Physics 303/573
Lecture 2
August 28, 2013
1
More on vectors
Last lecture, I tried to be abstract and general. Here are a few more concrete ways of
understanding some aspects of vectors.
1.1
The interpretation of the scalar product
The scalar product of tw
Physics 303/573
Calculus in three dimensions
September 15, 2014
1
Mathematical preliminaries
I will begin by reviewing multivariate calculus, including the denitions of the gradient,
curl, and divergence operators, and intuitive proofs of Stokes theorem a
Physics 303/573
Notes on the Harmonic Oscillator Part II
October 14, 2013
1
The forced harmonic oscillator
The key to understanding the forced harmonic oscillator is to use the linearity of the
equation. Suppose we have two solutions x1 (t) and x2 (t) tha
Physics 303/573
Two body system, central forces, and gravitation
December 1, 2013
1
Two body systems
Suppose we have a system with just two bodies; we may write the Lagrangian as
1
L = [m1 (r1 r1 ) + m2 (r2 r2 )] U (r1 , r2 )
2
(1.1)
Suppose further that
Physics 303/573
Charge in a Magnetic Field
September 9, 2013
1
Physical setup
A particle with charge q in a magnetic eld B experiences a force perpendicular to its
velocity vand to the magnetic eld:
F = qv B
(1.1)
(this is called the Lorentz force law). U
Physics 303/573
Lecture 1
August 27, 2014
1
Introduction
Physics is a description of the world around us; thousands of years of experience has shown
that the appropriate language for detailed quantitative descriptions is mathematics; this rst
became clear
Physics 303/573
Lecture 9
October 7, 2013
1
Motivation
In this lecture, we will develop variational techniques that let us reformulate Newtons laws in
a dierent languageas Lagrangian and Hamiltonian mechanics. These reformulations make
it possible to nd t
Physics 303/573
Sample Midterm
October 21, 2015
1
Formulas
Summation convention: repeated indices are implicit summations. qi pi
i qi p i
, etc.
Euler-Lagrange: If qi (t) obeys the set of dierential equations
L
d L
=
for all i
dt qi
qi
S=
L dt is extrem
Physics 303/573
Midterm Solution
October 29, 2013
1. Consider the force
F = (x2 y 2a )x + b(x2 y 2 )y y
x
(a) Are there any values of a, b so that this force if conservative? If so, nd all such values.
We compute the curl
F:
[(x2 y 2a )x +(x2 y 2 )y y ]
Physics 303/573
Practice Midterm
October 22, 2013
1
Formula page
Summation convention: repeated indices are implicit summations. qi pi
i qi pi
, etc.
If qi (t) obeys
L
d L
=
dt qi
qi
then the action S =
L is extremized.
If there exist transformations of
Physics 303/573
Projectile Motion
September 10, 2013
1
General comments
Consider an object moving under the force of gravity and encountering air resistance.
In general, Newtons law take the form
F = mg f (v) = ma
z
v
(1.1)
where v = |v| is the magnitude
Physics 303/573
Lecture 10
October 15, 2014
1
Review of the General Euler-Lagrange Equations
In the previous lecture, we found that the functional1
t2
S[qi ] =
dt L(qi , qi , t)
(1.1)
t1
(called the action functional) is extremized when the generalized co
Physics 303/573
Projectile Motion
September 7, 2013
1
General comments
Consider an object moving under the force of gravity and encountering air resistance.
In general, Newtons law take the form
F~ = mg
z f (v)
v = m~a
(1.1)
where v = |v| is the magnitude
Physics 303/573
Lecture 2
September 7, 2011
1
More on vectors
Last lecture, I tried to be abstract and general. Here are a few more concrete ways of
understanding some aspects of vectors.
1.1
The interpretation of the scalar product
The scalar product of
Physics 303/573
Lecture 1
September 7, 2011
1
Introduction
Physics is a description of the world around us; thousands of years of experience has
shown that the appropriate language for detailed quantitative descriptions is mathematics;
this first became c
Physics 303/573
Lecture 1
August 26, 2013
1
Introduction
Physics is a description of the world around us; thousands of years of experience has
shown that the appropriate language for detailed quantitative descriptions is mathematics;
this first became cle
Physics 303/573
Caluclus in three dimensions
September 7, 2013
1
Mathematical preliminaries
I will begin by reviewing multivariate calculus, including the definitions of the gradient,
curl, and divergence operators, and intuitive proofs of Stokes theorem
Physics 303/573
Practice Midterm
October 17, 2013
1. In two dimensions, consider the potential U = 14 r4 (cos(2)2
(a) What is this in Cartesian coordinates? What force does it give rise to?
(b) Add a second force F2 = x(y 2a y 2 )
x + (b + 1)(x2 y 2 )y y
Physics 303/573
Practice FinalReview session: Friday, December 7, 4:30-? P118.
Real final is at: December 12, 4:00-6:30 P118. This is quite a bit longer than the real final.
1
Formulas
Summation convention: repeated indices are implicit summations. qi pi
Physics 303/573
Lecture 10
October 14, 2013
1
Review of the General Euler-Lagrange Equations
In the previous lecture, we found that the functional1
Z t2
S[qi ] =
dt L(qi , qi , t)
(1.1)
t1
(called the action functional) is extremized when the generalized
Physics 303/573
Practice Midtermproblem 4.
October 23, 2014
4. Consider two beads of mass m constrained to move on a vertical circular hoop of radius
a and joined by a rigid massless rod of length .
(a) What are all the constraints on the 4 coordinates de
Physics 303/573
Practice Midterm
October 21, 2014
1. In two dimensions, consider the potential U = 1 r4 (cos(2)2
4
(a) What is this in Cartesian coordinates? What force does it give rise to?
(b) Add a second force F2 = x(y 2a x2 ) + (b + 1)(x2 y 2 )y y to
Physics 303/573
Systems with more than one particle
September 11, 2013
1
Center of mass
1.1
Denitions
Suppose that we have a system of n-particles with masses mi , i = 1.n, and positions ri . It
is very useful to introduce the weighted average of the posi
Physics 303/573
Motion of Rigid Bodies
December 3, 2013
1
Introduction
A rigid body is a collection of particles whose separation and relative orientation is xed. Its position
can be described entirely by specifying the position of its center of mass and