LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – NOVEMBER 2010

    PH 1814 / 1809  – CLASSICAL MECHANICS

 

 

 

Date : 03-11-10                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART A

Answer ALL questions                                                                                               (10×2 = 20)

1. What are cyclic coordinates?
2. Prove that d/dt (mT) = where T is the kinetic energy of the particle.
3. State and express Hamilton’s variational principle.
4. Give two differences between the Lagrangian and Hamiltonian methods of obtaining the equations of motion.
5. Using the definition of . Show that .
6. What are Euler’s angles?
7. Show that p = 1/Q and q = PQ² is canonical
8. Show that [p_x,L_z] = -p_y
9. Show that the Hamiltonian is a constant of motion if it is not an explicit function of time.
10. What is Hamilton’s principal function S?

PART B

Answer any FOUR questions                                                                                                (4×7.5 = 30)

11. Show that the charged particle in an electromagnetic field has a potential .
12. Reverse the Legendre’s transformation to derive the properties of L(q,,t) from H(q,p,t) treating the q_i as independent quantities and show that it leads to the Lagrangian equation of motion
13. Set up the Lagrangian for a particle of mass m in a central force using polar coordinates (r, q) and hence obtain the differential equation of orbit in the form: d²u/dq² + u = -m/l² d/du V(1/u)
14. Prove the invariance of Poisson’s brackets under a canonical transformation.
15. Solve by the Hamilton Jacobi method the motion of a particle in a plane under the action of a central force V(r) to obtain the equation of the orbit.

 

PART C

Answer any FOUR questions                                                                                                (4×12.5 = 50)

16 a. Show that the Lagrange’s equation can be derived from Hamilton’s principle for a conservative holonomic system.                                                                                                                              (6)

6. A particle of mass m moves in one dimension such that it has the Lagrangian
   L=m²/12 + mV(x) – V²(x) where V is some differential function of x. Find the equation of motion for x.                                   (6.5)

17 a. Using the definition of the Hamiltonian H(q,p,t) =  – L(,t) obtain Hamilton’s canonical equations.                                                                                                                            (7.5)

1. The Lagrangian for a system of one degree of freedom is,

L = m/2(sin²wt + qwsin2wt +q²w²). What is the Hamiltonian of the system?                              (5)

18  a.Give an account of the theory of canonical transformations.                                                          (6.5)

b.Obtain the transformation equations for the generating functions F₄(p,P,t).                        (6)

19   a. For the Kepler’s problem in action-angle variables assume the expression for the action integral as J_r = [2mE + 2mk/r – (J_q + J_j )² / 4p²r² ]^1/2.dr. Solve this integral to show that t² µ a³ where t is the time period of any planet with semi-major axis ‘ a ‘ about the Sun.                                                                         (8)

4. A particle moves in periodic motion in one dimension under the influence of a potential V(x) = F|(x)| where F is a constant. Using the action-angle variables find the period of the motion as a function of the particle energy.          (4.5)
5. Write notes on,
6. i) Applications of Coriolis Effect              ii) Infinitésimal canonical transformations

 

 

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