Page 85 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

72 Classical methods

Therefore a solution of the equation is given by
u
Z
p
S = S (x, u)= S (u)= 2g (s)ds .
0
We next solve
p
0
u (x)= H v (u (x) ,S u (u (x))) = S u (u (x)) = 2g (u (x))
which has a solution given implicitly by
u(x)
Z
ds
= x.
p
u(0) 2g (s)
Setting v (x)= S u (u (x)), we have indeed found a solution of the Hamiltonian
system
⎧
u (x)= H v (u (x) ,v (x)) = v (x)
0
⎨
v (x)= −H u (u (x) ,v (x)) = g (u (x)) .
⎩
0
0
Note also that such a function u solves
u (x)= g (u (x))
00
0
which is the Euler-Lagrange equation associated to the Lagrangian f.
2.5.1 Exercises

N
Exercise 2.5.1 Write the Hamilton-Jacobi equation when u ∈ R , N ≥ 1,and
generalize Theorem 2.19 to this case.
p p 2
Exercise 2.5.2 Let f (x, u, ξ)= f (u, ξ)= g (u) 1+ ξ . Solve the Hamilton-
Jacobi equation and find the stationary points of
Z
b
0
I (u)= f (u (x) ,u (x)) dx .
a
Exercise 2.5.3 Same exercise as the preceding one with f (x, u, ξ)= f (u, ξ)=
2
a (u) ξ /2 where a (u) ≥ a 0 > 0. Compare the result with Exercise 2.2.10.

2.6 Fields theories
As already said we will only give a very brief account on the fields theories and
we refer to the bibliography for more details. These theories are conceptually
important but often difficult to manage for specificexamples.