Physics 106a Final Solutions
December 17, 2013
Problem 1: Five quick questions
(a) The position of the center of mass (x horizontal, y vertical, down) is
Rcm = (l sin + R sin , l cos + R cos ).
(1)
From this we can get for Vcm = Rcm
2
Vcm = l 2 2 + R 2 2
Physics 106a Problem Set 5: Solutions
November 21, 2013
Problem 1: Generating functions
A type-2 generating function for a canonical transformation (q, p) (Q, P ) is F2 (q, P ) = q 2 eP .
(a) The type-2 generating function gives
F2
=2qeP ,
q
F2
=q 2 eP .
Physics 106a, Classical Mechanics Fall Term, 2013
Final Exam
Do not read beyond this page until you are ready to start the exam!
Due date: Due by 4pm in 149 Bridge on Friday 13 December. Late exams will require extenuating circumstances; otherwise, no cre
Lecture 3. Lagrangian Mechanics: Introductory Examples
(Dated: October 6, 2015 [Revised after lecture])
After reviewing Newtons laws last week, we start discussing Lagrangian Mechanics this
week. We will study two examples, in which we introduce how to ac
Lecture 1. Overview and Review of Newtonian Mechanics
(Dated: September 29, 2015 [Revised 9/30/2015])
Today I will rst give an overview of analytical mechanics, and then review Newtonian
dynamics.
I.
WHAT IS ANALYTICAL MECHANICS?
A.
Newtonian Dynamics
New
Lecture 2. Review of Newtonian Mechanics
(Dated: October 1, 2015)
Last time, we reviewed the Newtons laws, and momentum conservation of a system of
particles. We also discussed the work done by a force on a single particle, as well as the
concept of a con
Physics 106a: Assignment 1
4 October, 2013; due 4pm Friday, 11 October in the Ph106 In Box in 149 Bridge
Remember: only identify your solution set with your class id number, not your name.
Variational Approach and Lagrangian Dynamics
Reading
I am teaching
Physics 106a
Problem Set 6
Fall 2015
Problem Set 6
November 13; due at 5PM on November 20, 2015.
The total number of points is 50. Please email me if you nd any typos. Some of these problems were
from previous years please do not search for solutions!
1.
Physics 106a
Problem Set 5
Fall 2015
Problem Set 5
November 6; due at 5PM on November 13, 2015.
The total number of points is 50. Please email me if you nd any typos. Some of these problems were
from previous years please do not search for solutions!
1. C
Lecture 4. Lagrangian Mechanics: General Formulation
(Dated: October 8, 2015)
I.
CONTINUED FROM LAST TIME
A.
Summary
During the last lecture, we rewrote the Newtons law into the following form, by dotting
both sides of Newtons equations to virtual displac
Lecture 5. Variational Principle
(Dated: October 13, 2015)
I.
REVIEW OF THE LAST LECTURE
A.
Lagrangian Mechanics
During the last lecture, we considered Lagrangian Mechanics for systems with holonomic
constraints. In particular, we have
r j ( q k , t), . .
Lecture 6. Variational Principle
(Dated: October 15, 2015)
I.
REVIEW OF THE LAST LECTURE
A.
Variational Calculus
Last lecture, we introduced the following type of functional, which depends on a function
y( x) in the following way:
L
I [ y] =
F ( y( x), y
Lecture 13. Central Force
(Dated: November 10, 2015.)
I.
REVIEW FROM LAST TIME AND SUMMARY OF THIS LECTURE
Last time, we studied central forces. In particular
1. In the time domain, the evolution of the eccentric anomaly is very simple.
2. We studied the
Physics 106a
Midterm Exam
Fall 2015
Midterm Exam
October 23, 2015
Policy
1. Covarage: This exam covers the rst three weeks of lecture. The main reference is
Hand and Finch, Chapters 1 and 2.
2. Time Limit: You can work on this exam for up to 4 hours. In a
Lecture 12. Central Force
(Dated: November 3, 2015.)
I.
REVIEW FROM LAST TIME
We discussed phase space, showing that phase space trajectory is another way to represent the evolution of a system at various states. We then discussed central force interactio
Lecture 11. 1-D motion and central force
(Dated: November 3, 2015.)
