2016 Spring Semester (104-2), Homework Set 5
(to hand in with the next homework)
1. Please show that, the eigenvalues of a Hermitian 2 2 matrix
a0 + a3 a1 ia2
a0 1l + a1 1 + a2 2 + a3 3 =
a1 + ia2 a0 a3
p
are = a0 a21 + a22 + a23 . Note that the 1l is the

Classical Mechanics Homework: Lagrangian Mechanics
I.
A.
THE PROBLEMS
Cylindrical Tube Rolling Down an Incline
Consider a cylindrical tube of mass M and radius R, rolling down an incline (Note that we are talking about a
tube, whose Icm = M R2 , not a sph

Make sure you understand everything weve covered in class, and make sure you understand the examples illustrated
in the textbook.
1. Go through the derivations we did in class for damped oscillations.
2. In the case of forced oscillations, the EOM reads
x

2016 Spring Semester (104-2), Homework Set 3
1. Learn examples 11.3 in the textbook. Make sure you can obtain the Ic matrix (the lower-case superscript c
means the origin is located at the corner) with good understanding. Recall that we have also worked o

Between an Inertial Frame, and a Rotating and Non-inertial Reference Frame
I.
Relativity is not considered in the following discussion. We shall follow the Galilean transformation exclusively.
Say a vector is recognized in an inertial frame S 0 (fixed fra

2016 Spring Semester (104-2), Homework Set 4
1. Learn the derivation of the Euler Equations for rotational motions, and section 11.9 in the textbook. Make
sure you understand that the torque (or better denoted as ( )fixed ), is a physical vector observed

Make sure you understand everything weve covered in class, and make sure you understand the examples illustrated
in the textbook.
1. In class we have seen the 2-dimensional oscillator, and find that in fact its equations of motion (EOMs) are
simply two de

2016 Spring Semester (104-2), Homework Set 4
1. Learn the derivation of the Euler Equations for rotational motions, and section 11.9 in the textbook. Make sure
you understand that the torque (or better denoted as ( )fixed ), is a physical vector observed

Classical Mechanics:
The simple pendulum, with a fixed or moving support, on a 2-D plane
I.
THE PENDULUM PROBLEM
In class we have talked about the thought framework of Hamiltons (variational) principle, and the Euler-Lagrange
equations that follow. (Note:

2016 Spring Semester (104-2), Homework Set 3
1. Learn examples 11.3 in the textbook. Make sure you can obtain the Ic matrix (the lower-case superscript c
means the origin is located at the corner) with good understanding. Recall that we have also worked o