GP 25 Web Book

Day 3
The New Math and Some Physics

Bruno considered geometry fine for children but a waste of time for adults.  He had a vague sense that its self-imposed limitations hindered science.  He was far more progressive in his science. He advocated the same physical laws for the Earth and the heavens.  He considered motions to be relative, emphasizing that the Earth moves. He did not envision that increasing the scope of mathematics would unleash the power of relativity and unified physics.

Relativity of motion.  We are used to thinking as the Earth as fixed, terra firma.  Only a tort lawyer would say that the defendant’s barn struck his drunken plaintiff’s car.  This natural view of absolute motions retarded the progress of medieval science. Yet, even children sense that motions are relative.

Sally and Billy have discovered for themselves that motion is relative. They cannot tell how fast the car is moving from the game of catch. They can only tell when the velocity of the car changes.

Both Bruno and Galileo used a moving ship as their example of the relativity of motion. They noted that a weight dropped from the mast falls in the same place whether the ship is moving or not.  Their point is that, like the passengers of a ship, the inhabitants of the Earth experience no sensation of the movement of the Earth. Galileo took it one step further; one can simply add up velocities.  The velocity of the weight observed on the shore is the velocity of the ship plus the velocity of the weight as observed from the ship.

Galileo was correct to the level that we can observe in our daily lives.  We have already seen that the motion of the Earth in its orbit causes the aberration of starlight.  Other subtle effects became evident with developments in mathematics and physics.

Flexible mathematics.  In classical geometry, one is allowed to have a compass and a straight edge.  These are assumed to work perfectly so that lines have no width.  Proofs are exact, but they are limiting.  One can easily divide an angle in two, but it is impossible to divide an angle into three parts.  (One can get very close with repeated bisections, but never exact.)

One is limited to straight lines and circles. Being limited to circles straight jacketed Copernicus.  The orbits of the planets are in fact not circular and he still needed a morass of epicycles to describe them.  Galileo used circular orbits but made no detailed calculations of planetary position.

Kepler abandoned the assumption of circular orbits by using ellipses.  They are the curves that one gets cutting a cylinder like a sausage oblique to its axis.  One also gets an ellipse if one cuts a cone through its axis, like with a carrot.  One may draw an ellipse with a loop of string and two pins (Figure 1).  The length of the loop stays constant as one goes around.  Mathematically, the two pinpoints are the foci (plural of focus) of the ellipse.  The sum of the distances from each focus to a point on the ellipse stays constant.  A circle is an ellipse with both foci in the same place.

Figure 1: You make an ellipse when you cut a carrot.  Mathematically, an ellipse has major axis a and minor axis b.  The sum of the distances X and Y from the foci F1 and F2 remains constant.  This lets one drawn an ellipse with a loop of string around two tacks.  This was quite messy with ink in the days before computer plotting.

Kelper found out from analysis of reams of data that:

(1) The orbits of the planets are ellipses with the Sun at one focus.

(2) The planet sweeps out an equal area with lines going to the Sun over equal times. (See Figure 2)

(3) The square of the period (year) of a planet is proportional to the cube long axis of the ellipse.

Unlike epicycles, these laws are predictive.  One can see if a comet, for example, follows them.  The deviations of the actual planetary orbits from these laws would provide the keys to the development of physics.   A more general way of representing curves would be needed to do this.

Figure 2: Kepler’s second law is easily visualized geometrically. A planet orbits in an ellipse with the Sun at one focus.  The other focus has only mathematical significance. The planet sweeps out equal shaped areas (shaped) in equal intervals of time.  It thus moves fast when it is near the Sun.

René Descartes (His name translates “Of the maps.”) provided the tool, which may seem mundane to us.  In two-dimensions, one represents positions, like a map.  The coordinates might be north and east.  In three dimensions, one needs three perpendicular coordinates, north, east, and up would do locally.  One can pick the origin wherever one chooses, usually for convenience.  One can move back and forth between coordinate systems with different origins and different coordinate directions.  The full power of relativity comes from requiring that physical laws not change when one changes coordinate systems,

Figure 3: Graphs let one see the results of physical experiments and here simple calculus. The distance traveled by a roller on a ramp is a function of the time since it started from rest.  The object moves a distance X in time t.