Midterm teaching survey
Thank you for participating the teaching review and for your thoughtful responses. I
will make the following improvements in my teaching.
Give summary at the begi
Lecture 10. Theory of Small Vibrations
(Dated: October 27, 2015. Version 2.)
I.
REVIEW OF THE LAST LECTURE AND CONTINUATION
A.
E-L Equation in Matrix Form
Last time, we showed that we can write down the small vibration problem in to a matrix
form:
t + v =
Lecture 9. Theory of Small Vibrations
(Dated: October 22, 2015)
I.
REVIEW OF THE LAST LECTURE
Last time, we discussed two coupled harmonic oscillators. From the Lagrangian,
L=
1
ml 2 2 2 mgl 2
2
( 1 + 2 )
(1 + 2 ) kl 2 (1 2 )2
2
2
2
(1)
which is quadrati
Lecture 8. Harmonic Oscillators
(Dated: October 22, 2015)
I.
REVIEW OF THE LAST LECTURE
A.
Simple Harmonic Oscillator
Last week, we studied a simple harmonic oscillator. As we have argued, this can be a
good approximation for a 1-D system near its equilib
Lecture 7. Harmonic Oscillators
(Dated: October 20, 2015)
I.
REVIEW OF THE LAST LECTURE
In the past two weeks, we discussed Lagrangian dynamics. We can summarize this
subject in the following.
We rst construct the action, which is a functional of the traj
Physics 106a
Problem Set 3
Fall 2015
Problem Set 3
October 16, 2015; Due at 5 PM on October 23, 2015
The total number of points is 50. Please email me if you nd any typos. Some of these problems were
from previous years please do not search for solutions!
Physics 106a
Problem Set 4
Fall 2015
Problem Set 4
October 30 2015; due at 5PM on November 6, 2015.
The total number of points is 50. Please email me if you nd any typos. Some of these problems were
from previous years please do not search for solutions!
Physics 106a: Assignment 3
18 October, 2013; due 4pm Friday, 25 October in the Ph106 In Box in 149 Bridge
Constrained dynamics and small oscillations
Reading
Constrained dynamics: HF 2.6-8, T 7.3,7.10. You should work through some problems on the simpler
Physics 106a: Assignment 4
1 November, 2013; due 4pm Friday, 8 November in the Ph106 In Box in 149 Bridge
Hamiltonian Formulation
Reading
Hand and Finch Chapter 5. You should also review Chapter 4 on central forces if you did not do this for the midterm.
Physics 106a: Assignment 6
15 November, 2013; due 4pm Friday November 22 in Room 149 Bridge
Reading
I do not entirely like Hand and Finchs treatment in Chapter 7, so my notes are more extensive than usual. In particular I
nd 7.5 unnecessarily complicated.
Physics 106a: Assignment 7 Rotational Motion
15 November, 2013; due 4pm Friday December 6 in Room 149 Bridge
Reading
Read Hand and Finch Chapter 8 on rigid body rotations, except 8.5 which I think is confused (see
my notes instead), and 8.9 which is rathe
Physics 106a: Assignment 5
8 November, 2013; due 4pm Friday, 15 November in the Ph106 In Box in 149 Bridge
Canonical Transformations etc.
Reading
Hand and Finch Chapter 6. Taylor does not cover these topics. The discussion in Goldstein et al. is considera
Physics 106a, Classical Mechanics Fall Term, 2013
Midterm
Do not read beyond this page until you are ready to start the exam!
Material: The exam covers the material of Lectures 1-8.
Due date: Due by noon in 149 Bridge on Monday November 4. Late exams will
Physics 106a: Assignment 2
11 October, 2013; due 4pm Friday, 18 October in the Ph106 In Box in 149 Bridge
Constraints, Virtual Work, and Lagrangian Dynamics
Reading
You can now read the complete chapters 1 and 2 in Hand and Finch or 6 and 7 in Taylor on L
Physics 106a: Problem Set 2 Solutions
October 29, 2013
Problem 1: Degrees of freedom and constraints
(a) There are M + 1 degrees of freedom. One argument is as follows. The position of the rst particle
gives 2 degrees of freedom. After this, there is 1 an