1. Physics 321 Hour 15 The Calculus of Variations

2. Bottom Line A simple derivative is useful for finding the value of a variable that maximizes or minimizes a function. The calculus of variations is useful for finding the function that maximizes or minimizes a quantity that depends on the function. You don’t need to reproduce this section, but it is an important tool in physics that you should put forth some real effort to understand.

3. The Life Guard Problem You are a life guard on a beach and see someone needing your help. You want to get there as fast as possible, where do you jump in the water?

4. Example SnellsLaw.nb

5. Finding the Shortest Path between Points x y P1P1 P2P2 (x1,y1)(x1,y1) (x2,y2)(x2,y2) y=y(x)y=y(x)

6. The path length is x y P1P1 P2P2 (x1,y1)(x1,y1) (x2,y2)(x2,y2) y=y(x)y=y(x)

7. Finding the Shortest Path between Points α L(α)L(α) α=0 Is y the shortest path?

8. Finding the Shortest Path between Points α L(α)L(α) α=0 Is y the shortest path?

9. Finding the Shortest Path between Points

10. A Useful Theorem

11. Distance Between Two Points

13. 0

15. The Action Integral