Lecture 2 - The clockwork Universe.

Laws of motion, in the earth and the heavens.. from an apple to the moon in its orbit.

Galileo , Kepler, and Newton was fruition of Ionian Dream.

Demonstration that the Laws of Nature were comprehensible to human beings and that the universe is orderly and ruled by mathematics.

Laws of nature are simple, beautiful, and comprehensible, and superstition started to recede and the age of reason was born.

Galileo - laws of motion on earth. Inertia. Laws of falling bodies, parabolic shape.. swinging pendulum.

Kepler - looked at the heavens… deciphered laws of planetary motion.

Planets moved in an ellipse, not in circles as greeks thought.

Speed changed… Whipped around sun and slowed down far away.

Period of motion of planets based on distance.

Newton - He is really "The Man"… Towering genius.

He did have an apple tree. Newton saw an apple drop.

His genius was to see the same force that pulled an apple to the ground was the same force that pulled the moon around the earth and earth around the sun.

Single operating principle.

24 years old invented calculus

Force dropped off inverse square.

3 laws of motion… Law of inertia, F=ma, action and reaction.

Calculus how things move according to force law.

Force law was his axiom and

Calculus logical tool for deducing the consequences.

Imagine watching a movie of a planet around sun.

Kepler and Galileo could give the storyline of movie.

Newton found a frame by frame description of the movie.

Newton given a frame (state) what is the logical next scene in the movie

(updating the state) and instant later. Difference between one frame and the next.

First deterministic laws of nature.

A systems future is pre-determined by its present and by the laws of nature.

If you rewound the movie and play again, future plays out the same way every time.

It is appealing because it is so orderly.

Clear why it is called the clockwork. All you have to do is set the gears in motion and the clock knows how to tell time.

Things unfold by itself

Banished disorder, and Free will…

Newtons laws made be do it.. Everything was predetermined.

If you could say where every atom was, then the future would be set forever.

No room for morality.

We see chaos and disorder everywhere?

Clockwork system cannot be totally correct.

Chaos does exist, but it is not fundamental. Reflects only our imperfect understanding.

Laplace - talking to Napolean. Laplace - I have no need for God.

Probability theory - practical tool for handing

Stop gap measure. Practical but not fundamental.

Chaos is more deeply engrained in the structure of the universe.

Give you everything Laplace wants , the system is still

Deterministic but still unpredictable… Sounds like a paradox.

Determinism and unpredictability at the same time.

Laws of motion, in the earth and the heavens.. from an apple to the moon in its orbit.

Galileo , Kepler, and Newton was fruition of Ionian Dream.

Demonstration that the Laws of Nature were comprehensible to human beings and that the universe is orderly and ruled by mathematics.

Laws of nature are simple, beautiful, and comprehensible, and superstition started to recede and the age of reason was born.

Galileo - laws of motion on earth. Inertia. Laws of falling bodies, parabolic shape.. swinging pendulum.

Kepler - looked at the heavens… deciphered laws of planetary motion.

Planets moved in an ellipse, not in circles as greeks thought.

Speed changed… Whipped around sun and slowed down far away.

Period of motion of planets based on distance.

Newton - He is really "The Man"… Towering genius.

He did have an apple tree. Newton saw an apple drop.

His genius was to see the same force that pulled an apple to the ground was the same force that pulled the moon around the earth and earth around the sun.

Single operating principle.

24 years old invented calculus

Force dropped off inverse square.

3 laws of motion… Law of inertia, F=ma, action and reaction.

Calculus how things move according to force law.

Force law was his axiom and

Calculus logical tool for deducing the consequences.

Imagine watching a movie of a planet around sun.

Kepler and Galileo could give the storyline of movie.

Newton found a frame by frame description of the movie.

Newton given a frame (state) what is the logical next scene in the movie

(updating the state) and instant later. Difference between one frame and the next.

**He needed to find a way to integrate the infinitely many frames to yield a continuous movie!**

Hence part of Newtons Method is Integrating the differential equations

(not even real movies are continuous - flicker fusion.. the illusion of continuity).Hence part of Newtons Method is Integrating the differential equations

First deterministic laws of nature.