With regard to astronomy, one picks 3 perpendicular coordinate directions on the celestial sphere of fixed stars.  Putting the Sun at the center, one can represent the position of each planet with these three coordinates.  One needs only 5 parameters to describe the orbit of each planet with Kepler’s laws, three to describe the long axis of the ellipse and two to describe the short axis (which is perpendicular to the long axis).  This is a lot more elegant than a myriad of epicycles.

In the meantime, Galileo had been studying the dynamics of falling bodies.  To slow them down, he cleverly let them roll down a ramp.  He found that a roller that went (to make it easy in modern units) 1 meter by the end of the first second from rest, went 4 meters by the end of the next second, 9 meters by the end of the third, and so on. The distance traveled was proportional to the time squared (Figure 3).  The average velocity over each second increased linearly as 1, 3, 5 meters per second.  In three dimensions, he found that the vertical velocity of a dense object in flight decreased (with upward velocity positive) linearly with time.  The horizontal velocity stayed more or less constant.  For a real cannon ball, one had to correct some for air resistance.

Galileo and Kepler were not particularly receptive to each other’s innovations.  (It would have been dangerous for Galileo to openly support Kepler, a Protestant, by name.) Galileo retained the medieval notion of circular motions being natural in the heavens.  He did not apply his laws of falling bodies to the planets.  As legend has it, a falling apple jolted Isaac Newton into doing just that. Newton’s contribution involved the instantaneous change in quantities. I illustrate the method by graphing the results of the ramp experiment (Figure 3).  My intent is to illustrate what calculus does without having to teach it.  The distance traveled increased with the square of time.  The slope of the curve of position versus time is the instantaneous velocity.  We have already seen that the average velocity (over a second) increases linearly with time.  The instantaneous velocity also increases linearly with time.  The area under the graph of the velocity versus time gives the distance traveled.  Remembering that this area is one-half of the base times the height, we find that the distance traveled is proportional to the square of the time.  We can also plot the slope of the velocity curve, that is, the change of velocity per time, called the acceleration.  In this experiment, it is constant.  Galileo got to this point.

Newton generalized this example into calculus in 3-D.  To keep things simple, I stay in 1-D and use modern notation.  To begin with the instantaneous velocity, I find the position (called X) at time 1 and later time 2.  The difference (a differential called X) is (time 2) squared minus (time 1) squared.  We get some simplification in this special case, because this quantity factors into (time 2 + time 1) times (time 2 - time 1).  As we are after the instantaneous velocity, we assume the difference between time 2 and time 1 to be quite small, the differential t.  The sum (time 1 + time 2) is then just 2 times the time t.  The velocity is this change divided by t.  In mathematical notation, the velocity V in meters per second at time t (in seconds) is

In calculus, one assumes that differentials are very small so that the velocity is the derivative dX/dt.  Similarly, the acceleration is the derivative of the velocity with respect to time, dV/dt., which is here 2 meters per second each second, or more compactly, 2 meters per second squared (Figure 4).

Figure 4: This graph illustrates simple calculus. The acceleration of the object on the ramp is constant, 2 meter per second per second.  The velocity at time t is proportional to the blue shaded area.

Figure 5: More simple calculus.  The velocity of the object increases linearly with time (top).  The distance traveled at time t is proportional to the blue shaded area.  We can compute the blue shaped area by dividing up into rectangles of length t.  In calculus, one assumes that the intervals are very small.

As already noted, we can get the position if we know the history of the velocity.  Distance equals rate times time.  To do this we add up the change of position for each time interval (Figure 5).  That is,

where the sum sign capital sigma tells us that we need to add up the product of the velocity and length of each time interval, which I have denoted by a subscript i.  In calculus, the intervals are assumed to be very small and the sum is replaced by an integral.  The position at some time ƒ after the starting time defined as zero 0 is:

The differential dt reminds us that we are taking time steps.  The integral sign (the funny s in books printed before about 1800) tells us that we need to add up velocity steps between the starting time 0 and the final time f.

