CLASSICAL MECHANICS
Homework 1: solutions
1. (a) Write the dening equation of a plane in Cartesian, cylindrical, and spherical coordinates
(b)Write the dening equation of a cylinder in Cartesian, cylindrical, and spherical
coordinates
(c)Write the dening
CLASSICAL MECHANICS
Homework 1: solutions
1. (a) Write the dening equation of a plane in Cartesian, cylindrical, and spherical coordinates
(b)Write the dening equation of a cylinder in Cartesian, cylindrical, and spherical
coordinates
(c)Write the dening
CLASSICAL MECHANICS
Homework 5
1. A particle of mass m is moving in 3-dimensional space where we have established a
force eld described by a potential U (x2 + y 2 + z 2 , y/x), where x, y, z are the standard
Cartesian coordinates.
(a) Using spherical coor
CLASSICAL MECHANICS
Homework 7: solutions
1. I have said that Hamiltons principle requires that the action acquires a minimum at
the classical path followed by the system. However, I never gave you any condition
that checks which of the critical functions
CLASSICAL MECHANICS
Exam 1
1. (25pts) Consider the functional
b
f (y, y , x) dx ,
F [y (x)] =
a
subject to constraint
b
g(y, y , x) dx = 0 .
G[y (x)] =
a
Find the equation that determines the critical functions of F [y (x)].
2. (25pts) A particle of mass
CLASSICAL MECHANICS
Exam 1
1. (25pts) Consider the functional
b
f (y, y , x) dx ,
F [y (x)] =
a
subject to constraint
b
g(y, y , x) dx = 0 .
G[y (x)] =
a
Find the equation that determines the critical functions of F [y (x)].
SOLUTION: See Hand and Finch o
CLASSICAL MECHANICS
Exam 1: solutions
1. (25pts) For a system with Lagrangian L = L(q1 , q2 , q1 , q2 , t) and a constraint
A1 (q, t) dq1 + A2 (q, t) dq2 + B (q, t) dt = 0 ,
(a) state the conditions such that
J=
L
L
q1 +
q2 L
q1
q2
is an integral of mot
University of Central Florida
Department of Physics
Spring 2016
PHY 3220
Homework Assignment # 1
All problem numbers refer to chapter 1 of the textbook.
1. (10) 1.1
2. (10) 1.4
3. (10) 1.10
4. (10) 1.12
5. (10) 1.18
6. (10) 1.26
7. (10) 1.27
8. (10) 1.30
COP 3223 C Programming Assignment 3
Arrays
This assignment is designed in small progressive parts, with the intention of developing the skill of working with arrays.
Do each part separately. Include in your submission the code and output for Part I, then
Exam 2 Overview
18 questions:
11 multiple choice, 4 tracing, 3 coding
Time allotted: 75 minutes
Things to bring: pencil/pen, ID card
Exam 2 Review Topics
Declaring 1D Arrays
<data type> <variable name> [<size>];
Which of the following lines of C correctly
Assignment 2 C Programming
More data types, arithmetic, selection.
Part I
You can find out how many seconds have elapsed since Jan 1, 1970 using the time() function.
#include <time.h>
now = time(NULL);
/ now is more than a billion seconds (which data type
CLASSICAL MECHANICS
Homework 4
1. Recall the particle on the circular rotating hoop studied in class. As it was shown,
after eliminating the constraints the Lagrangian of the system is
L=
1
1
mR2 2 + mR2 2 sin2 mgR cos .
2
2
This implies an eective posten
CLASSICAL MECHANICS
Homework 3: solutions
1. In one of the lectures it was pointed out that, given a system and some generalized
coordinates describing it, we often prefer to introduce new coordinates that transform
the Lagrangian to one without cross ter
CLASSICAL MECHANICS
Homework 2: Solutions
1. Lagrange multipliers: Using lagrange multipliers nd the extrema of the function
f (x, y ) = 49 x2 y 2
subject to the constraint
(x, y ) = x + 3y 10 = 0 .
Give a geometric interpretation.
SOLUTION: We introduce
CLASSICAL MECHANICS
Homework 3: solutions
1. In one of the lectures it was pointed out that, given a system and some generalized
coordinates describing it, we often prefer to introduce new coordinates that transform
the Lagrangian to one without cross ter
CLASSICAL MECHANICS
Homework 4
1. Recall the particle on the circular rotating hoop studied in class. As it was shown,
after eliminating the constraints the Lagrangian of the system is
L=
1
1
mR2 2 + mR2 2 sin2 mgR cos .
2
2
This implies an eective posten
CLASSICAL MECHANICS
Homework 5
1. A particle of mass m is moving in 3-dimensional space where we have established a
force eld described by a potential U (x2 + y 2 + z 2 , y/x), where x, y, z are the standard
Cartesian coordinates.
(a) Using spherical coor
CLASSICAL MECHANICS
Homework 7: solutions
1. I have said that Hamiltons principle requires that the action acquires a minimum at
the classical path followed by the system. However, I never gave you any condition
that checks which of the critical functions
CLASSICAL MECHANICS
Homework 8
1. In HW 2 I mentioned supercially the concept of an operator. In this problem, I
return to that concept and I combine it with another idea I have described in class
while talking about the relation of Lie algebras and Lie g
CLASSICAL MECHANICS
Homework 9
1. In class, I showed to you how to prove Nthers theorem for a holonomic system of one
o
degree of freedom. Generalize it (by showing all calculations) to a holonomic system
of N degrees of freedom.
2. Explain which conserva
CLASSICAL MECHANICS
Homework 10
1. Solve problem 1 of chapter 9 of Hand & Finch. This is the problem I was planning to
solve on the blackboard but, due to limited time, I skipped and said that I will place
in the homework.
2. Laplace-Runge-Lenz vector: Fo
CLASSICAL MECHANICS
Homework 11
1. In the case of the hard spheres model, let m1 = m2 . Consider the particles distinguishable, despite the fact that they have equal mass for example, one could be
blue and the other red. Find the dierential cross sections
CLASSICAL MECHANICS
Exam 1
1. (25pts) Consider the functional
b
f (y, y , x) dx ,
F [y (x)] =
a
subject to constraint
b
g(y, y , x) dx = 0 .
G[y (x)] =
a
Find the equation that determines the critical functions of F [y (x)].
2. (25pts) A particle of mass
CLASSICAL MECHANICS
Exam 1: solutions
1. (25pts) For a system with Lagrangian L = L(q1 , q2 , q1 , q2 , t) and a constraint
A1 (q, t) dq1 + A2 (q, t) dq2 + B (q, t) dt = 0 ,
(a) state the conditions such that
J=
L
L
q1 +
q2 L
q1
q2
is an integral of mot
CLASSICAL MECHANICS
Homework 2: Solutions
1. Lagrange multipliers: Using lagrange multipliers nd the extrema of the function
f (x, y ) = 49 x2 y 2
subject to the constraint
(x, y ) = x + 3y 10 = 0 .
Give a geometric interpretation.
SOLUTION: We introduce