Download Q1 Show that the solution of the two body problem is a - UR-CST

KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY
INSTITUT DES SCIENCES ET TECHNOLOGIE
Avenue de l'Armée, B.P. 3900 Kigali, Rwanda
INSTITUTE EXAMINATIONS – ACADEMIC YEAR 2012-2013
FACULTY OF SCIENCE
END OF SEMESTER EXAMINATION
SECOND YEAR SEMESTER II
MAT 3226 CLASSICAL MECHANICS
TIME
: 2 HOURS
MAX.MARKS =60
DATE: ….. /…../2013
INSTRUCTIONS:
1. This Question Paper has 4 Questions in two sections;
2. The section A is compulsory and carries 30 marks
3. Answer to any 2 Questions in section B showing all working clearly.
4. Start every new question from a fresh page
5. Every question carries 15 marks in section B
6. Do not forget to write your Registration Number
7. Write all your answers in the booklet provided
8.
Do not write any answers on this question paper.
Section A:
Q1.
i.
ii.
iii.
Verify the conservation laws for a simple harmonic oscillator. (6 marks)
In the system with constraints, prove that the work done by the constraint force in
any possible displacement is zero. (Hint: Consider that the mass is moving on a
smooth fixed surface f ( x, y, z )  0 in three dimension space). (6 marks)
Show that the Lagrange’s equation of motion of a particle defined by
L(q, q , t ) 
iv.
v.
1
mq 2  V (q, t ) is equivalent to the Newton’s equation if q is a cartesian
2
coordinate. (6 marks)
Considering a motion with a Lagrangian which does not depend explicitly on the
time, establish the Jacobi’s integral of the motion. (6 marks)
Verify if the transformation (q, p)  (Q, P) produced by the generating function
F2 ( P, q)  P tan q is canonical transformation or not. (6 marks)
Section B: Answer to any 2 questions in this section
.
Q2. Find the Lagrangian and Hamiltonian for a motion of a bead of mass m sliding
smoothly on a wire in the shape z  f (x) , the z –axis and x -axis are respectively
vertical and horizontal (Make a sketch). (15 marks)
Q3.
Using the Newton’s second law, deduce that for a single particle with constant
mass, the equation of motion implies the differential equation
dT
 F .v where T stands
dt
for the kinetic energy, F the force and v the velocity. Moreover, show that if the mass
varies with the time the corresponding equation becomes
d
(mT )  F . p where p is the
dt
linear momentum. (15 marks)
Q4. A single particle moves in space under a conservative potential. Establish the
Jacobi’s integral of the motion in ellipsoidal coordinates u , v,  defined in terms of the
usual cylindrical coordinates r , z ,  by the relations
r  a sinh v sin u
z  a cosh v cos u
 
(15 marks)