Page 90 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Fields theories 77

Since ξ → f (x, u, ξ) is convex, then the function E is always non negative.
Note also that since u (x)= Φ (x, u 0 (x)) then
0
0
0
E (x, u 0 (x) , Φ (x, u 0 (x)) ,u (x)) = 0, ∀x ∈ [a, b] .
0
Now using the definition of exact field we get that, for every u ∈ X,
0
f (x, u (x) , Φ (x, u (x))) + f ξ (x, u (x) , Φ (x, u (x))) (u (x) − Φ (x, u (x)))
d
= f (x, u, Φ) − pΦ + pu = S x + S u u = [S (x, u (x))] .
0
0
dx
Combining these facts we have obtained that
Z b
I (u)= f (x, u (x) ,u (x)) dx
0
a
Z b ½ d ¾
0
= E (x, u (x) , Φ (x, u (x)) ,u (x)) + [S (x, u (x))] dx
a dx
Z
b
d
≥ [S (x, u (x))] dx = S (b, u (b)) − S (a, u (a)) .
a dx
Since E (x, u 0 , Φ (x, u 0 ) ,u )= 0 we have that
0
0
I (u 0 )= S (b, u 0 (b)) − S (a, u 0 (a)) .
Moreover since u 0 (a)= u (a), u 0 (b)= u (b) we deduce that I (u) ≥ I (u 0 ) for
every u ∈ X. Thisachievesthe proofofthe theorem.
The quantity
Z b
d
[S (x, u (x))] dx
dx
a
is called invariant Hilbert integral.

2.6.1 Exercises
N
Exercise 2.6.1 Generalize Theorem2.21tothe case where u :[a, b] → R ,
N ≥ 1.
Exercise 2.6.2 Generalize Hilbert Theorem (Theorem 2.27) to the case where
N
u :[a, b] → R , N ≥ 1.
Exercise 2.6.3 (The present exercise establishes the connection between exact
field and Hamilton-Jacobi equation). Let f = f (x, u, ξ) and = H (x, u, v) be as
in Theorem 2.10.