CSIR NET Maths previous year question papers

Suppose a dynamical system has the Lagrangian L = (q̇$_{1})^{2} + (q$̇$_{2})^{2} + (q_{1})^{2} + q$̇$_{1}q$̇$_{2}$. If $p_{1}$ and $p_{2}$ are momenta conjugate to $q_{1}$ and $q_{2}$ respectively, then which of the following statements is true?

Suppose f, g are smooth functions of generalized coordinates $q_{1}, q_{2}, \dots, q_{n}$, the associated conjugate momenta $p_{1}, p_{2}, \dots, p_{n}$, and time t. Let [f, g] denote the Poisson bracket of f and g. Suppose H is a Hamiltonian of the system. Then which of the following statements are true?

A mechanical system is described using generalized position q and generalized momentum p. Let Q and P denote new generalized position and generalized momentum variables respectively, generated by the generating function $F(q,P)=q^{2}e^P$, and Q,P are canonical coordinates. Let G(p,Q) be a function such that G(2,e)=0, and it generates the same canonical coordinates Q,P. Then which of the following statements are true?

Consider a particle of mass $m_{1}$ moving along a horizontal line L that is perpendicular to a vertical wall. Let x denote the distance of the particle from the wall. Suppose a simple pendulum of length l having mass $m_{2}, m_{2} \ne m_{1}$, is attached to the particle, hanging below L with $\theta$ measured from the downward vertical. If the pendulum oscillates in a plane containing L, then the equations of motion in terms of the generalized coordinates x and $\theta$ are (g denotes the acceleration due to gravity)

Two blocks of equal mass m are connected by a flexible inelastic cord of mass M. One block is placed on a smooth horizontal table, the other block hangs over the edge. The total potential energy of the entire cord is given by (−Mg$/2l)x^{2}$, where x is the distance of the hanging block from the edge of the table, l is the length of the cord, and g is the gravitational acceleration. Then

Consider a solid circular cylinder of radius 2 meters and height 3 meters of uniform density. If the density of the cylinder is $\rho$ kg/meter$^{3}$, then the moment of inertia (in kg meter$^{2})$ of the cylinder about a diameter of its base is

Consider a particle of mass m which is moving on a surface due to gravity. Suppose the Lagrangian of the particle in the cylindrical coordinates is given by $L = (m/2)[(1 + 4\alpha^{2}r^{2})$ṙ$^{2} + r^{2}\theta$̇$^{2}] - m\alpha$gr$^{2}$, where $\alpha$ is a positive constant, and g is the acceleration due to gravity. If the particle is in circular motion, then

Consider the system of two particles with total kinetic energy T = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta$, and Lagrangian L = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta + 2$gl $\cos \theta$, where $x, \theta$ are generalized coordinates, and g, l are positive constants. Then the non-zero frequency of the normal mode of the system with small oscillations (|$\theta$| ≪ 1) is

Consider a solid torus of constant density $\rho$, formed by revolving the disc $(y - b)^{2} + z^{2} \le a^{2}, x = 0$ about the z-axis, where 0 < a < b. Then the moment of inertia of the solid torus about the z-axis is

Let g denote the acceleration due to gravity and a > 0. A particle of mass m glides without friction on the cycloid $x = a(\theta - \sin \theta), y = a(1 + \cos \theta), 0 \le \theta \le 2\pi$. The equation of motion of the particle is

Let $q_{1}, q_{2}$ be generalized coordinates and $p_{1}, p_{2}$ their conjugate momenta. Let a and b be such that $Q_{1} = q_{1}, P_{1} = a p_{1} + 16 p_{2}, Q_{2} = p_{2}, P_{2} = 2q_{1} + b q_{2}$ is a canonical transformation. Which of the following statements are true?