A systems future is pre-determined by its present and by the laws of nature.

If you rewound the movie and play again, future plays out the same way every time.

It is appealing because it is so orderly.

Clear why it is called the clockwork. All you have to do is set the gears in motion and the clock knows how to tell time.

Things unfold by itself

Banished disorder, and Free will…

Newtons laws made be do it.. Everything was predetermined.

If you could say where every atom was, then the future would be set forever.

No room for morality.

We see chaos and disorder everywhere?

Clockwork system cannot be totally correct.

Chaos does exist, but it is not fundamental. Reflects only our imperfect understanding.

Laplace - talking to Napolean. Laplace - I have no need for God.

Probability theory - practical tool for handing

Stop gap measure. Practical but not fundamental.

Chaos is more deeply engrained in the structure of the universe.

Give you everything Laplace wants , the system is still

Deterministic but still unpredictable… Sounds like a paradox.

Determinism and unpredictability at the same time.

## Newtonian Physics lasted for over 200

Late 1600s to late 1800s

Motion of bodies, teresstrial and celestial

Many applications of Newtons Laws by applying spurned the industrial revolution

Trillions of atoms tugging and vibrating but all stuck in their relative position

Local freedom, global cohesive Form a continuum where molecules can start to move more freely but still bound in

2.

3.

The Newtonian is the essence of Modernity

Uses DIfferential Equations to give you the whole movie, frame by frame.

Electricity and Magnetism ALSO Newtonian because it uses Newtons Method and is deterministic.

Telegraphs, electric lights, radios, telephones, television all of it built on this understanding that we need to discover the differential equations that govern electricity and magnetism. That is the Newtonisn approach.

Indirectly influenced politics and philosophy.. Voltaire and Locke... Everything

Enlightenment, Age of Reason

Influence of Newtonian approach even in declaration of indepence.

We hold these truths to be self evident... Self evident truth (undeniable).

Euclidian... His approach to geometry... certain axioms, which were self evident.

Build on axioms using logic to deduce theorems, proofs and corollaries to deduce the consequences of the axioms.

Principia written in Euclidian framework of axioms in which he deduced theorems and collaries.

Jefferson channels Newton.. Cause and Effect...

We are entitled by the laws of Nature and Natures god

Newtons Laws and their generalizations, that the Universe is LAWFUL, that is what gives us our freedom.

Testiment to the domination of Newtonian... Deterministic model of the world.

Motion of bodies, teresstrial and celestial

Many applications of Newtons Laws by applying spurned the industrial revolution

**Earth**(structure)- 1.__Elasticity and Structural Theory__- ORDER Buildings, Skyscrapers and bridges, roads, highways, etcTrillions of atoms tugging and vibrating but all stuck in their relative position

**Water**(connecting) - BALANCE - Strange ATTRACTOR.Local freedom, global cohesive Form a continuum where molecules can start to move more freely but still bound in

2.

__Fluid Mechanics and Aerodynamics__- Planes, Boats, Submarines,**Air/Gas**(motion/energy) - CHAOS - Freedom/Independence - except when colliding.3.

__Kinetic Theory/Thermodynamics__which gave rise to steam engine and combustion engine and eventually electric power.The Newtonian is the essence of Modernity

Uses DIfferential Equations to give you the whole movie, frame by frame.

Electricity and Magnetism ALSO Newtonian because it uses Newtons Method and is deterministic.

Telegraphs, electric lights, radios, telephones, television all of it built on this understanding that we need to discover the differential equations that govern electricity and magnetism. That is the Newtonisn approach.

**Find the underlying differential equations with the confidence that that is how Nature will be described.**Indirectly influenced politics and philosophy.. Voltaire and Locke... Everything

Enlightenment, Age of Reason

Influence of Newtonian approach even in declaration of indepence.

We hold these truths to be self evident... Self evident truth (undeniable).

Euclidian... His approach to geometry... certain axioms, which were self evident.

Build on axioms using logic to deduce theorems, proofs and corollaries to deduce the consequences of the axioms.

Principia written in Euclidian framework of axioms in which he deduced theorems and collaries.

