Page 91 - INTRODUCTION TO THE CALCULUS OF VARIATIONS
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78 Classical methods
(i) Show that if there exists an exact field Φ covering D,then
S x + H (x, u, S u )= 0, ∀ (x, u) ∈ D
where
S u (x, u)= f ξ (x, u, Φ (x, u))
S x (x, u)= f (x, u, Φ (x, u)) − S u (x, u) Φ (x, u) .
(ii) Conversely if the Hamilton-Jacobi equation has a solution for every
(x, u) ∈ D,prove that
Φ (x, u)= H v (x, u, S u (x, u))
is an exact field for f covering D.
