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Classical Mechanics                             June 7, 2019

                                   

                                                             Tutorial-III

• For a relativistic particle in three dimensions the Lagrangian is given by $$ L = - Mc^2 \sqrt{1- \dot{\vec{r}}{\,\,^2}/c^2} $$ where $\vec{r}=(x,y,z)$ is the position vector of the particle.
  1. Obtain the expressions for canonical momenta, $p_x,p_y,p_z$, conjugate to $x,y,z$
  2. Express the velocities in terms of the momenta. Use your results to show that relativistic Hamiltonian is $$ H = \sqrt{\vec{p}^{\,\, 2} c^2 + M^2 c^4} $$
  3. Obtain equations of motion using Poisson brackets.
• The Lagrangian for a symmetric top, in terms of Euler angles $\theta,\phi,\psi$ is given by $$ L = {1\over2} I_1 \big(\dot{\theta}^2 + \dot{\phi}^2 \sin^2\theta \big) +{1\over2} I_3 (\dot{\psi} + \dot{\phi} \cos\theta)^2 - mgL\cos\theta $$
  1. Compute the canonical momenta conjugate to $\theta,\phi,\psi.$
  2. Show that the Hamiltonian, in terms of momenta $p_\theta, p_\phi$ and $p_\psi$ the Euler angles, is given by $$ H= \frac{p_\theta^2}{2I_1} + \frac{p^2_\psi}{2I_3} + \frac{(p_\phi-p_\psi \cos\theta)^2}{2I_1\sin^2\theta} + mgL\cos\theta. $$
• The Hamiltonian or a particle in two dimensions, written in plane polar coordinates is \begin{equation} H = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2} + V(r,\theta) \end{equation}
  1. Obtain the Hamilton's equations of motion for the coordinates \(r, \theta\) and canonical momenta \(p_r,p_\theta\).
  2. Derive expression Lagrangian for this system using your results in part (a).

TIME :3hrs                   ELECTRODYNAMICS                                MM:100                                         

                               End Semester Examination

                                       ATTEMPT ANY FIVE QUESTIONS.

1. Find the potential due to a sphere carrying a uniform polarization \(\vec{P}\). The centre of the sphere is at the origin and radius of the sphere is \(R\). What is the electric field at a point (i) inside the sphere (ii) outside the sphere?
2. 1. Give an example of a charge configuration such that its dipole moment is zero and quadrupole moment depends on the choice of origin.
   2. An alpha particle travels in a circular path of radius $0.45$m in a magnetic field with $B=1.2$w/m$^2$. Calculate  (i) its speed (ii) its period of revolution, and (iii) its kinetic energy.Mass of alpha particle = \(6.64424. 10^{-27}\)kg \(\approx 4\times M_p= 4\times938.27\) MeV.
   3. A circular coil is formed from a wire of length $L$ with $n$ turns. The coil carries a current $I$ and is placed in an external uniform magnetic field $B$. Show that maximum torque developed is $\displaystyle\frac{IBL^2}{4n\pi}$.
   4. Give examples of at least six results/concepts that require modifications in time varying situation.                                                                            [4+5+5+6]
3. 1. Derive an expression for electrostatic energy of a charge distribution and hence show that electric field carries energy density \(\frac{\epsilon_0}{2}|\vec{E}|^2\).
   2. A conducting spherical shell carries a charge \(Q\), compute its electrostatic energy and hence obtain an expression for the capacitance of the shell.\hfill[10+10]
4. A rod of mass $m$ and length $\ell$ and resistance $R$ starts from rest and slides on two parallel rails of zero resistance as shown in figure 1. A uniform magnetic field fill the area and is perpendicular and out of the plane of the paper. A battery of of voltage $V$ is connected as shown in the figure 1.
   1. Argue that the net EMF in the loop is $V = Bv\ell$ when the rod has speed $v$.
   2. Write down $F = m\big(\dfrac{dv}{dt}\big)$ and integrate it so show that \begin{equation*}\label{EQ01} v(t) =\frac{V}{B\ell}\Big(1- \exp\Big(- \frac{B^2\ell^2 t}{mR}\Big)\Big). \end{equation*} Hint: Find the limiting speed and separate that out from the total $v$.
   3. What happens when the direction of magnetic field is reversed?            [8+8+4] 
5. 1. Show that, in absence of charges and currents, the electric and magnetic fields obey wave equation.
   2. State and prove important properties of plane wave solutions.
   3. Obtain an expression for energy density and intensity of plane waves.      [6+8+6]
6. An infinite rectangular hollow pipe is bounded by the planes \(x=\pm a,y=0, y=b\). The pipe extends to infinity in positive as well as negative \(Z\)- directions. The sides \(y=0, x=\pm a\) are grounded and the the side \(y=b\) is held at constant potential \(\phi_0\). Show that the potential inside the pipe is  \begin{equation*} \phi(x,y) = \phi_0\Big\{\frac{y}{b} +\frac{2}{\pi} \sum_{n=1}^\infty\frac{(-1)^n}{n} \frac{\cosh (n\pi x/b)}{\cosh(n\pi a/b)} \sin (n\pi y /b) \Big\} \end{equation*} 

$\newcommand{\Lsc}{\mathscr L}$       

                                                Classical Mechanics                      July 4, 2009                                                                                                                           


End Semester Examination::PART-B

Instruction:-Attempt any four questions. 

1. Consider a particle of mass $m$ moving in two dimensions in a potential The equation of motion for small oscillations are given to be                                                   [4+4+4] \begin{eqnarray} m\ddot{x} =k \big( 5x + y \big)\\ m\ddot{y} =k \big( x + 5 y \big)\nonumber \end{eqnarray}
   1. Find the normal frequencies of vibration of normal modes.
   2. Obtain expressions for \(x,y\) in terms of normal coordinates.
   3. Write the Lagrangian \[{\Lsc} = \frac{m}{2}(\dot{x}^2+m\dot{y}^2) - \frac{k}{2}\big(5x^2+2xy +5y^2\big)\] in terms of normal coordinates and verify that it takes the form \[{\Lsc} = \frac{1}{2}(\dot{Q}_1^2+\dot{Q}_2^2) - \frac{1}{2}(\omega_1^2Q_1^2+\omega_2^2Q_2^2)\]
2. Consider pendulum with a spring shown in figure. It oscillates in vertical plane like any simple pendulum.
   1. What is number of degrees of freedom? Which generalized coordinates will you use?[2+2+8]
   2. Write the Lagrangian for the system.
3. A missile is fired from the origin with initial velocity \(u\) and at an angle \(\alpha\) with horizontal.                                                                                               [3+3+3+3] 
1. Write the Lagrangian for the missile.
2. Is there a cyclic coordinate? Which one? What is the conserved quantity associated with it?
3. Is there any other conserved quantity?
4. Using conserved quantities you have found, briefly indicate how a full solution can be obtained.
4. • Show that                                                                                       [3+3+6] \begin{equation*} Q=-p ,\qquad P=q + Ap^2 \end{equation*} (where A is any constant) is a canonical transformation,
     1. [(i)] by evaluating \([Q,P]_{q,p}\)
     2. [(ii)] by expressing \(pdq-PdQ\) as an exact differential \(dF(q,Q)\). Hence find the type one generating function of the transformation. To do this, you must first use the transformation to express \(p,P\) in terms of \(q,Q\).
   • Use the relation \(F_2=F_1+PQ\) to find the type 2 generating function \(F_2(q,P)\),and check your result by showing that \(F_2\) indeed generates the transformation.
5. Consider the motion in a spherically symmetric potential $$ V(r) = - V_0 \left( {3 R\over r } + {R^3\over r^3} \right)$$ If orbital angular momentum of the particle is given by $l^2 = 10 m V_0 R^2$, and answer the following questions.
   1. Plot the effective potential as a function of $r$                                       [3]
   2. What should be the energy of the particle so that it may move in a circular orbit? How many circular orbits are possible? Find the radius of the stable orbit.     [3+3+3]

Physics of Information

A Course being Offered Currently at Chennai Mathematical Institute

• Lecture Notes
• Home Work and Solutions
• Test and Examination Papers
• Suggested for Further Study
• Video Recording Links
• Lecture 1 -- Jan 25, 2022
• Review of Laws of Thermodynamics; Concept of EntropyCarnot Engine;Maxwell's Demon
• Lecture 2 --  Jan 28, 2022
• Maxwell's Demon 2;Work of Landauer and Bennet
• Lecture 3 -- Feb 1, 2022
• Landauer Bennet argument; erasure of information costs energyInformation and Entropy; Shannon Entropy; Joint  Entropy;Connection with Equilibrium Statistical Mechanics
• Lecture 4 -- Feb  3, 2022
• Erasure of Information; Different Protocols;Work Done in A Measurement; Classical Information Theory;Joint Entropy; Conditional Entropy;Mutual Information;An Example
• Lecture 5 -- Feb  8, 2022
• Quantum Information Theory Joint Entropy; Conditional Entropy; Mutual InformationMeasurement in Quantum Mechanics; Projective Measurement;Positive Operator Valued Measurement; Relative Entropy
• Lecture 6 --- Feb 10, 2022
• Measurement; Projective Measurement; Positive Operator Valued Measurement
• Lecture 7 --- Feb 15, 2022
• Change in entropy in a projective measurement; Purification;Concave property of von Neumann entropy
• Lecture 8 --- Feb 17, 2022
• Quantum Szilard Engine
• Lecture 9 --- Feb 22, 2022
• Lecture 10 Feb 24, 2022
• Lecture 11, Mar 1, 2022
• Lecture 12 Mar 4, 2022
• Lecture 13 Mar 8, 2022
• Lecture 14. Mar 15, 2022
• Lecture 15 March 24, 2022

 Books recommended

1. Michael A.Nielsen and Issac L. Chuang, "Quantum Information and Quantum Computing", Cambridge University Press, Cambridge (2000)
2. Mark M. Wilde,"Quantum Information Theory", Cambridge University Press Cambridge (2013)
3. Thomas M. Cover and Joy A. Thomas, "Elements of Information Theory" Wiley-Interscience 2nt Edn (2006)
4.  Harvey S Leff and Andrew F. Rex (ed.) "Maxwell's Demon2--  Entropy, Classical and Quantum Information Computing", Institute of Physics Publishers, Bristol (2003)

Link to You tube Video

Classical Mechanics               July 3, 2019
 $\newcommand{\Lsc}{\mathscr L}$

Test-I

• For a system the Hamiltonian is given to be \begin{equation*} H(q,p) = \frac{p^\beta}{\beta} + V(q) \end{equation*}
  1. Obtain Hamilton's equations of motion.\hfill[3]
  2. Show that the Lagrangian of the system is given by [4] \[L = \frac{q^\alpha}{\alpha} -V(q)\] where \(\alpha\) is given by \[ \frac{1}{\alpha} + \frac{1}{\beta}=1\]
• Consider a particle of mass $m$ moving in two dimensions in a potential The equation of motion for small oscillations are given to be \begin{eqnarray} m\ddot{x} = \frac{1}{2}\big( 3x + y \big)\\ m\ddot{y} = \frac{1}{2}\big(x + 3 y \big)\\ \end{eqnarray}                                                                                                                       [5+5+5]
  1. Find the normal frequencies of vibration in (b).
  2. Obtain expressions for \(x,y\) in terms of normal coordinates.
  3. Write the Lagrangian in terms of normal coordinates and verify that it takes the form \[{\Lsc} = \frac{1}{2}(\omega_1^2\dot{Q}_1^2+\omega_2^2\dot{Q}_2^2) - \frac{1}{2}(Q_1^2+Q_2^2)\]
• 1. Find a differential equation for \(f(P)\) so that the transformation[3+2+3] \begin{equation*} q=\frac{f(P)}{m\omega} \sin Q, \quad p= f(P) \cos Q \end{equation*} may be a canonical transformation.
  2. Solve the differential equation you get subject to condition \(f(0)=0\) and show that \[ f= \sqrt{2m\omega P}\]
  3. Obtain the type 1 generator for this transformation.