Mechanics
Physics 151
Fall 2003
Masahiro Morii
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Mechanics , Physics, Motion , Newtonian Mechanics ,LagrangianFormulation,Hamilton’s Principle,Angular momentum, Momentum Conservation , Force ,Potential Energy, Inertial System, kinetic energy , moment of inertia, Field theory ,Perturbation theory.
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Mechanics
Physics 151
Fall 2003
Masahiro Morii
Teaching Staff!
Lecturer: Masahiro Morii
!
Tuesday/Thursday 11:30 – 1:00. Jefferson 256
!
Section leaders: Srinivas Paruchuri and
Abdol-Reza Mansouri
!
Two or three 1-hour sections per week
!
Date/time to be announced
!
Please fill out the student survey
!
Course assistant: Carol Davis
!
She will have all course materials (problem sets, etc.)
Prerequisite Courses!
Physics 15a and 15b
!
Introductory Mechanics, Relativity and E&M
!
Mathematics 21a and 21b (or equivalent)
!
Multivariate Calculus, Linear Algebra and Diff. Equations
!
Take these prerequisites seriously
!
Without sufficient background, you’ll get lost quickly
!
If you
still
want to try, you need to get a written permission
from the Head Tutors (Howard Georgi, David Morin)
Textbook!
Classical Mechanics,
Goldstein, Poole and Safko
!
Required
!
Classic (literally) textbook. Originally published in 1950
!
A must-read for serious physicists
!
rd
edition came out in 2001
!
nd
edition still good (or
better
) – Get it if you can
!
Will follow this textbook closely
!
Except for skipping a few advanced materials
!
It’s a 600-page book written for graduate students
Homework!
Problem sets are distributed on Thursdays
!
Reports are due at the next week’s Thursday lecture
!
Typical format:
!
2–3 problems that will be discussed at sections
!
3–4 problems you solve and turn in report
!
Work together in groups
!
Groups will be assigned according to the Survey
!
Each of you must turn in your own report, though
Is This Course For Me?!
Yes if you have serious interest in Physics
!
Definitely if you want to try a career in Physics
!
I’ll explain what we will study in the next 15 minutes
!
Plus a mini-lecture at the end
!
If you are not sure at the end of this lecture
!
Come to my office (Lyman 239) and ask questions
!
I’ll be there everyday this week
Mechanics!
Mechanics concerns
!
Motion of objects
Velocity and acceleration
!
Cause of the motion
Force and energy
!
The objects move, but do not change their properties
!
Idealized particles and rigid bodies
!
Mass and moment of inertia are all what matters
!
Newton’s Three Laws of Motion
!
You remember them, right?
!
Principia
(1687) pretty much wrapped it up
Mechanics: A branch of physicalscience that deals with energy andforces and their effect on bodies (Webster’s)
Classical vs. Modern!
“Modern” in physics means “
th
century”
!
Quantum Mechanics
!
Relativity
!
Classical Mechanics = pre-Quantum Mechanics
!
We include special relativity as well as E&M
!
What happened between the 17
th
and 20
th
centuries?
Why Classical Mechanics?!
Three Good Reasons
!
Close connection to Modern Physics
!
Mastering CM gives you clearer view of QM
!
Powerful and versatile mathematical tools
!
Indispensable for advanced studies in physics
!
Reformulate familiar laws of physics using completelydifferent approaches
!
Cleaner and more general formalism
!
It’s just cool
!
But first, let’s go back to the 17
th
century…
Newtonian Mechanics!
In principle, Newton’s Equation of Motion predictedthe motion of any object(s) from the force
!
All you need is a big, fast computer
!
In reality, life was not so easy
!
Intel was not founded until 1968
!
More fundamentally, the force may not be known
!
It may depend on time, location, or even velocity
!
E.g.
2 objects attracting each other by gravity
!
Solving 3-body problem turns out to be impossible
Quest for more powerful mathematics
Lagrangian Formulation!
Newton’s Equation is about force
!
You start from
F
F
x
t
) for all particles
!
N
functions corresponding to 3
N
coordinates
!
Forget the force. Introduce something else
!
Lagrangian:
!
Lagrange’s Equation
!
Lagrangian does not depend on a coordinate system
!
Switching to a different set of coordinates is a snap
L
L q q
Coordinate
q
and its time derivative
d
L
L
dt
q
q
Everything about this system isembodied in a scalar function
L
m
F
a
Hamilton’s Principle!
Hamilton’s Principle derives Lagrange’s Equationfrom a simple rule:
Rather weird statement…
!
Newton’s Laws were found by induction
!
“It is so because it agrees with many observations”
!
Deriving
them from a principle means knowing
why it is so
!
Not quite that dramatic, but it does suggest deeper reason
!
Eventually connected to Feynman’s path integral
!
Besides, calculus of variations is a useful technique
The time integral of
L
is stationary for the path
taken by an actual physical system
2 1
δ
∫
Ldt
Success out of a Failure!
Quest for tools that solve the 3-body problem failed
!
Unless you count invention of computers
!
Byproducts (Lagrangians and Hamiltonians) turnedout to be the cornerstones of Quantum Mechanics
!
Development of QM was guided by analogies to Lagrangianand Hamiltonian formulations
!
Pioneers of QM grew up with Classical Mechanics
!
Classical Mechanics is the missing link betweenNewton and Schrödinger
!
It allows you to fully appreciate QM
What We Will Study!
Lagrange’s Equations, Hamilton’s Principle
!
Central force problem
!
Rigid body motion
!
Oscillation
!
Extension to special relativity
!
Hamilton Equation, Canonical transformations
!
Hamilton-Jacobi Equation
!
Advanced stuff
!
Classical chaos?
!
Perturbation theory?
!
Field theory?