A Tr
ip D
own
Mem
ory
Lane
Where We’ve Beentell them what you told them
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Theo
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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.
Theo
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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
Theo
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Principal Elements (1)
State Model of Physics Laws
• Determinism
• Reversibility
• Allowed vs Disallowed Forms
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Theo
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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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Theo
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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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Theo
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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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Theo
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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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Theo
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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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Theo
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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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Theo
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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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Theo
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Principal Elements (7.1)
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L = T ¡ V
ddt
³ @L@_qi
´= @L
@qi
Example: Lagrangian Mechanics
Leas
t Acti
on
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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