[2019CM/HMW-03]

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).