21. For the given transformation
1. Q = p and P = -q
2. P = q and Q = p
where p, q are canonically conjugate variables, which one of the following statements is true?
1. Q = p and P = -q
2. P = q and Q = p
where p, q are canonically conjugate variables, which one of the following statements is true?
22. The Lagrangian for a simple pendulum is given by $$L = \frac{1}{2}m{L^2}{{\dot \theta }^2} - mgL\left( {1 - \cos \theta } \right)$$
Hamiltonian equations are given by
Hamiltonian equations are given by
23. Two events are separated by a distance of 6 × 105 km and the first event occurs 1 s before the second event. The interval between two events is
24. The lagrangian of a particle of mass m moving in one dimension is $$L = \exp \left[ {\left( {\alpha t} \right)\frac{{M{{\dot x}^2}}}{2} - k{x^2}} \right]$$ where, $$\alpha $$ and k are positive constants. The equation of motion of the particle is
25. Although mass energy equivalence of special relativity allows conversion of a photon to an electron-positron pair, such a process cannot occur in free space because
26. The Lagrangian forthe Kepler problem is given by $$L = \frac{1}{2}\left[ {{{\dot r}^2} + {r^2}{{\dot \theta }^2}} \right] + \frac{\mu }{r}\,\,\,\,\,\,\,\left( {\mu > 0} \right)$$
where, $$\left( {r,\,\theta } \right)$$ denotes the polar coordinates and mass of the particle is unity, then
where, $$\left( {r,\,\theta } \right)$$ denotes the polar coordinates and mass of the particle is unity, then
27. For a simple harmonic oscillator, the Lagrangian is $$L = \frac{1}{2}{{\dot q}^2} - \frac{1}{2}{q^2},\,{\text{if }}A\left( {p,\,q} \right) = \frac{{p + iq}}{2}$$ and H(p, q) is the Hamiltonian of the system, the Poisson bracket, {A(p, q), H(p, q)} is given by
28. A bead of mass m slides along a straight frictionless rigid wire rotating in a horizontal plane with a constant angular speed ω. The axis of rotation is perpendicular to the wire and passes through one end of the wire. If r is the distance of the mass from the axis of rotation and v is its speed, then the magnitude of the Coriolis force is
29. If for a system of N particles of different masses m1, m2, . . . mN with position vectors $${\overrightarrow {\bf{r}} _1},\,{\overrightarrow {\bf{r}} _2},\,.\,.\,.\,{\overrightarrow {\bf{r}} _N}$$ and corresponding velocities $${\overrightarrow {\bf{v}} _1},\,{\overrightarrow {\bf{v}} _2},\,.\,.\,.\,{\overrightarrow {\bf{v}} _N}$$ respectively such that $$\sum\limits_i {\overrightarrow {{{\bf{v}}_i}} = 0,} $$ then
30. 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
The quantity that conserved is