[QUIZ/CM-05001]

The Lagrangian for several systems is given below, find cyclic coordinates and corresponding conservation laws. Write your answers in the table given below.

1. The Lagrangian for a particle in three dimension is $$ L = {m\over 2} ( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 ) $$
2. The Lagrangian for particle in three dimension is given by $$ L= {m\over 2} \left( \dot{\rho}^2 + \rho^2 \dot{\phi}^2 + {\dot{z}^2\over{1+(\rho/\lambda)^2 }}\right) -V(\rho) $$
3. The Lagrangian for a symmetric top, in terms of Euler angles $\theta,\phi,\psi$ is given by $$ L = {1\over2} I_1 \left(\dot{\theta}^2 + \dot{\phi}^2 \sin^2\theta \right) +{1\over2} I_3 (\dot{\psi} + \dot{\phi} \cos\theta)^2 - mgL\cos\theta $$
4. For a relativistic particle the Lagrangian is given by $$ L = - Mc^2 \sqrt{1- \dot{\vec{r}}^2/c^2} $$ find the relativistic momenta and the relativistic Hamitonian.