LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

AC 14

FIRST SEMESTER – NOV 2006

PH 1809 – CLASSICAL MECHANICS

 

 

Date & Time : 02-11-2006/1.00-4.00    Dept. No.                                                       Max. : 100 Marks

 

 

PART A                      ( 10×2 = 20)

1. What are cyclic coordinates? Show that the momentum conjugate to a cyclic coordinate

is a constant

2. Give an example of a velocity dependent potential
3. State and express Hamilton’s variational principle.
4. What are Euler’s angles?.
5. Show that the kinetic energy T for a torque free motion of a rigid body is

a constant of motion.

6. What is meant by canonical transformation?
7. Show that the generating function F₄ = pP generates a transformation that interchanges

momenta and coordinates.

8. Show that [q,H]_q,p = q dot and [p,H]_q,p = p dot
9. Express the Hamiltonian using Hamilton’s characteristic function W in polar coordinates

for a particle under a central force V(r).

10. Define action variable J and angle variable w.

 

PART B                     (4×7.5 = 30)

Answer any Four questions only

11a Establish the relation between the Lagrangian and the Hamiltonian   (4 marks).

b.Obtain the equations of motion of a simple pendulum using the Hamiltonian formulation.

(3.5 marks)

12. Obtain Hamilton’s equations of motion from the variational principle.
13. Solve the equation of orbit given : q = l ò   dr/r²   / [2m (E+ V(r) – l²/2mr² ]^½    +   q’

for an attractive central potential and classify the orbits in terms of e and E.

14a Obtain the tranformation equation for the generating function F₂(q,P,t)    (4.5 marks)

b Show that the transformation Q = q +  ip and P = q – iP is not canonical   (3marks)

15. Solve the harmonic oscillator problem by the HJ method.

 

PART C                     (4×12.5 = 50)

Answer any Four questions only

16a. Derive the general  form of Lagrange’s equation using D’Alembert  principle.  (8 marks)

1. A particle of mass m moves in one dimension such that it has the Lagrangian

L = m²x⁴/12 + mx²V(x) –V²(x) where V is some differentiable function of x. Find the

equation of motion for x.           (4.5 marks)

17a. Obtain Euler’s equations of  motion for rigid body acted upon by a torque N  (6 marks)

1. Solve the Euler’s equation of motion for a symmetric top I₁=I₂ ≠ I₃ with no torque

acting on it                                                                                  (6.5 marks)

18a. Show that the Poisson bracket is invariant under canonical transformation  (8 marks)

1. Prove that an infinitesimal canonical transformation does not change the value of the

Hamiltonian of a system.   (4.5 marks)

19. Solve the Kepler’s problem in action-angle variables.
20. Write notes on any TWO of the following
21. i) Constraints of motion
22. ii) Coriolis Effect

iii)  Hamilton Jacobi method.

 

 

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