Phys 325 Spring 2011 Lecture 17

# Phys 325 Spring 2011 Lecture 17 - Physics 325 Lecture 17...

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Physics 325 Lecture 17 Calculus of Variations (cont.) In the previous lecture, we found that the time it takes to go from A (the origin) to B is given by the integral over the path   0 1 22 2 0 1 2 2 0 2 1 2 B B B y ds t v dx dy gx dy dx dx gx    Finding the quickest path from A to B is now equivalent to finding the function y(x) that minimizes t . We denote the path with the shortest time () s yx , and let x be the difference between some trial path and the shortest-time path       s y x y x x  We must have     0 AB xx   since all paths begin at A and end at B, by definition. For example: y s (x) =y s -y x x y y(x)

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The time it takes to travel along the path y ( x ) is     1 2 2 0 1 2 B x s d y x x dx t dx gx       (17.1) since       s y x y x x  . We consider the family of functions       , s y x y x x   where   1 corresponds to the example above. Note that 0 s y    because y s and only depend on x . In addition x does not explicitly depend on , so 0 x Now,     , yx x We define     ,, d y x y x dx then       , s y x y x x and       , y x x d x dx  The integral in Equation (17.1) now becomes       1 2 2 0 1 2 B x s d y x x dx t dx gx 
The time is now a function of the parameter , and has an extremum (minimum) at 0 if   0 0 t  Since our integrand contains y s ( x ) , which we have already defined to be the shortest

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## This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys 325 Spring 2011 Lecture 17 - Physics 325 Lecture 17...

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