61. The Lagrangian of a particle' of mass m is $$L = \frac{m}{2}\left[ {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} \right] - \frac{V}{2}\left( {{x^2} + {y^2}} \right) + W\sin \omega t$$
where V, W and ω are constants, then conserved quantities are
where V, W and ω are constants, then conserved quantities are
62. The Lagrangian of a diatomic molecule is given by $$L = \frac{m}{2}\left( {\dot x_1^2 + \dot x_2^2} \right) - \frac{k}{2}{x_1}{x_2}$$ where, m is the mass of each atom and x1, x2 are displacements from equilibrium position and k > 0. The normal frequencies are
63. A planet moves around the sun in an elliptical orbit with semi-major axis a and time period T. T is proportional to
64. A particle of charge q, mass m and linear momentum $$\overrightarrow {\bf{p}} $$ enters an electromagnetic field of vector potential $$\overrightarrow {\bf{A}} $$ and scalar potential $$\phi $$. The Hamiltonian of particle is
65. A mass m is connected on either side with a spring each of spring constants k1 and k2. The free ends of springs are tied to rigid supports. The displacement of the mass is x from equilibrium position.
Which one of the following is true?
Which one of the following is true?