Page 82 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Hamilton-Jacobi equation 69

2.5 Hamilton-Jacobi equation

We now discuss the connection between finding stationary points of the func-
tionals I and J considered in the preceding sections and solving a first order
partial differential equation known as Hamilton-Jacobi equation.This equation
also plays an important role in the fields theories developed in the next section
(cf. Exercise 2.6.3).
Let us start with the main theorem.

1
Theorem 2.17 Let H ∈ C ([a, b] × R × R), H = H (x, u, v).Assume that
2
there exists S ∈ C ([a, b] × R), S = S (x, u), a solution of the Hamilton-Jacobi
equation
S x + H (x, u, S u )= 0, ∀ (x, u) ∈ [a, b] × R , (2.14)
1
where S x = ∂S/∂x and S u = ∂S/∂u. Assume also that there exists u ∈ C ([a, b])
asolutionof
u (x)= H v (x, u (x) ,S u (x, u (x))) , ∀x ∈ [a, b] . (2.15)
0
Setting
v (x)= S u (x, u (x)) (2.16)
1
1
then (u, v) ∈ C ([a, b]) × C ([a, b]) is a solution of
⎧
u (x)= H v (x, u (x) ,v (x))
0
⎨
(2.17)
⎩ 0
v (x)= −H u (x, u (x) ,v (x)) .
2
Moreover if there is a one parameter family S = S (x, u, α), S ∈ C ([a, b] × R × R),
solving (2.14) for every (x, u, α) ∈ [a, b] × R × R, then any solution of (2.15)
satisfies
d
[S α (x, u (x) ,α)] = 0, ∀ (x, α) ∈ [a, b] × R ,
dx
where S α = ∂S/∂α.
Remark 2.18 (i) If the Hamiltonian does not depend explicitly on x then every
solution S (u, α) of
∗
∗
H (u, S )= α, ∀ (u, α) ∈ R × R (2.18)
u
leads immediately to a solution of (2.14), setting

∗
S (x, u, α)= S (u, α) − αx .
(ii) It is, in general, a difficult task to solve (2.14) and an extensive bibliog-
raphy on the subject exists, cf. Evans [43], Lions [69].