Physics 321

Hour 15The Calculus of Variations

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.

The Life Guard ProblemYou 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?

ExampleSnellsLaw.nb

Finding the Shortest Path between PointsThe path length is

x

y P1

P2

(x1,y1)

(x2,y2)

y=y(x)

𝑑𝑠=√𝑑𝑥2+𝑑𝑦 2¿𝑑𝑥 √1+(𝑑𝑦𝑑𝑥 )2

¿𝑑𝑥 √1+𝑦 ′ 2

𝐿=∫𝑃1

𝑃2

√1+𝑦 ′2𝑑𝑥

Finding the Shortest Path between PointsThe path length is

x

y P1

P2

(x1,y1)

(x2,y2)

y=y(x)

𝐿=∫𝑃1

𝑃2

√1+𝑦 ′2𝑑𝑥

Take another function subject to.

What does that tell us about Y(x)?

Finding the Shortest Path between Points

α

L(α)

α=0

Is y the shortest path?

Finding the Shortest Path between Points

α

L(α)

α=0

Is y the shortest path?

It is – if the same thing works for every function subject to .

Finding the Shortest Path between PointsIf we choose an arbitrary path ,where , then is the shortest path if

for all functions .

A Useful Theorem

Integration by parts give us:

Let

Distance Between Two Points

Distance Between Two Points

Distance Between Two Points

0

Distance Between Two Points

The Action Integral

r s

The Action Integral