LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – mathematics

FourTh SEMESTER  – APRIL 2003

MT  4801/ M  1026   mechanies – II

16.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

 

Answer ALL the questions

 

1. a)  Explain the term ‘ABERRATION’.  Also derive the relativistic formula

 

for aberration in the form      (8)

(OR)

 

1. Show that the operator  is an invariant for

Lorentz  transformation.                                                                       (8)

 

2. a)   State ‘ETHER’ Hypothesis.  Explain the  Michelson- Morley experiment

and give the conclusion.                                                                         (17)

 

(OR)

1. b) Show that Lorentz  transformations forma group      .                            (17)

 

3. a) Obtain the transformation formula for mass in the form (8)

(OR)

 

1. If a body of mass m disintegrates while at rest in to two parts of rest masses m₁ and m₂, show that the energies E₁ and E₂   of the parts are

given by       (17)

 

4. a) Derive the equation E = m C², Deduce that p² – is an invariant under

Lorentz  transformation.                                                                      (17)

 

 

 

 

 

 

 

 

 

 

 

 

 

(OR)

1. Obtain the transformation formula  for force components in the

(17)

 

 

 

5. a)    Explain ‘contravariant vectors’,   covariant vectors,

‘contravariant tensors’  and ‘covariant tensors’.                                   (8)

 

(OR)

1. If a vector has components, on cartesion coordinates then

the components in polar coordinates are and if the components be

then the polar coordinates  components are         (8)

 

6. a)      Define fundamental tensors and show that gmg is a Covariant  tensor of

rank two. Also transform ds² = dx² + dy² + dz² in polar and cylindrical

coordinates.                                                                                            (17)

 

(OR)

1. b) Define Christoffel’s 3-index symbols of the first and second kind.  Also

calculate christoffel’s symbols corresponding to the metric

ds² = dr² + r²dq² + r²sin²q df².                                                                (17)

 

7. a) Define ‘Energy Tensor’.  Show that the equationfor m = 4

gives the equation of continuity in Hydrodynamics.                                  (8)

 

 

 

 

 

 

 

 

 

(OR)

1. Obtain isotropic polar coordinates and Cartesian coordinates.

Also Deduce that the velocity of light at distance r₁ from the origin is

 

(8)

 

8. a) Obtain the schwarzchild  line element in the neighbourhood of  an attracting particle

 

in the from       (17)

 

(OR)

1. b)     Derive the differential equation to the planetary orbits in the

 

form  .                                    (17)

 

 

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