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p605hw4f08wsoln - Phys 605 Homework 4 Due 5pm Monday...

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Phys 605. Homework 4 Due 5pm, Monday, October 6, 2008 Problem 4-1: [10 pts.] A general surface of revolution may be described in cylindrical polar coordinates ( r , φ , z ) by the function r = r ( z ). The function r ( z ) and its derivative dr/dz r 0 ( z ) are given. Use variational calculus to find the equation for the curve that is the shortest path between two points on this surface. (a) Show clearly that the expression for the differential path length on the surface that is the result of displacement dz and is given by ( ds ) 2 = (1 + r 0 2 )( dz ) 2 + r 2 ( ) 2 . (b) Take z as the independent variable and show that the curve, φ ( z ), that is the shortest path between two points on the surface is given by φ ( z ) = φ 0 + k Z z z 0 p r 0 2 ( z 0 ) + 1 r ( z 0 ) p r 2 ( z 0 ) - k 2 dz 0 . (c) Thus, show that the shortest path between two points on the surface of a cylinder of radius a is a section of a helix. (d) [Bonus 5 pts.] Clearly show that the above procedure leads to an infinite number of distinct helices that all pass through the same two points. They can’t all be the shortest path.
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