PHY 305/601 Classical Mechanics Assignment No. 1 1. Consider a particle of mass m constrained to move on a frictionless cylinder of radius R, given by the equation ? =R in cylindrical polar coordinates (? , ? , z). Besides the force of constraint, the only force on the mass is force F=-kr directed toward the originUsing z and ? 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 n n ? . ?. 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. Fig. 1 4. A point mass glides without friction on a cycloid, which is given by x= a(v-sin? ) and y=a(1+cos? ) with 0? v? 2?. 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 ??? for which x ? ? 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 holds with the zero variation of both qi and q? i ? ? ? ? ? ?? at the end points, then the corresponding Euler-Lagrange equations are ? ? ? d 2 ? ?L ? ? ? d ? ?L ? ? ? ?L ? 0, i ? 1,2….. n ?? dt ? ? q ? ?q dt 2 ? ? qi ? i ? ? ? i ?
Apply this result to the Lagrangian m ?? k 2 L ? q q? q 2 2 7. Three point masses m1, m2 and m3 are fixed to the ends of two massless rods and glide without friction in circular tyre of radius R, which stands vertically in the gravitational filed of the Earth as shown in fig. 2. Find the equation of motion by means of Lagrange multipliers and determine the equilibrium position. Fig. 2. 8. A particle of mass m moves without friction under the action of gravitation on the inner surface of a paraboloid, which is given by, x2+y2=ax.
Determine the Lagrangian and equation of motion. 9. A mass m is suspended by a spring with spring constant k in the gravitational field. Besides the longitudinal vibration, the spring performs a plane pendulum motion as shown in fig. 3. Find the Lagrangian and derive the equation of motion. . Fig. 3. 10. Solve the Atwood machine problem using the method of lagrange multipliers. 11. Figure 4 shows a solid cylinder with centre G and radius a rolling on the rough inside surface of a fixed cylinder with centre O and radius b > a.
Find the Lagrange equation of motion. Fig. 4. 12. Let S be a system shown in fig. 5. The rail is smooth and the prescribed force F(t) acts on a particle P2 as shown. Gravity is absent and find the Lgranje equations of motion and solve it. The system moves under the prescribed force F(t). Fig. 5. 13. Discuss (a) Homogeneity of time and conservation of energy (b) Homogeneity of space and conservation of linear momentum (c) Isotropy of space and conservation of angular momentum