Galileo's laws of Motion:

Aside from his numerous inventions, Galileo also laid down the first accurate laws of motion for masses. Galileo measured that all bodies accelerate at the same rate regardless of their size or mass.

Key among his investigations are:

• developed the concept of motion in terms of velocity (speed and direction) through the use of inclined planes.
• developed the idea of force, as a cause for motion.
• determined that the natural state of an object is rest or uniform motion, i.e. objects always have a velocity, sometimes that velocity has a magnitude of zero = rest.
• objects resist change in motion, which is called inertia.

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Kepler's laws of Planetary Motion:

Kepler developed, using Tycho Brahe's observations, the first kinematic description of orbits, Newton will develop a dynamic description that involves the underlying influence (gravity)

• 1st law (law of elliptic orbits): Each planet moves in an elliptical orbit with the Sun at one focus.

Ellipses that are highly flattened have high eccentricity. Ellipses that are close to a circle have low eccentricity.
• 2nd law (law of equal areas): a line connection the Sun and a planet (called the radius vector) sweeps out equal areas in equal times

Objects travel fastest at the low point of their orbit, and travel slowest at the high point of their orbit.
• 3rd law (law of harmonics): The square of a planet's orbital period is proportional to its mean distance from the Sun cubed.

The 3rd law is used to develop a ``yardstick'' for the Solar System, expressing the distance to all the planets relative to Earth's orbit by just knowing their period (timing how long it takes for them to go around the Sun).

Although successful, Kepler's laws remained a set of empirical rules without a dynamical basis. The link between these laws and the physical world would be established about 50 years later by Isaac Newton (1642-1727).

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Newton:

Newton's Explanation of Kepler's Laws

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.

momentum experiment

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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

Well fine, if gravity is an attractive force, then how come the moon doesn't come crashing into the earth, or the earth into the sun. What causes an orbit to be stable?

An orbit is the balance between inertial and gravitational forces. That is, the earth is continually falling toward the sun, but inertia also wants the earth to keep moving in a straight line. When these two forces are in balance a stable orbit results:

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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

But how can you show this from first principles?

Well only an R^-2 force law can reproduce Kepler's Third Law:

• Step 1
• Step 2

and for something really cool

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:

Newton's development of 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.

Synposis of the Principia

Universality and the Scale of the Universe