Jefferson channels Newton.. Cause and Effect...

We are entitled by the laws of Nature and Natures god

Newtons Laws and their generalizations, that the Universe is LAWFUL, that is what gives us our freedom.

Testiment to the domination of Newtonian... Deterministic model of the world.

## Three Cracks in the Newtonian Paradigm and the Foundations of Determinism

1. Einsteins Theory of Relativity

Came about because of the inconsistencies of Newtons Laws of Motion and Differentials Equations and Maxwells Equations of Electicity and Magnetism.

Einsteins thought experiment of riding on a photon... Using Newtonian logic, you would be moving and the EMAG waves would be static and not propagate... But maxwells equation insist on this undulating wavelike motion and propagation at the speed of light. Laws of physics should be the same for all inerital frames.

Resolution of inconsistencies was radical... Overthrowing Newtonian ideas of absolute time and absolute space.

2. Quantum Mechanics -grew out of an attempt to explain the strange phenomenon of light and atomic radiation. The behavior of the electron did not make sense in a newtonian way. Electron should spiral into nucleus... So things like electrons popping teleporting and many... Also did away with determinism altogether.

Spoke of a world of probability where nothing is certain, but only probabilities.

3. Chaos grew out of the same problem that created newtonian physics.

How did we get from Clockwork to Chaos.

Planetary Motion

1) Law of Gravity between two objects. Inverse Square Law

2) Mathematical Soothsayer... F=ma (allows him to predict into the future for any system of interest).

Position and velocity at one instant and using Newtons law of gravity and F=ma we can built a successive picture frame by frame (like a movie) of how the planet travels).

Key assumption solving the 2-body problem

New place, new speed, everything constantly recalculating, and frames put together...

Good clear coherent model of reality.

Wizardy of newton involves an ingenuous trick. inverse square is nonlinear (rubber band and spring are linear... simpliest possible force... very easy differential equations... Simple harmonic oscillator.

Newton use transformation to convert nonlinear to linear.

u=1/r ... How it moves relative to the angle and not respect to time.

Comes out to be a simple harmonic oscillator... Double pendulum behaves linearly for small angles.

Logical next step is solve the 3 body problem

Improved astronomical observations saw not perfect ellispses, and suspected

Nobody could solve it to transform it to a linear problem.

Figured a good approximation

Perturbation theory... Breaks a problem in its main contributions..

Primary , secondary, tertiarty ,etc.

Zeroth order is sun, next orders are tugs of other planets.

Able to predict existence of a new planet, Neptune,,,

Based on wobbles of Uranus... Perturbation theory in reverse.

Marvelous vindication of Newtonian approach and pertubation theory.

Lets go back to 3 body problem and just solve...

Simplify the problem

Take two big planets very big, and a small speck of dust

Restricted circular 3 body problem (same plane)

2 big planets rotating and put a small speck in the same plane.

Why was it so hard...

Difficult deep mathematical obsticale. and that turned out to be the case.

Speck orbiting demonstration..

Go into rotating frame of reference... Looks like a pair of fixed planets.

Use a computer to integrate and plot differential equation

Very complicated

Seeds of Chaos... A phenomenon yet unimagined..

Chaos in the contemporary sense of a random looking, unpredictable behavior of system that is nevertheless governed by non-random and deterministic laws.

Came about because of the inconsistencies of Newtons Laws of Motion and Differentials Equations and Maxwells Equations of Electicity and Magnetism.

Einsteins thought experiment of riding on a photon... Using Newtonian logic, you would be moving and the EMAG waves would be static and not propagate... But maxwells equation insist on this undulating wavelike motion and propagation at the speed of light. Laws of physics should be the same for all inerital frames.

Resolution of inconsistencies was radical... Overthrowing Newtonian ideas of absolute time and absolute space.

2. Quantum Mechanics -grew out of an attempt to explain the strange phenomenon of light and atomic radiation. The behavior of the electron did not make sense in a newtonian way. Electron should spiral into nucleus... So things like electrons popping teleporting and many... Also did away with determinism altogether.

