Physics 325 HW 8 Solutions - Homework Assignment#8...

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Homework Assignment #8 Solutions - Physics 325, Fall 2010 1) Show that the shortest distance between two points on a plane is a straight line. Solution : Following the work we did in class, lets define our two points to be in the x-y plane as A = ( x a , y a ) and B = ( x b , y b ). The incremental element of arc length on a path between those two points can be written as: ds = dx 2 + dy 2 The total length of a path between the two points is: S = B A ds = B A dx 1 + ( dy dx ) 2 = B A dx 1 + y 2 We now have an integral equation of the form I = B A f ( y, y , x ) dx that we would like to minimize. In our case, f ( y, y , x ) = 1 + y 2 and we can apply the Euler equation to find the extremum. We note that ∂f ∂y = 0 and ∂f ∂y = y 1 + y 2 So the Euler equation becomes: d dx y 1 + y 2 = 0 We have y 1 + y 2 = a = constant Rearranging, this becomes: y = a 1 - a 2 Performing a separation of variables and integrating, we find: y = a 1 - a 2 x + b, b = constant and we can impose our boundary conditions to find: y = y b - y a x b - x a x + y a x b - y b x a x b - x a
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