The Lagrangian of a free particle in spherical polar coordinates is given by $$L = \frac{1}{2}m\left[ {{{\dot r}^2} + r{{\dot \theta }^2} + {r^2}{{\dot \phi }^2}{{\sin }^2}\theta } \right]$$
The quantity that conserved is

A mass m is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius r₀ and angular speed ω₀ by an applied force $$\overrightarrow {\bf{F}} $$ communicated through an inextensible thread that passesthrough a hole on the surface as shown in figure given below. Then, this force is suddenly doubled.

The magnitude of the radial velocity of the mass

The speed of a particle whose kinetic energy is equal to its rest mass energy, is given by (c is the speed of light in vacuum)

The Hamiltonian corresponding to the Lagrangian $$L = a{{\dot x}^2} + b{{\dot y}^2} - kxy$$     is

A particle is moving in an inverse square field. If the total energy of the particle is positive, then trajectory of particle is