MPH-006 Solved Assignment 2026

Description

MPH-006 Solved Assignment 2026 Available

Assignment Code: MPH-006/TMA/2026

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) The Lagrangian for a harmonic oscillator if given as L= 1 2 m( dot x ^ 2 + dot y ^ 2 )+ 1/2 * m * omega ^ 2 * (x ^ 2 + y ^ 2)

The transformed coordinate under rotation about the Z-axis is given as. x’ = x * sin theta + y * cos theta

Determine the corresponding y’ so that the transformation leaves the Lagrangian invariant

b) Determine the Hamiltonian and the Hamilton’s equation of motion:

i) The Lagrangian of a bead of m moving without friction along a wire bent in the shape of a parabola in the X-Y pane with y= x ^ 2 | given by. L= 1 2 m(1+4x^ 2 ) dot x ^ 2 – mg * x ^ 2

ii) The Lagrangian of a relativistic particle with charge e is given as: L= – m * c ^ 2 * sqrt(1 – (v/c) ^ 2) – e*phi +e overline A * overline v

c) Write down the expression for the Routhian for a bead of mass m sliding due to gravity on an elliptical wire for which the Lagrangian is given as Hence derive the equation of motion. L = 1/2 * m(a ^ 2 * sin^2 theta + b ^ 2 * cos^2 theta) * theta ^ 2 – mgb * sin theta

d) If G = J_{y} = 4p_{x} – x*p_{z} using Poisson bracket show that G generates a roation in X-Z plane.

2. a) The transformation of a 1D harmonic oscillator is given as Q = a/g and P = alpha*p * q ^ 2

1) Determine the Hamitonian and the value of a so that the transformation is canonical.

Obtain the generating function F_{1}(q, Q) Using F (q, Q) obtain F_{2}(q, P)

b) The Hamiltonian of a system with two degrees of freedom is given as

H(q_{1}, q_{2}, p_{1}, p_{2}) = (p_{1} ^ 2)/2 + (p_{2} ^ 2)/2 + sin(q_{1} + 2q_{2})

Determine the new Hamiltonian K

PART B

3. a) A simple pendulum with a bob of mass m and a string of length to with Hamiltonian

H = (rho_{0} ^ 2)/(2m_{0}) – m*g_{0} * cos theta

Construct Hamilton-Jacobi equation and obtain the characteristic function in integral form. Write the new coordinates P,Q.

b) A particle of mass m moves under a potential V(r) = (- k) / r, k > 0 The total energy of the system is where/is the angular momentum. E = 1/2 * m * r ^ 2 + (l ^ 2)/(2m * r ^ 2) – k/r

i) Obtain the radial action and the angular action variables. Hence, express the total action.

ii) Determine the radial and the angular frequencies and explain their significance

4. a) Derive the expression for total energy for a heavy symmetric top.

b) Obtain the expression for the rotational kinetic energy of a rigid body when the rotation is referred to its principal axes of inertia, for a spherical top (a uniform solid sphere).

c) A rigid body consisting of a uniform plate with x = y = a and mass M, obtain the moments of inertia (I x .I ny .I zz ) and hence the principal moments of inertia.