Page 17 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

4 Introduction

Example: Newton problem. We seek for a surface of revolution moving
in a fluid with least resistance. The problem can be mathematically formulated
as follows. Let n = N =1,
3
ξ
f (x, u, ξ)= f (u, ξ)= 2πu 2
1+ ξ
and
( )
Z b
0
(P) inf I (u)= f (u (x) ,u (x)) dx : u (a)= α, u (b)= β = m.
a
We will not treat this problem in the present book and we refer to Buttazzo-
Kawohl [18] for a review.

Example: Brachistochrone.The aim is to find the shortest path between
two points that follows a point mass moving under the influence of gravity. We
place the initial point at the origin and the end one at (b, −β),with b, β > 0.
We let the gravity act downwards along the y-axis and we represent any point
along the path by (x, −u (x)), 0 ≤ x ≤ b.
In terms of our notation we have that n = N =1 and the function, under
p 2 √
consideration, is f (x, u, ξ)= f (u, ξ)= 1+ ξ / 2gu and
( )
b
Z
0
(P) inf I (u)= f (u (x) ,u (x)) dx : u ∈ X = m
0
© ª
1
where X = u ∈ C ([0,b]) : u (0) = 0,u (b)= β and u (x) > 0, ∀x ∈ (0,b] .The
shortest path turns out to be a cycloid.
Example: Minimal surface of revolution. We have to determine among
all surfaces of revolution of the form

v (x, y)= (x, u (x)cos y, u (x)sin y)

with fixed end points u (a)= α, u (b)= β one with minimal area. We still have
n = N =1,
q
2
f (x, u, ξ)= f (u, ξ)= 2πu 1+ ξ
and
( )
b
Z
(P) inf I (u)= f (u (x) ,u (x)) dx : u (a)= α, u (b)= β, u > 0 = m.
0
a