Questions on Lagrangian Dynamics Document Preview:

PHYS20401 Lagrangian Dynamics Terry WyattExample Sheet 2:1. In Lecture 4 we wrote down in spherical polar coordinates (r;;) the Lagrangianfor a pointlike particle of mass m moving in a time-independent conservative po-tentialV (r;;). Demonstrate explicitly that Lagrange’s equation is valid for thecoordinate . (Hint: Look back at your notes for Lecture 3, where we demon-strated explicitly that Lagrange’s equation is valid for the coordinate in planepolar coordinates. Apply the same basic method to the coordinate here.)2. Here are some simple problems to help you practice solving systems involvingangular coordinates using Lagrangian methods. Write down the Lagrangian andthen apply Lagrange’s equation(s) to obtain the equation(s) of motion. Whereappropriate calculate the frequency of small oscillations about the equilibriumposition. In each case please draw a diagram indicating clearly the direction ofany coordinates you dene. By considering simple limiting cases or other methodstry to cross check your answers. The systems to be considered are as follows:(a) A simple pendulum consists of a massm attached to a massless rod of lengthl.Analyse in terms of the angular displacement of the pendulum bob from theequilibrium position. In the rst instance, do NOT make the approximationthat 1. Having obtained the general solution, make the approximationthat 1 and show that your solution reduces to SHO. Can you obtain thesame equation of motion by applying the conservation of energy?(b) A simple pendulum consists of a massm attached to a massless rod of length2l. A mass m lies at the point of support and can move on a horizontal line1lying in the plane in which m moves (as shown in the following gure).2(c) A thin rod of mass m and length 2l is suspended from one end and swingsin one dimension. Note: it is informative to use try out three alternativemethods for calculating the kinetic energy of the rod:i. Calculate and use the…

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