Calculus is extremely useful in all quantitative science.  Scientists represent physical laws in terms of instantaneous change in time and space, called a differential equation.  They can predict the subsequent behavior of a system from its initial conditions and their differential equation.  They also have rules, like the conservation of energy, that involve adding up everything in their system.  They use integrals to do this.

In Newton’s case, all he needed to get the positions of the roller or of a planet was a rule to determine the velocity at each time.  Using the calculus that he just invented and simple special cases like the roller, he proposed:

(1) Every body at rest stays at rest and every body in motion stays in straight motion unless it is acted on by a force.

This rid the world of the medieval concept that circular motion was natural in the heavens and that force was needed to keep a body moving.  It also included the concept that velocity and rest are relative.

(2) The acceleration of a body is inversely proportional to the body’s mass and directly proportional to the forces acting on it.

The first law is the special case of the second law where there are no forces. The weight that we measure with a scale with springs is a force.  The balance in a doctor’s office measures mass by comparing your mass to a known mass.

(3) Every action is opposed by an equal and opposite reaction, or there are no unbalanced forces.

For example, the force of water on oars balances the force the oars exert on the water.  The forces acting between two bodies in isolation are equal and opposite so no net force acts on the two bodies.  These forces do not accelerate the center of mass of the two bodies.

Newton proposed the law of universal gravitation to complete his physical description of planetary orbits.

The gravitational force between two bodies acts along the line between the two bodies and is proportional to the product of their masses and inversely proportional to the square of their distance of separation.

The forces are equal and opposite as in the third law.  Equivalently the force per mass on one object is proportional to the mass of the other and the inverse square of the distance.  The acceleration of the first body is proportional to this quantity because of the second law.  The acceleration from gravity decreases with the square of distance just like the brightness of a light.

Using calculus, Newton derived Kepler’s laws with the correct assumption that the Sun has almost all the mass of the solar system.  Derivation of the elliptical orbit is complicated, but the other two laws have simple illustrations.  Kepler’s second law of equal areas is the conservation of angular momentum.  An ice skater spins more rapidly when she pulls in her arms.  This is equivalent to saying that gravitational force acts on a line between the Sun and a planet.

Figure 6: Graphical demonstration of Kepler’s third law.  The planet orbits in a circle of radius R at an angular rate .  Its speed remains constant, but the velocity changes over the interval t.  Note the change in velocity, the acceleration a is in the direction between the planet and the star.

Kepler’s third law has a simple graphic derivation for circular orbits (Figure 6).  To allow myself to use the small angle formula, I express angles in radians and the rate of rotation in radians per second, which I denote by .  The planet covers an angle times t over the small time interval t.  The speed (magnitude of the velocity) S does not change as the planet orbits, but the direction of the velocity direction does.  Using the small angle formula again, the change in velocity is the speed S times the angle xt.  Note that the change is radial inward toward the Sun.  The gravitational force is proportional to the product of the mass of the Sun M times the gravitational constant G and inversely proportional to the square of the planet-Sun distance R.  Once again using the small angle formula, the speed is xR. Equating the force per mass to the acceleration gives the equation

The planet revolves 2 times radians revolution so its year Y is 2 times divided by the rotation rate .  We get Kepler’s third law with the radius making this substitution in the acceleration equation

One does not need to know the mass of the Sun to apply this equation, only the product GM.  In fact, the gravitational constant is difficult to measure because it is hard to get a big enough mass in the laboratory to produce a precisely measurable gravitational force.  The constant was not measured reliably until the early 1800s.  It is still the least accurately measured physical constant.

The Moon provided the start to Newton’s thinking because its orbit is mainly determined by the mass of the Earth.  He conclusively unified astronomy and physics.  Newton got the Moon’s orbital acceleration from the length of the month and he knew the acceleration of gravity at the Earth’s surface.  He knew the radius of the Earth’s surface and the distance to the Moon from the Earth’s center.

