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Physics 325 Spring 2011 Homework 7

# Physics 325 Spring 2011 Homework 7 - Physics 325 Spring...

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Physics 325 Spring 2011 Homework 7A Name: ____________________________ Due: Apr 6, 2011 by 9am (lecture or 325 box) 1. (10 points) A soap bubble placed between two centered hoops will take the shape with the minimum area. Use the calculus of variations to find the curve ( ) y x that defines the surface of revolution. Hint: Assume that the surface is circular everywhere in the x z plane. The problem then reduces to a two-dimensional problem with an element of surface area written as 2 xds , where ds is an element of length in the x y plane. 2. (10 points) In relativity theory, velocities can be represented by points in a certain “rapidity space” in which the distance between two neighboring point is 2 2 2 2 2 1 ds dr r d r where r and are polar coordinates, and we consider just a two-dimensional space. (An expression like this for the distance in non-Euclidean space is often referred to as the metric of the space.) Use the Euler-Lagrange equation to show that the shortest path from the origin to any other point is a straight line.

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Physics 325 Spring 2011 Homework 7 - Physics 325 Spring...

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