Spoke of a world of probability where nothing is certain, but only probabilities.

3. Chaos grew out of the same problem that created newtonian physics.

How did we get from Clockwork to Chaos.

Planetary Motion

1) Law of Gravity between two objects. Inverse Square Law

2) Mathematical Soothsayer... F=ma (allows him to predict into the future for any system of interest).

Position and velocity at one instant and using Newtons law of gravity and F=ma we can built a successive picture frame by frame (like a movie) of how the planet travels).

Key assumption solving the 2-body problem

New place, new speed, everything constantly recalculating, and frames put together...

Good clear coherent model of reality.

Wizardy of newton involves an ingenuous trick. inverse square is nonlinear (rubber band and spring are linear... simpliest possible force... very easy differential equations... Simple harmonic oscillator.

Newton use transformation to convert nonlinear to linear.

u=1/r ... How it moves relative to the angle and not respect to time.

Comes out to be a simple harmonic oscillator... Double pendulum behaves linearly for small angles.

Logical next step is solve the 3 body problem

Improved astronomical observations saw not perfect ellispses, and suspected

Nobody could solve it to transform it to a linear problem.

Figured a good approximation

Perturbation theory... Breaks a problem in its main contributions..

Primary , secondary, tertiarty ,etc.

Zeroth order is sun, next orders are tugs of other planets.

Able to predict existence of a new planet, Neptune,,,

Based on wobbles of Uranus... Perturbation theory in reverse.

Marvelous vindication of Newtonian approach and pertubation theory.

Lets go back to 3 body problem and just solve...

Simplify the problem

Take two big planets very big, and a small speck of dust

Restricted circular 3 body problem (same plane)

2 big planets rotating and put a small speck in the same plane.

Why was it so hard...

Difficult deep mathematical obsticale. and that turned out to be the case.

Speck orbiting demonstration..

Go into rotating frame of reference... Looks like a pair of fixed planets.

Use a computer to integrate and plot differential equation

Very complicated

Seeds of Chaos... A phenomenon yet unimagined..

Chaos in the contemporary sense of a random looking, unpredictable behavior of system that is nevertheless governed by non-random and deterministic laws.

## Chaos Found and Lost - The Amazing Mind of Poincare

Generally regarded as one of the greatest mathematicians to have ever lived, Poincare was the first to really stumble on Chaos and Fractals lurking in the 3-body problem.

Generalized geometry and founder of topology

Extremely intuitive and visual, but couldn't draw.

Poincare Conjecture.

The implication is that a system governed by deterministic laws can nevertheless behave unpredictably.... Chaos creeps into Clockwork... Also late 1800's like Relativity and Quantum Mechanics, but this is less remembered.

Poincare discovery of implications of 3 body problem was Triggered by an international contest.

Approach he took was inventive... using pictures instead of problems... Visualize solution.

Tremendous insight... Method lives on today and Chaologists use it today.

Moves like a pendulum, tick tock back and forth

Newton F=ma

You tell me the force and I will tell you the acceleration.

Force keeps changing. Calculating the motion is integrating frame by frame what the acceleration and motion is.

Elliptic functions... Involved in solution of the pendulum.

Picture of the differential... Pendulum moves in an imaginery world called state space or phase space.

Trajectory through phase space.

A differential equation is like the logic behind a movie, it would tell you logically how one scene in a movie logically leads to another scene...

In terms of that analogy you could think of a state as being like a single frame in the movie. It tells you everything you need to know about what is going to happen next.

(chaos like a suspence with many twists and unexpected turns).

State space is the collection of all possible states.

All possible frames or scenes you could imagine in the motion of the pendulum.

The state of a pendulum is all the information you need to know in order to determine how the pendulum will move in the next instant. Totality of information needed to predict the future.

Move pendulum to the side... One thing is Angle... But not enough... Need to know how fast.

Or you could push it.

Initial position or angle and initial velocity are all that is needed.

Newtons second equation F=ma will predict the future.

Its state requires two numbers (angle and velocity)

We need 2 axis for state space.

Two things going on at the same time...

Familiar vs Powerful ways to look at it.

Pendulum swinging vs State Space

Animation...