Newton had a general method for describing the motions of the planets without using his new calculus (Figure 7).  One needs only to add up the forces per mass on one planet from the Sun and the other planets and use this acceleration. The acceleration acts over the time interval t changing the velocity.  One then does this for the Sun and all the other planets to assign new velocities.  One then uses the computed velocities to get new positions after time t and then repeats getting new velocities.  The procedure works if one takes small enough time steps.  This was a major problem with hand calculations in Newton’s time and still a problem if one wishes to model orbits over millions of years on a computer.  Mathematical astronomers have put considerable effort into using the fact that Kepler’s laws are approximately correct to make calculations more efficient.

Figure 7: Graphical illustration of numerical calculation of planetary orbit.  At time 0, the velocity is known.  One obtains the position at time 1 by moving Vt.  One obtains the forces at the new position and changes the velocity by the acceleration times the time interval, mathematically at.  One gets the position at point 2 using the new velocity.  One continues alternately updating velocity and acceleration. Small time and spaces steps are needed for the procedure to work accurately.  This is a limitation even with modern computers. With calculus, it is sometimes possible to do the calculation with simple functions, like for a planet in circular orbit.

Please keep things simple!  The non-mathematically inclined reader will probably already be complaining about equations by this point.  Still, Newton’s laws are simple.  There are only seven free parameters for each planet at the starting time of the calculation, 3 for its position, 3 for its velocity, and 1 for its mass.  Once we have these for all the planets, we can compute the positions of the planets at any time in the future or even the past.

That is, we can not only make testable predictions about the future, but we can examine data from antiquity. If one discovers another object, like Halley’s comet, one can compute its orbit and show that it is the same comet that was seen many times in the past and predict its future arrivals.  Astronomers have done this with thousands of asteroids, comets, and spacecraft. They routinely model the orbits of multiple star systems and extrasolar planets.

If we proceeded with the planets as point masses as discussed above, we would find out that our predictions are not perfect. Some of the imperfection occurs because we do not have exact estimates of the masses and the starting conditions.  If we have been careful in estimating errors in our data, it is a simple though cumbersome matter to predict how accurate our predictions should be.  Typically the actual inaccuracy of the predictions exceeds the expected errors.  Because we have a physical model, we can learn a lot from these discrepancies.  With epicycles, we would be able to do nothing but add still more epicyles and tweak the ones we already have.

We would find that the orbital distance of the Moon increases with time and on average the rotation of the Earth slows down.  This is due to tides that dissipate energy.  The net effect is to transfer angular momentum of the Earth to the Moon.  In general, the planets need to be represented as finite rotating bodies.  Once Newton did this, he got a physical explanation for the procession of equinoxes on the Earth in terms of the forces on a spinning object.  This conclusively showed that the Earth rotates on its axis.  Astronomers have since included these more cumbersome effects in their predictions.

Historically, there where two more serious problems with planetary predictions that led to major discoveries.  Once the planet Uranus was discovered, astronomers just could not predict its orbit right.  The answer turned out to be that the planet Neptune exists still farther away from the Sun. Two mathematical astronomers independently predicted its position using Newton’s laws.  They were right on. Neither could astronomers get Mercury’s orbit right.  The blinding search for other planets near the Sun came up fruitless.  This aided the discovery of the theory of general relativity.

We have seen a progression from epicycles to general relativity.  This may not look like a simplification but it is. The number of rules and special restrictions decreased with each step and the number of possible predictions increased.  One can still use epicycles to represent motions.  (For the mathematical, epicycles are a form of infinite series.)  Ptolemaic astronomy has special rules for each planet and makes zero general predictions, like those that pointed to Neptune.  From then on each step retained the previous step as a useful approximation.  Kepler’s laws are the case of Newton’s laws where all the mass is in the Sun and the planets are small point masses.  Newton’s laws are the special case of general relatively where the velocities are low and the masses small.  General relativity removed the restriction, known from the time of Newton, that Newton’s laws did not apply to an accelerating coordinate system.

One famous prediction of general relativity is that light does not travel in a straight line, an example of the curvature of space around masses.  This effect was confirmed by carefully observing the position of stars during an eclipse of the Sun. The calculations required considerable sophistication to get the actual very precise prediction, but the concept of curvature is simple enough to express in a single compact equation. In contrast, a vague pronouncement that something weird would happen during the eclipse is useless to science.

Copyright Notice

back to top