IGNOU MPH 6 SOLVED ASSIGNMENT

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MPH 6 2025 - English

Tutor Marked Assignment CLASSICAL MECHANICS II

Course Code: MPH-006

Assignment Code: MPH-006/TMA/2025

Max. Marks: 100

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

PART A

1. a) (i) What is Legendre transformation of a function? Express the Legendre transformation of a Lagrangian to obtain the Hamilton's function H.

(ii) The Hamiltonian of a simple harmonic oscillator is given as:

Show that the velocity vector is tangential to the curve defined by the Hamiltonian. Draw a labeled diagram of the velocity vector for the phase trajectory for energy E.

b) Consider a particle of mass m moving in a plane attracted to the origin due to the potential k/r:

(i) Choose an appropriate coordinate

(ii) Write down the Lagrangian, the momenta conjugate to your choice of coordinates, the Hamiltonian and the action for the system.

(ii) Show that the action is invariant under translation of time. What conser vation law does this yield?

c) Consider a Lagrangian L where q is cyclic. Show that the momentum conjugate to q is conserved under the translation q→q+a.

2. a) (i) What is a canonical transformation? Why do we use canonical transformation?

Give an example where a canonical transformation becomes useful.

(ii) The Hamiltonian of a harmonic oscillator if given by 

Solve the problem of the harmonic oscillator using canonical transformation with the generator 

 

b) Using symplectic condition for canonical transformation, show that the transformation is canonical

Obtain the generating function for the transformation.

c) State Liouville's theorem and write the mathematical expression. Show that if pdepends on q.p through the Hamiltonian 

 

PART B

3. a) Consider a damped harmonic oscillator whose Lagrangian is given by:

(i) Write the equation of motion and hence obtain the corresponding Hamiltonian.

(ii) Using Hamilton-Jacobi equation associated with the Hamiltonian, solve the equations of motion.

b) For a particle moving in central force field, the Hamiltonian of the system is given by

 

 Assuming the motion to be elliptical, obtain action variables J_r and J_o.

c) Use the Hamilton-Jacobi method to find Hamilton's principal function W for a particle in the three dimensional isotropic oscillator well with a potential  Hence obtain the corresponding momentum  and the corresponding action variables 

4. a) Using appropriate labelled diagram, obtain the kinetic energy of a free symmetrical top of mass M.

b) Consider a uniform square plate of length x ya and mass m. Obtain the moment of inertia along  z-axis

MPH 006 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment

CLASSICAL MECHANICS II

Course Code: MPH-006

Assignment Code: MPH-006/TMA/2026

Max. Marks: 100

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:

The transformed coordinate under rotation about the Z-axis is given as:

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 is given by:

ii) The Lagrangian of a relativistic particle with charge e is given as:

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.

d) If , 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:

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

ii) Obtain the generating function F₁(q, Q). Using F₁(q, Q), obtain F₂(q, P).

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

Determine the new Hamiltonian K.

PART B

3. a) A simple pendulum with a bob of mass m and a string of length l₀ with Hamiltonian

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 . The total energy of the system is

where l is the angular momentum.

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 and mass M, obtain the moments of inertia (I_xx, I_yy, I_zz) and hence the principal moments of inertia.

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