Assignment No. 1
1. Consider a particle of mass m constrained to move on a frictionless cylinder of radius R, given by the equation ρ=R in cylindrical polar coordinates (ρ, φ, z). Besides the force of constraint, the only force on the mass is force F=-kr directed toward the originUsing z and φ as generalized coordinates find the Lagrangian L, solve Lagrange’s equations and describe the motion. 2. Show that the kinetic energy of any holonomic mechanical system has the form.
T a jk (q) q ( j ) q(k )
j 1 k 1
3. Consider a double pendulum (fig. 1) made up of two masses, m1 and m2 and two lengths l1 and l2. Find the equation of motion.
4. A point mass glides without friction on a cycloid, which is given by x= a(v-sinθ) and y=a(1+cosθ) with 0≤v≤2π. Determine the Lagrangian and solve the equation of motion.
5. 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 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.). 6. The term generalized mechanics has come to designate a variety of classical mechanics in which the Lagrangian contains time derivatives of qi higher than the first order. Problems
x f x, x, x, t have
been referred to as ‘jerky’ mechanics. Such
equations of motion have interesting applications in chaos theory. By applying the methods of the calculus of variations, show that if there is a Lagrangian of the form
L q, q, q, t and Hamilton’s principle...