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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 7

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online