Notes for Lecture 19
Inertia Tensor, Non-linear Physics
19.1
Inertia tensor
Before going on to the non-linear physics, let us look at some typical inertia tensors
and their derivations.
1
19.1. INERTIA TENSOR
G.-H. (Sam) Gweon
2
Phys. 105, UCSC, F2011
NOT
Greens function method
Phys. 105, UCSC, F2011
Greens function method for SHO
This is an optional topic the method of Greens function for solving the SHO
problem highly recommended for your reading, if you have time. Specically, this
shows how one can obta
Notes for Lecture 6
Small Oscillations
We consider small oscillations around a stable equilibrium point. We assume that
the second derivative of the potential is non-zero. The resulting motion is a simple
harmonic motion (SHM), an extremely important kind
Notes for Lecture 5
Conservation principles and 1D
motions
5.1
Conservation theorems
It is not possible to overemphasize the importance of the principle of conservation. In
classical mechanics, these principles can be thought as consequences of Newtons la
Notes for Lecture 4
Lorentz force
4.1
Example 2.10 of text
Motion of a charge in a uniform B eld.
B = B0 z . (This is a more common convention. The textbook chooses the y
direction.) The Lorentz force F = q v B, where q is the electric charge of the
parti
Notes for Lecture 3
Perturbation, air resistance (cont.)
3.1
Perturbation
As a general strategy to solve a real world problem, the perturbation theory is very
important. Please read Appendix A (and your note taken during class), which describes the rules
Problems for rigid body
Phys. 105, UCSC, F2011
Problems for your study (especially the rst two problems). NOT due.
Problem 1 Consider a disk with a small hole cut out from it. It is a thin disk, whose
thickness is negligible. The radius of the disk is R a
Homework 8
Phys. 105, UCSC, F2011
Due Nov. 29, Tuesday Dec. 1, Thursday.
70 points total. 30 points for the extra credit (but you are advised to understand all
problems).
Problem 1 (20 points) Find the center of mass for each of the following many particl
Notes for Lecture 7
Driven SHO
7.1
Particular solution, complementary function
Suppose that a force, F (t), drives a SHO.
m + bx + kx = F (t)
x
2
x + 2 x + 0 x = f (t)
Lx = f (t)
def
f (t) = F (t)/m
def
L =
d
d2
2
+ 2
+ 0 is a linear operator acting on x(
Notes for Lecture 9
Symmetry and conservation
Conservation principles are very fundamental in physics. Most conservation principles
that we know arise from symmetry.
9.1
Symmetry principles
Symmetry is important. It really is. A well-known Nobel laureate
Notes for Lecture 18
Coupled Oscillators and Rigid
Body
Let us look at couple more examples about coupled oscillators. Then, we cover very
basic stu on rigid body.
18.1
Coupled oscillator example 2
This example is a model for the CO2 molecule. We are inte
Notes for Lecture 17
Coupled Oscillators, cont.
You should read the previous lecture note carefully, if you have not.
Essentially, the problem of coupled oscillators is to reduce a problem that seems
extremely complicated to a problem that seems extremely
Notes for Lecture 16
Coupled Oscillators
We discuss one of the most important general topics of mechanics, the problem of
coupled oscillators. We will approach this problem using a simple example, as dened
below.
16.1
Newtonian EOM method
This method alwa
Notes for Lecture 15
Scattering
Here, we describe two particle collisions and then introduce the concept of the scattering crossection. The terms collision and scattering are used interchangeably
in this note.
15.1
Elastic collisions between two particles
Notes for Lecture 14
Kepler problem and many body
problem
14.1
Elliptical orbits
14.1.1
Turning points and angular momentum conservation
Let us come back to the discussion of elliptical orbits, applicable to comets as well as
planets. In an elliptical orb
Notes for Lecture 13
Central force and the Kepler
problem
Central force problems are important. All planet motion problems are of this kind.
Also, crude but useful classical mechanics models of atoms and nucleons are of this
kind. Crude, because classical
Notes for Lecture 11
Eective potential, gravity
11.1
Eective potential
Let us consider a problem for which the Lagrangian is not explicitly dependent on
time. So, the Hamiltonian is conserved. Suppose that the Hamiltonian can be written
as 1 mx2 + f (x) w
Homework 7
Phys. 105, UCSC, F2011
Due Nov. 17, Thursday.
Problem 1 (20 points) Consider a cylindrical shell with uniform density and innite
length. The cylinder parallel to the z direction. Mass exists between radii a
and b (a < b) while no mass exists fo
Homework 6
Phys. 105, UCSC, F2011
Due Nov. 10, Thursday.
Problem 1 (10 points) Mechanical similarity.
(a) Show that scaling the Lagrangian
L (q, q, t) = CL(q, q, t)
where C is a non-zero constant, does not change physics either, in the sense
that the Hami
Homework 5
Phys. 105, UCSC, F2011
Due Nov. 1, Tuesday.
Problem 1 (5 points) A general property of the Lagrangian formalism. Consider a
Lagrangian L, and another Lagrangian L , which is related to the original one by
L (q, q, t) = L(q, q, t) + f (q, q, t)
Notes for Lecture 1
Newtons laws
In general, my notes are not complete. You should read both my notes and the
textbook.
Chapter 1 materials will be spread around and parts of them will be taken up
during my lectures. We do that in this lecture, for exampl
Problems for rigid body and collisions
Phys. 105, UCSC, F2010
Problems for your study. NOT due.
Problem 1 Consider a disk with a small hole cut out from it. It is a thin disk, whose
thickness is negligible. The radius of the disk is R and the radius of th
Homework 7
Phys. 105, UCSC, F2010
Due Nov. 23, Tuesday.
Problem 1 (20 points) Consider the example that was studied in Lecture 14 and
summarized in pages 9 and 10 of LN 14.
(a) Assume that the initial conditions are: x1 = 2a, x2 = 0, and no initial
veloci
Homework 6
Phys. 105, UCSC, F2010
Due Nov. 16, Tuesday.
Problem 1 (30 points) Consider problem 5 of the pre-midterm. It involves three
masses: m (mass on string, pulled by gravity), M (block on a frictionless
horizontal surface, connected to mass m by a m
Homework 5
Phys. 105, UCSC, F2010
Due Nov. 2, Tuesday.
Problem 1 (10 points) Fermats principle and the PoLA. Let us consider a
photon, the quantum of light, with the angular frequency . While the study
of photons is denitely not a realm of classical mecha
Homework 4
Phys. 105, UCSC, F2010
Due Oct. 26, Tuesday.
Problem 1 (10 points) [Reading this problem is as important/easy as doing this
problem.] Here, we shall consider a subtle but important point about what
we discussed in class as a key measure of chao
Homework 3
Phys. 105, UCSC, F2010
Due Oct. 19, Tuesday.
Problem 1 (10 points) Near an unstable equilibrium point, x0 , of a one dimensional
motion, the potential energy can be written as U (x) = U0 1 k (x x0 )2 , where
2
k is a positive constant. We assum