LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3812 – CLASSICAL MECHANICS

 

 

Date : 06/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer ALL the questions:

 

1. a. i. The quantity exerted by the outside agent that causes the change of position is called ———
2.   I get up in the morning and go to work – denotes ————–type of motion.

iii. I get up in the morning and go to work but evening I’m back at home  – denotes ————–type

of motion.

1. The generalized displacement is —————————-
2. Holo means —————-in Greek.

OR

1. Derive the equation of motion of Atwood’s machine  [ 5 ]

 

1. .i. State and prove the principle of Virtual Work and deduce D’Alembert’s principle
2. An inextensible string of negligible mass hanging over a smooth peg at A connects another

mass m₁on a frictionless inclined plane of angle q to another mass m₂ . Use  D’Alembert’s

principle to prove that the mass will be in equilibrium if  m₂  = m₁sinq.

OR

1. Classify constraints and state the principles involved in choosing the generalised coordinates and

classify the constraints with reasons for any one of  the following cases

1. A bead moving on a circular wire.
2. A sphere rolling down a rough inclined plane without slipping.

iii. The molecules moving inside a gas container.                                                                      [15]

 

2. a. i. An ignorable coordinate is one which ———————-
3.  In a conservative system,  H = ——– + V

iii. In  variation , energy is ————————

1. In – variation process, System point is speeded up or slowed down in order to make the total

travel time along every path ——————

OR

1. Find the Routh’s function for the motion for the motion of a projectile.  Hence deduce equation of

motion.                                                                                                                                            [5]

 

1. State Hamilton’s principle and deduce Lagrange’s equation from Hamilton’s principle and hence

find the equation of motion of  one dimension Harmonic oscillator.                                              [7+8]

OR

1. i. Derive the Hamiltonian function
2. Derive Hamilton canonical equation of motion.

iii.Give the physical significance of Hamilton’s function                                                           5+5+5]

 

3.a. i.

ii.

iii. The time taken by a light ray to travel between two points is —————.

1. If the dynamical system have one degree freedom, then pdq – PdQ = —————
2. generates an ————– transformation.  

OR

1. b. Show that is a canonical transformation

                                                                                                                  [5]

1. State and prove Integral Invariant theorem of Poincare

.

OR

1. Discuss about the motion of a top

[7+8]

 04.a. i. The solution of H (q₁, q₂ q_3,…. q_n , ) + = 0 is known as ———————-

1. In any dynamical system, the collection of points is called a—————–

iii.  =

1. If q_i is cyclic, then p_i a ———————–

v.

OR

1. State and prove Liouvilli’s theorem. [5] 

                                                     

1. Derive the conservation theorem of angular momentum using Infinite decimal

contact transformation                  

OR

d.Derive the Hamilton – Jacobi equation for the Hamilton’s principle function S. and deduce that

[15+5]                                                                                                      

05.a.i.  The  Complete integral W of Hamilton –Jacobi equation is called  ——-

1. Separation of variables in Hamilton Jacobi’s equation is possible only if ——————

iii. For a conservative dynamical system in which the generalized coordinates are q , f  cyclic, then

the solution is given by ————-

1. Action integral denoted by A is defined to be —————–
2. If W_k denotes characteristic function, then  J_k = is known as —————

OR

1. Discuss the motion of a particle moving in a plane under the action of central

force using Hamilton – Jacobi equation.                                                                                                [5]

1. Derive the Hamilton – Jacobi equation for the Hamilton’s characteristic function

OR

1. Discuss Kepler’s problem using action angle variable.

[15]

 

 

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