By moving around in state space, you can change the state of the pendulum.

Dot moving around in an orbit

Closed loop is a visual version of periodic motion of a pendulum.

Different Trajectories in state space correspond to different motions of the pendulum.

Represents different movies...

Three qualitatively different trajectories (need hyperbolic functions.. very complicated to solve.

1) Whirling

2) Simple Pendulum - small angles

3) In between - almost wants to whirl.

Key Questions - can we use state space to predict how a system can behave.

Imagine what the movie will look like before we see it.

All of it is animated by Newtons law?

Newtons law determines a vector for each point in state space.

At each point in state space Newton gives you an arrow... "go there next"... then "go there next".

Arrows on state space are Newtons Equations

Vector fields... like weather maps

Tells you how to update the state from one moment to the next.

Poincare Blundered

New analysis found chaos...

Poincares Nightmare

Infinitely fine mesh... Looping back and forth...

Different states or boxes in the state space very close together can have totally different orbits.

Two almost identical states can have two very different trajectories.

Obstacle that prevented the solution of the 3 body problem.

1890...New QM and Relativity... 3 body problem old fashioned.

Little impact in the next 70 years.

Generalized geometry and founder of topology

Extremely intuitive and visual, but couldn't draw.

Poincare Conjecture.

The implication is that a system governed by deterministic laws can nevertheless behave unpredictably.... Chaos creeps into Clockwork... Also late 1800's like Relativity and Quantum Mechanics, but this is less remembered.

Poincare discovery of implications of 3 body problem was Triggered by an international contest.

Approach he took was inventive... using pictures instead of problems... Visualize solution.

Tremendous insight... Method lives on today and Chaologists use it today.

Moves like a pendulum, tick tock back and forth

Newton F=ma

You tell me the force and I will tell you the acceleration.

Force keeps changing. Calculating the motion is integrating frame by frame what the acceleration and motion is.

Elliptic functions... Involved in solution of the pendulum.

Picture of the differential... Pendulum moves in an imaginery world called state space or phase space.

Trajectory through phase space.

A differential equation is like the logic behind a movie, it would tell you logically how one scene in a movie logically leads to another scene...

In terms of that analogy you could think of a state as being like a single frame in the movie. It tells you everything you need to know about what is going to happen next.

(chaos like a suspence with many twists and unexpected turns).

State space is the collection of all possible states.

All possible frames or scenes you could imagine in the motion of the pendulum.

The state of a pendulum is all the information you need to know in order to determine how the pendulum will move in the next instant. Totality of information needed to predict the future.

Move pendulum to the side... One thing is Angle... But not enough... Need to know how fast.

Or you could push it.

Initial position or angle and initial velocity are all that is needed.

Newtons second equation F=ma will predict the future.

Its state requires two numbers (angle and velocity)

We need 2 axis for state space.

Two things going on at the same time...

Familiar vs Powerful ways to look at it.

Pendulum swinging vs State Space

Animation...

By moving around in state space, you can change the state of the pendulum.

Dot moving around in an orbit

Closed loop is a visual version of periodic motion of a pendulum.

Different Trajectories in state space correspond to different motions of the pendulum.

Represents different movies...

Three qualitatively different trajectories (need hyperbolic functions.. very complicated to solve.

1) Whirling

2) Simple Pendulum - small angles

3) In between - almost wants to whirl.

Key Questions - can we use state space to predict how a system can behave.

Imagine what the movie will look like before we see it.

All of it is animated by Newtons law?

Newtons law determines a vector for each point in state space.

At each point in state space Newton gives you an arrow... "go there next"... then "go there next".

Arrows on state space are Newtons Equations

Vector fields... like weather maps

Tells you how to update the state from one moment to the next.

Poincare Blundered

New analysis found chaos...

Poincares Nightmare

Infinitely fine mesh... Looping back and forth...

Different states or boxes in the state space very close together can have totally different orbits.

Two almost identical states can have two very different trajectories.

Obstacle that prevented the solution of the 3 body problem.

1890...New QM and Relativity... 3 body problem old fashioned.

Little impact in the next 70 years.