(Q1) A mass m slides down the smooth incline surface of a wedge of mass M. The wedge can move on a smooth horizontal surface. Find the Lagrangian of the system and the equation(s) of motion.
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- A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity with the Lagrangian L = e2γt(1/2 * mq˙2 − V (q)) which gives the desired equation of motion. (a) Consider the following generating function: F = eγtqP - QP.Obtain the canonical transformation from (q,p) to (Q,P) and the transformed Hamiltonian K(Q,P,t). (b) Let V (q) = (1/2)mω2q2 be a harmonic potential with a natural frequency ω and note that the transformed Hamiltonian yields a constant of motion. Obtain the solution Q(t) for the damped oscillator in the under damped case γ < ω by solving Hamilton's equations in the transformed coordinates. Then, write down the solution q(t) using the canonical coordinates obtained in part (a).The inclination angle of a particle of mass m is adjustable, located on the moving edge. Inclined plane is horizontal at time t = 0 is in position. At t>0 moveable edge of the inclined plante is lifted by constant angular velocity of w to allow mass m to start to moving. Write down the Lagrangian equation of the mass m.determine the general solution to thegiven differential equation (D2 − 2D + 2)2(D2 − 1)y = 0.
- For a one dimensional system, x is the position operator and p the momentum operator in the x direction.Show that the commutator [x, p] = ihUsing Lagrangian formalism, solve the following problem: A solid cylinder is released from rest to roll without slipping down a ramp of slope ϕ. a) What is the best choice of generalized coordinates to be used?Consider the “Foucault pendulum”, as shown below. Foucault set up his 1851 spherical pendulum (of mass m and length L) experiment in the Pantheon dome of Paris, showing that the plane of oscillation rotates and takes about 1.3 days to fully revolve around. This demonstrated the extent to which Earth’s surface is not an inertial reference frame (e.g., role of the Coriolis force). Your task here is to determine (but not solve) the equations of motion.