Page 73 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

60 Classical methods

2
Proof 1. Observe firstthatfor any u ∈ C ([a, b]) we have, by straight
differentiation,
d
[f (x, u, u ) − u f ξ (x, u, u )]
0
0
0
dx
∙ ¸
d
0 0 0 0
= f x (x, u, u )+ u f u (x, u, u ) − [f ξ (x, u, u )] .
dx
By Theorem 2.1 we know that any solution u of (P) satisfies the Euler-Lagrange
equation
d
0
0
[f ξ (x, u (x) ,u (x))] = f u (x, u (x) ,u (x))
dx
hence combining the two identities we have the result.
Proof 2. We will use a technique known as variations of the independent
variables and that we will encounter again in Chapter 5; the classical deriva-
tion of Euler-Lagrange equation can be seen as a technique of variations of the
dependent variables.
−1
Let ∈ R, ϕ ∈ C ∞ (a, b), λ =(2 kϕ k L ∞) and
0
0
ξ (x, )= x + λϕ (x)= y.
Observe that for | | ≤ 1,then ξ (., ):[a, b] → [a, b] is a diffeomorphism with
ξ (a, )= a, ξ (b, )= b and ξ (x, ) > 0.Let η (., ):[a, b] → [a, b] be its inverse,
x
i.e.
ξ (η (y, ) , )= y.
Since ξ (η (y, ) , ) η (y, )=1 and ξ (η (y, ) , ) η (y, )+ ξ (η (y, ) , )=0,
x y x
we find (O (t) stands for a function f so that |f (t) /t| is bounded in a neighbor-
hood of t =0)
¡ ¢
2
0
η (y, )= 1 − λϕ (y)+ O
y
η (y, )= −λϕ (y)+ O ( ) .

Set for u a minimizer of (P)

u (x)= u (ξ (x, )) .
Note that, performing also a change of variables y = ξ (x, ),
Z b ¡ ¢

0
I (u )= f x, u (x) , (u ) (x) dx
a
Z b
0
= f (x, u (ξ (x, )) ,u (ξ (x, )) ξ (x, )) dx
x
a
Z b
¡ ¢
= f η (y, ) ,u (y) ,u (y) /η (y, ) η (y, ) dy .
0
y y
a