Himachal Pradesh University (HPU) 2012-2nd Year M.A Mathematics HP University, Classical Mechanics, M.Sc thetics - Question Paper

(b) For a particle moving on a sphere. show that the generalized forces arc given by Q, a fflj (i iin Q, Qj ^a0. where m is the mass and a is ihe radius of the sphere and 0. v are Eukrian angles.

SF.CnON-I!

A,. Derive Hamiltons equations and discuss ihe properties of ihe Hamiltonian and Hamilton's equation of motion.

(a)    ffcscuss the stationary value of a definite integral.

(b)    Define Canonical transformation. Show that the transformation    +9^s).Qun^_,j [_s canonical.

6. (a) For a spherical pendulum, show that the Hamiltonian

H T + V (p ³ +cotcc¹$pJ\- mga sinO, 2ma '    

where p, are generalized momenta and m. a are respectively, the mass and radius of the pendulum.

SECTION-III

7. fg).- State and prove modified Hamilton's principle.

(b) For a particle of mass m moving in a plane with

potential energy , where K is a constant and r is the

distance from the origin, show that the HamiliooJacobi equation can be obtained as

(r. 0) are polar coordinates.

State and prove the principle of least action.

(b) Define Lagrange's brackct and show that Lagrange's *** bracket is canonical invariant.

9.. jW Derive H.irailton-Jacobi equation.

(b) Show that for a free partkk of unit moss moving on a straight line, the Hamiltoo-Jacobi equation reads as

dS ifJsV

5T ⁺ 2W *-