A Tr

ip D

own

Mem

ory

Lane

Where We’ve Beentell them what you told them

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The Original Theoretical Minimum

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Lev Davidovich Landau, 1908 - 1968

One of the great Russian physicists of the 20th century

Tested prospective students in theoreticalphysics.

43 students passed, the 2nd of whom washis famous collaborator, Ilya Lifshitz.

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Our Theoretical Minimum

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Leonard Susskind, 1940-

Prominent American physicist atStanford University: “brilliant imaginationand originality”

Received many awards and honors(but not yet The Big One)

Devotes substantial efforts to a series oflectures aimed at the physics-oriented public

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Susskind’s Target Audience

“The courses are specifically aimed at

people who know, or once knew, a bit

of algebra and calculus, but are more

or less beginners.”4

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Principal Elements (1)

State Model of Physics Laws

• Determinism

• Reversibility

• Allowed vs Disallowed Forms

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Mathematical Infrastructure (I)

• Spaces

• Trigonometry

• Vectors

Principal Elements (2)

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x

y

z

Spac

es, T

rigon

omet

ry, a

nd V

ecto

rs

Vector Components

• We often represent a vector by its x, y, and z components.

• We define , , and to be unit vectors pointing in the x, y, and z directions, respectively.

• We specify by the three scalars, , , and , thus

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Principal Elements (3)

Description of motion (kinematics)

Where particles move but not why:

• position

• velocity (and speed)

• acceleration8

Parti

cle

Moti

on (K

inem

atics

)

Units

• position has the units (m)

• velocity has the units (m/s)

• acceleration has the units (m/s2)

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Principal Elements (3.5)

Mathematical Infrastructure (III)

Differential calculus

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_W = dW (t)dt

Diff

eren

tial C

alcu

lus

Limits

We were computing

where stands for some (small) amount of time.

The true speed is the limit of as goes to zero. We write this as

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Diff

eren

tial C

alcu

lus

What Have We Done?

We’ve computed

or

the (first) derivative of .

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Principal Elements (4)

Mathematical Infrastructure (IV)

Integral calculus

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Q =Ra

b q(x)dx

Inte

gral

Cal

culu

s

Goal

Compute the signed area under some portion of an arbitrary curve

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Inte

gral

Cal

culu

s

Animatedly

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Inte

gral

Cal

culu

s

At Any Given Level

The true area, , is given by

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Principal Elements (5)

Dynamics of motion

Forces and their effects

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d(m~v)dt = m~a = ~F

Dyn

amic

s

General Dynamics

• Aristotle

• Newton

• Einstein

where

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Dyn

amic

s

Isaac Newton

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1642-1727

Philosophiæ Naturalis Principia Mathematica

Dyn

amic

s

Newtonian Dynamics

•Deterministic• Reversible• Right

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Dyn

amic

s

An Aside on Units

Fundamental units are

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Unit Measured in…

Length meters

Time seconds

Mass kilograms

Dyn

amic

s

Units of Observed Quantities

Quantity Units Measured in…position metersvelocity meters/secondacceleration meters/second2

force kilogram-meters/second2

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Principal Elements (6)

Mathematical Infrastructure (V)

Partial differentiation (just more

differentiation)

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p(x;y) = @P (x;y)@y

Ener

gy

Conservation of Energy

We have shown that conservative forces always conserve energy.

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Principal Elements (7)

Extremum principles

• the whole rest of the course

• the Promised Land

• the heart of classical mechanics

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Leas

t Acti

on

Formulating (cont.)

Second approach requires minimizing the action:

where

is the Lagrangian.

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Leas

t Acti

on

Minimizing the Action,

• Actually only imposing stationarity

• A little miraculous since depends on everywhere

• General solution described by Euler-Lagrange equations

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Principal Elements (7.1)

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L = T ¡ V

ddt

³ @L@_qi

´= @L

@qi

Example: Lagrangian Mechanics

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So What’s the Point?

• Lagrangian bundles everything about a system’s dynamics into one package.

• Very straight-forward to change coordinates, a common operation.

• Easy to work out equations of motion for complex problems by routine differentiation.

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