Newtonian Dynamics

Newtonian dynamics really consists of two parts. One is dynamics, which refers to the description of the motion of particles in the presence of forces, whatever the origin of the forces may be. The second is the part which describes the forces acting on particles due to the presence of other bodies around them, i.e., gravitation.

Newton's laws

Let us review again what Newton was aware of when he embarked on his description of the interactions of matter. There was the work Galileo had done on mechanics. The experimental results that Galileo handed down to Newton was:

1. All bodies tend to continue in its state of rest or uniform rectilinear motion unless disturbed by an external force.
2. Falling bodies obey the law of constant acceleration. That is, the rate of change of velocity is a constant when the force acting on the body is constant. And it increases or decreases with the force.
3. All bodies of the same mass, regardless of constituent material have the same acceleration for a given force.

Newton codified these observations into a set of three laws, known by his name.

• Newton's First Law: This is essentially a re-statement of Galileo's statement of it. It states that a body continues at rest or in uniform motion along a straight line unless it is acted upon by a force. This is also known as the law of inertia.
• Second Law: This states that the acceleration of a body is proportional to, and is in the direction of, the force acting on it. The constant of proportionality is the inertia of the body and is called its mass. This statement is codified as, F = m a.
• Third Law: This states that action on a body and its reaction are equal and opposite. So if a body applies a force on another, the reaction of the second body on the first is equal in magnitude and opposite in direction to the force applied by the first on the second.

The first law is infact more important than it may seem. It is more than a statement of the obvious. It helps the observer refer to physical phenomenon. Suppose you picked up a ball and let it go. At the instant you let it go, it has zero velocity, but it doesn't stay still, it falls down. Remember this is before Gravitation was understood. So there was no reason to think of the existence of a force. So is the first law violated then? After Newton's first law we can say, no, infact the appearance of the violation of first law indicates that there are forces present. We can then use it to define ideal observers who would see the ball stand still. Obviously, these observers would be crashing down to Earth at the same rate as the ball was. The frames of reference fixed on these ideal observers are called inertial frames.

Gravitation

Now recall again the three laws of Kepler that Newton had in hand when he started on his endeavour.

1. The planets move in ellipses with the sun in one of the foci.
2. The line joining the planet and sun sweeps out equal amounts of area in equal time intervals.
3. Consider the average radius r of an orbit of a planet. The cube of this radius is proportional to the square of the time period (i.e., time taken by the planet to return to a point on its orbit).

Newton realised that in order to explain the motion of the planets he had to hypothesize the following. That all bodies in the Universe attracted each other. The reasoning for this is elegant and entirely non-trivial. If planets are rotating around the Sun then they are obviously not in uniform motion rectilinear motion. Newton's first law implies they must have a force acting on them. It stands to reason that this force is applied on them by the Sun. But the Sun isn't unique in this. The moon revolves around the Earth. So by the same argument the Earth applies a force on the moon. And there are the satellites of Jupiter. Then Newton avoided unnecessary complications in defining this force, and made a bold hypothesis. He said all bodies in the Universe attracted each other through a force he called Gravitation. This immidiately solved an huge slew of problems. For one it became clear why we were not being blown off the Earth's surface as it careened through space. We were stuck to it by gravity! Now Newton didn't wish to distinguish between live humans and Earth when it came to physical phenomenon, perhaps because of the influence of Cartesian theories of the detachment of mind and physical reality which encouraged the explanation of physical phenomenon strictly in terms of physical interactions between constituent particles. So he suggested that we attracted the Earth just as it attracted us. In fact that is required by Newton's third law. We apply on Earth a force just as strong as the one that it applies on us. Now clearly a feather falls far more lightly to the Earth than does a metal ball which is heavier. So Newton speculated that the force on the falling body must be proportional to its mass. But again we can think of the Earth being pulled to the feather rather than vice versa. So the force must also be proportional to the mass of Earth. But then the Sun's mass must be larger than Earth's if it is forcing so many large planets including Earth to rotate around it. Why isn't the Moon orbiting around the Sun? Newton speculated that this is because the Moon is much closer to the Earth than to Sun. So the gravitational force between two objects must be inversely related to the distance between them. Kepler's second law can be shown to imply that the force between two particles may depend only on the distance between them and not on their orientation. Such forces are called central forces. The other two Kepler's laws can be used to show that the dependance can only be such that the force falls as the inverse square of the distance between the particles. So for two particles of masses and , seperated by a distance d, the gravitational force on them is,

where, G is the constant of proportionality, called the Gravitational constant. The force is attractive, acts equally on both particles and along the line connecting the two pulling them towards each other. This law is called the Gravitation or the inverse squared law.

This is a very special kind of force law to which we are restricted by Kepler's first law. Joseph Bertrand, born in 1822 in Paris, showed that there are only two kinds of central forces that have closed orbits, i.e., particle trajectories where the repeats a fixed pattern, like a circle, ellipse or the figure 8. One of them is the Newton's inverse squared law and the other is when the force is still attractive, but directly proportional to the distance between them, . This law was first used by Hooke, to describe the restoring force on stretched materials, like a rubber band! He was born in 1635 in the wind swept Isle of Wight in England. He is most famous perhaps for discovering plant cells, although physicists know him mostly for his force law mentioned above. A true renaissance scientist, he invented the universal joint, the iris diaphragm, and an early prototype of the respirator; invented the anchor escapement and the balance spring, which made more accurate clocks possible; served as Chief Surveyor and helped rebuild London after the Great Fire of 1666; worked out the correct theory of combustion; assisted Robert Boyle in working out the physics of gases; worked out the physics of elastic materials; invented or improved meteorological instruments such as the barometer, anemometer, and hygrometer; and so on. Along with Edmund Halley, he suggested (before Newton published his work but presumably after Newton had worked them out) that the force between the heavenly bodies was proportional to the inverse square of the distance between them. However he couldn't solve the motion of the planets because calculus hadn't been publicised by Leibniz yet (although Newton had already invented it). When Halley (the world's first actuary and meteorologist, as well as the discover of the recurrent nature of comets) induced Newton to publish his work, he plunged Hooke and Newton into a bitter quarrel over attribution. The bitterness while rooted in the intransigence of either individual is also understandable in view of the cruelly competitive world they inhabited. In an environment vitiated by patronage, attribution was more than a matter of individual pride.

It is tragic that Kepler died before the publication of the Principia in 1687, because one of the most painful things he had to do was to drop the requirement that planets move in circles. A deeply theological man he was greatly pained at having to drop the beautiful and elegant music of the spheres that was driven by Ptolemy's Prime mover from outside the celestial sphere. Much of his early work on explaining planet orbits in terms of geometric figures, which were rooted in the Pythagorean vision of harmonia ("fitting together") lost its meaning in view of his laws. How great a joy it would have been to him if he could have seen the same harmony return in such blazing purity. For as Newton showed, not only is the gravitational force uniform on spheres (i.e., all bodies distributed in a sphere around a gravitating mass, like the sun, feel the same force) but the flux of the acceleration is constant on spheres as well. The acceleration of any body is given by Newton's second law as force per unit mass, a = f/m. If the force be due to the gravitational pull of a particle of mass M then the acceleration is, at a distance r. But the surface area, A, of a sphere of radius r is , so the flux of the acceleration, , a constant! Harmony of the spheres returns with a depth and significance even greater than before.