Gravity

Newton:

Newton expanded on the work of Galileo to better define the relationship between energy and motion. In particular, he developed the following concepts:

change in velocity = acceleration -> caused by force
inertia = resistance to change in velocity and is proportional to the mass of the object
momentum = quantity of motion energy and is equal to mass times velocity
law of conservation of momentum = total momentum (mass x velocity) of an interaction is conserved -> is the same before and after

Example: Cars and Trucks on Ice!

A corollary to Newton's ideas was the so called Clockwork Universe model. A concept that states that the total momentum of the Universe is conserved, interactions redistribute the momentum, but the total never changes. In this model, God only starts the clock (initial cause), then it runs by itself for the rest of time.

Newton's laws of motion:

1st law: a body remains at rest or moves in a straight line of constant velocity as long as no external forces acts on it
2nd law: a body acted on by a force will accelerate such that force equals mass times acceleration (F=ma)
3rd law: for every action there is an equal and opposite reaction

For example, from Newton's 1st law we know that an object travels in a straight line unless acted upon by an external force. A circular orbit is clearly not a straight line, what is the force? Newton showed that the planets are acted on by the force of gravity arising from the Sun. Each orbit is a constantly changing velocity where gravity adds a small ``delta-vee'' at each moment. This ``delta-vee'' is what produces the elliptical curvature that is the orbit.

Also, from Newton's 2nd law, when a baseball player hits a ball he applies a force, F, to the ball of mass m. Let's say he hits a tennis ball of mass of one-tenth the mass of a regular baseball (1/10m). What is the resulting acceleration? Ten times the acceleration of a regular baseball and, therefore, ten times the final velocity and ten times the distance hit.

You are trapped on a lake of ice with a sandbag. Remembering Newton's 3rd law, how do you escape?

Newton's Law of Universal Gravitation:

Galileo was the first to notice that objects are ``pulled'' towards the center of the Earth, but Newton showed that this same force (gravity) was responsible for the orbits of the planets in the Solar System.

Objects in the Universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their distances

All masses, regardless of size, attract other masses with gravity. You don't notice the force from nearby objects because their mass is so small compared to the mass of the Earth. Consider the following example:

With vector calculus, Newton was able to develop a cosmology which included the underlying cause of planetary motion, gravity, completed the solar system model begun by the Babylonians and early Greeks. The mathematical formulation of Newton's dynamic model of the solar system became the science of celestial mechanics, the greatest of the deterministic sciences.

Although Newtonian mechanics was the grand achievement of the 1700's, it was by no means the final answer. For example, the equations of orbits could be solved for two bodies, but could not be solved for three or more bodies. The three body problem puzzled astronomers for years until it was learned that some mathematical problems suffer from deterministic chaos, where dynamical systems have apparently random or unpredictable behavior (see below).

Differential Gravitational Forces (Tides):

Tides are caused by the interaction of a body's motion around a planet or the Sun and the internal force of gravity.

Water Tides:

Water tides are caused by the fact that the water on the Earth's surface is more easily deformed by tidal forces than the rocky crust. And the strength of tides are dependent on three factors:

location on the Earth's surface
orientation of the Sun and the Moon (both has approximately equal tidal influence on the surface of Earth)
geographic features (shape of bay, inlets, etc.)

Roche Limit:

What happens when the tidal forces become greater than the internal gravity of an object? The object is torn apart. This occurs when a moon approaches too close to its primary. The tidal forces increase as R, the distance between the planet and the moon, becomes smaller until the moon is disrupted into numerous small bodies. This is the origin to the rings around Saturn and other Jovian worlds.

In celestial mechanics, the Roche limit is the distance within which a celestial body, held together only by its own gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings whereas outside the limit material tends to coalesce.

Three Body Problem and Complexity:

Deterministic laws, such as Newton's laws of motion, imply predictability only in the idealized limit of infinite precision. The Universe itself cannot know its own workings with absolute precision, and there cannot predict what will happen next in every detail. Deterministic chaos seems random because we are necessarily ignorant of the ultrafine details and so is the Universe itself.

The behavior of complex systems is not truly random, it is just that the final state is so sensitive to the initial conditions that it is impossible to predict the future behavior without infinite knowledge of all the motions and energy (i.e. a butterfly in South America influences storms in the North Atlantic).

Even games with simple rules can produce complex behavior, as shown in the following example:

Although this is `just' a mathematical game, there are many examples of the same shape and complex behavior occurring in Nature.