Page 75 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

62 Classical methods

We have seen in the preceding sections that a minimizer of I,ifitissufficiently
regular, is also a solution of (E). The aim of this section is to show that, in
certain cases, solving (E) is equivalent to finding stationary points of a different
functional, namely
b
Z
0
J (u, v)= [u (x) v (x) − H (x, u (x) ,v (x))] dx
a
whose Euler-Lagrange equations are
⎧
u (x)= H v (x, u (x) ,v (x))
0
⎨
(H)
v (x)= −H u (x, u (x) ,v (x)) .
⎩
0
The function H is called the Hamiltonian and it is defined as the Legendre
transform of f,which is defined as
H (x, u, v)= sup {vξ − f (x, u, ξ)} .
ξ∈R

Sometimes the system (H) is called the canonical form of the Euler-Lagrange
equation.
We start our analysis with a lemma.

2
Lemma 2.8 Let f ∈ C ([a, b] × R × R), f = f (x, u, ξ), such that
f ξξ (x, u, ξ) > 0, for every (x, u, ξ) ∈ [a, b] × R × R (2.4)

f (x, u, ξ)
lim =+∞, for every (x, u) ∈ [a, b] × R. (2.5)
|ξ|→∞ |ξ|
Let
H (x, u, v)= sup {vξ − f (x, u, ξ)} . (2.6)
ξ∈R
2
Then H ∈ C ([a, b] × R × R) and
H x (x, u, v)= −f x (x, u, H v (x, u, v)) (2.7)

H u (x, u, v)= −f u (x, u, H v (x, u, v)) (2.8)
H (x, u, v)= vH v (x, u, v) − f (x, u, H v (x, u, v)) (2.9)

v = f ξ (x, u, ξ) ⇔ ξ = H v (x, u, v) . (2.10)