Chapter 7, Problem

Two mass points of mass m1 and m2 are connected by a string passing through a hole in a smooth table so that m1 rests on the table surface and m2 hangs suspended. Assuming m2 moves only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for for the system and, if possible, discuss the physical significance any of them might have. Reduce the problem to a single second-order differential equation and obtain a first integral of the equation. What is its physical significance? (Consider the motion only until m1 reaches the hole.)

A point mass is constrained to move on a massless hoop of radius a fixedin a vertical plane that rotates about its vertical symmetry axis with constantangular speed ω. Obtain the Lagrange equations of motion assuming the only external forces arise from gravity. What are the constants of motion? Show that if ω is greater than a critical value ω0, there can be a solution in which the particle remains stationary on the hoop at a point other than at thebottom, but that if ω < ω0, the only stationary point for the particle is atthe bottom of the hoop. What is the value of ω0?

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).