CLASS: M.Sc. MATHEMATICS
11N/ 419
St. JOSEPH’S COLLEGE (AUTONOMOUS) TIRUCHIRAPPALLI – 620 002
SEMESTER EXAMINATIONS – NOVEMBER 2011
TIME: 3 Hrs.
MAXIMUM MARKS: 100
SEM
SET
PAPER CODE
TITLE OF THE PAPER
I
2011
10PMA1105
CLASSICAL DYNAMICS
SECTION – A
Answer all the questions:
10 x 2 = 20
1.
2.
3.
4.
5.
6.
Define degrees of freedom and give an example.
Define holonomic system.
Define ignorable coordinates.
State Kepler problem.
Write the difference between impulsive constraints and constraint impulses.
What do you mean by gyroscopic system?
7.
What is meant by geodesic problem?
8.
Write down the n  n Hessian determinant for the transformations
9.
How is Pfaffian differential forms written?
10.
Write the Hamilton – Jacobi equation in terms of S and H.
vi 
F
(i  1 to n ) .
x i
SECTION – B
Answer all the questions:
11.
a.
5 x 7 = 35
State and prove principle of virtual work.
OR
12.
b.
Derive the Lagrangian form of d’Alembert’s principle.
a.
Find the differential equations of motion for a spherical pendulum of length l.
OR
13.
b.
A mass-spring system is attached to a frame which is translating with a uniform velocity
v0. Let l0 be the unstressed spring length and use the elongation x as the generalized coordinate. Find the Jacobi integral.
a.
State and prove the principle of linear impulse and momentum.
OR
14.
b.
State and prove the principle of angular impulse and momentum.
a.
Find the stationary values of the function f = z subject to constraints
1 = x2 + y2 + z2 – 4 = 0
2 = xy – 1 = 0
OR
15.
b.
Derive Hamilton’s equations.
a.
State and prove Jacobi’s theorem.
OR
b.
State and prove Stackel’s theorem.
SECTION – C
Answer any THREE questions:
3 x 15 = 45
16.
Derive an expression for rotational kinetic energy interms of the angular velocity.
17.
A double pendulum consists of two particles suspended by massless rods. Assuming that all
motion takes place in a vertical plane, find the differential equation of motion. Linearise these
equations assuming small motions.
18.
Four rigid uniform rods, each of mass m and length ‘l’ are joined by p in joinly at their end to
form a rhombus ABCD. Initially the system is motionless in a square configuration with point

F , solve for its velocity
C directly above point A. (i) If point A is given a vertical impulse
immediately after the impulse (ii) Suppose that the system falls with velocity v0 and point A
 1 and q 2 immediately after the impact.
strikes the floor in elstically. Solve for q
19.
State and prove Brachistochrone problem.
20.
Use the Hamilton – Jacobi method to analyse the Kepler problem.
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