Page 59 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

46 Classical methods

precise definition). However if (u, ξ) → f (x, u, ξ) is convex for every x ∈ [a, b],
then every solution of (E) is automatically a minimizer of (P).
In Section 2.3 we show that any minimizer u of (P) satisfies a different form
of the Euler-Lagrange equation. Namely for every x ∈ [a, b] the following differ-
ential equation holds:

d
0
0
0
0
[f (x, u (x) , u (x)) − u (x) f ξ (x, u (x) , u (x))] = f x (x, u (x) , u (x)) .
dx
This rewriting of the equation turns out to be particularly useful when f does
not depend explicitly on the variable x. Indeed we then have a first integral of
(E) which is
0
0
f (u (x) , u (x)) − u (x) f ξ (u (x) , u (x)) = constant, ∀x ∈ [a, b] .
0
In Section 2.4, we will present the Hamiltonian formulation of the problem.
Roughly speaking the idea is that the solutions of (E) are also solutions (and
conversely) of
⎧
⎨ u (x)= H v (x, u (x) ,v (x))
0
(H)
⎩
v (x)= −H u (x, u (x) ,v (x))
0
where v (x)= f ξ (x, u (x) ,u (x)) and H is the Legendre transform of f,namely
0
H (x, u, v)= sup {vξ − f (x, u, ξ)} .
ξ∈R
In classical mechanics f is called the Lagrangian and H the Hamiltonian.
In Section 2.5, we will study the relationship between the solutions of (H)
with those of a partial differential equation known as Hamilton-Jacobi equation

(HJ) S x (x, u)+ H (x, u, S u (x, u)) = 0, ∀ (x, u) ∈ [a, b] × R .

Finally, in Section 2.6, we will present the fields theories introduced by Weier-
strass and Hilbert which allow, in certain cases, to decide if a solution of (E) is
a (local or global) minimizer of (P).
We conclude this Introduction with some comments. The methods presented
in this chapter can easily be generalized to vector valued functions of the form
N
u :[a, b] −→ R ,with N> 1,to different boundary conditions, to integral
constraints, or to higher derivatives. These extensions will be considered in the
exercises at the end of each section. However, except Section 2.2, the remaining
part of the chapter does not generalize easily and completely to the multi di-
n
mensional case, u : Ω ⊂ R −→ R,with n> 1; let alone the considerably harder
n
N
case where u : Ω ⊂ R −→ R ,with n, N > 1.