(i) State and prove Hamilton's principle and use it to prove that the shortest distance between two points in space is a straight line joining them. (ii) Use Hamiltonian mechanics to find the differential equation for planetary motion, moving under force f(r) = -\frac{k}{r^2} and prove that the areal velocity is constant.
Discuss the mechanics of a system of point particles with special emphasis on the conservation theorems. How can we extend the results to a system with continuous mass distribution ?
A bead slides on a wire in the shape of a cycloid described by the equations x = a (\theta - \sin \theta) y = a (1 + \cos \theta) \quad \text{with} \quad 0 \le \theta \le 2\pi. Find the Lagrangian and equation of motion.
Write the components of the velocity 4-vector of a particle in an inertial frame S. How does the four velocity transform to another inertial frame S' ? Obtain the Einstein's law of addition of velocities from the above transformations.
What is the significance of the null result of Michelson-Morley experiment ? In their experiment Michelson and Morley set l_1 = l_2 = 11\text{ metres}. The wavelength of light used was 5.5 \times 10^{-7}\text{ m}. The orbital velocity of the earth is taken to be 30\text{ kms/sec}. Estimate the fringe shift to be expected.
Using Euler's equations for a force free motion of a rigid body, show that the Kinetic energy remains constant throughout the motion of the rigid body.
What are cyclic coordinates for a system of particles ? Discuss the significance of the Hamilton's principle. Show how Lagrange's equations can be derived from Hamilton's principle.
A force field is given by \vec{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k} Is it a conservative field ? If so, what is the scalar potential ?
Describe briefly the Michelson-Morley experiment and discuss the significance of its null result in the context of the special theory of relativity.
Find the velocity and momentum of a particle having rest mass m_0 and kinetic energy equal to two times of its rest mass energy.
Solve Euler's equation of motion for a rigid body to show that the torque-free motion of a spherical top is a uniform rotation about an axis fixed in space.
A particle of mass m is moving with the velocity \vec{v} on the surface of the earth. Discuss the nature of its motion in both the hemispheres due to the Coriolis force \vec{F} = -2\,m\,\vec{\omega} \times \vec{v}.
Discuss that the motion of a particle in a central field gets restricted to a plane provided its angular momentum is nonzero. What will happen if the angular momentum is zero?
The Lagrangian for a particle has the form L = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) - U(r) Obtain its form in plane polar coordinates and then get expression for the Hamiltonian using appropriate transformation.
Derive the law of addition of relativistic velocities. Use it to prove that under the Lorentz transformation no two velocities can add upto more than the value of the speed of light.
A particle of rest mass M moving at a velocity u collides with a stationary particle of rest mass m. If the particles stick together, show that the speed of the composite ball is equal to u\alpha M/(\alpha M + m), where \alpha = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} .
Discuss the torque-free motion of a symmetric rigid body and hence estimate the period of precession of Earth's polar axis. [Use \frac{I_3 - I_1}{I_3} = 0.33, where I_1, I_2 and I_3 are the moments of inertia about the three principal axes of the Earth]
Deduce the minimum energy of a gamma ray photon (in MeV), which can cause electron-positron pair production.
State Hamilton's principle for a system of particles. If I and L represent the action integral and the Lagrangian function, respectively, write down the mathematical form of Hamilton's principle and explain clearly the significance of the same.
Using the concept of D'Alembert's principle, show that the generalized force can be defined as Q_j = \sum_i \bar{F}_i \cdot \frac{\partial \bar{r}_i}{\partial q_j} , where \bar{r}_i is the Cartesian coordinate of the i^{\text{th}} particle experiencing external force \bar{F}_i and q_j stands for the generalized coordinate. Discuss the significance of the above expression.
