opposite reaction
  • f(action) = f(reaction)
• Universal Law of Gravitation
  • All objects in the Universe attract each other with a force that varies directly as product of their masses and inversely as square of their separation from each other
  • f(gravity) = m(1)m(2)/{r(12)[E(2)]}
  • Gravity is always attractive and never repulsive
  • For two homogeneous spheres, r(12) is distance between centers, not surfaces; thus one can think of bodies attracting each other as if they were mathematical points
  • Based on concept of "action-at-a-distance"
• Explaining planetary motion
  • Planet's velocity is in direction tangent to orbit at any instant of time; for circular orbit, velocity perpendicular to direction toward Sun
  • Sun by its gravitational attraction continually exerts an attractive force on planets
  • Attractive gravitational force responsible for accelerating planet toward Sun
  • Acceleration toward Sun alters instantaneous velocity so that it is not tangent to orbit but inclined slightly toward Sun
  • This new velocity is again altered by acceleration with consequence that planet moves in closed orbit about Sun
  • Motion of planets is independent of fact that Sun is an extended body
  • Turn off Sun's gravitational attraction, planets' momentum would not change and planets would move in straight lines across space
  • Planets Neptune and Pluto discovered, 1846 and 1930, respectively, because of their gravitational influence on Uranus
• "I feign no hypotheses"
  • Theory of gravitational forces, every particle attracting every other particle, amazingly successful in providing the cause for which observed changes in motion were the effect
  • Newton, his contemporaries, and his successors bothered by how one could account for gravity itself
  • How does gravity work? Does some intervening medium (later called ether) exist which transmits gravitational pull in a mechanical fashion?
  • Very question shows how firmly human mind is committed to mechanical models and unsatisfied a mathematical argument leaves our emotions
  • Newton's response
    • "But hitherto I have not been able to discover the cause of those properties of gravity from phenomena [observations], and I feign no hypotheses.... To us it is enough that gravity does really exist, and act according to laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies and of our sea."
  • This famous statement, echoing Galileo's admonitions, exempts science from providing fundamental causes (why questions) and accepts mathematical relations as explanation (how questions)
  • Privately, Newton did in fact believe some material agent existed to explain "action-at-a-distance" as evidenced by letter of 1693 to Richard Bentley
    • "That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers."