[2019CM/TEST-01]

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.