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EXTREMAL FIELDS AND GLOBAL MINIMALS The proof is based on the fact that the minimal γ and its translates T qp γ : t → x(t + q) − p do not intersect. Proof. First part of the proof. 1) It is enough to show that the sequence x(j)/j for j ∈ Z, converges. 4 with T = 1 and A = |x(j + 1) − x(j)| + 1, follows for t ∈ [j, j + 1], j > 0 |x(t) − x(j)| < c0 (|x(j + 1) − x(j)| + 1) and | x(t) x(j) − | ≤ t j ≤ ≤ ≤ x(t) − x(j) 1 1 + x(j)( − )| t t j |x(j)| (t − j) |x(t) − x(j)| |+ j j t |x(t) − x(j)| |x(j)| 1 + j j t x(j + 1) − x(j) |x(j)| 1 c0 | |+ j j t | if we assume, that α = limj→∞ x(j)/j exists, we have lim | t→∞ x(t) x(j) − |=0.

5. THE HAMILTONIAN FORMULATION 35 The means however, that x˙ = gx (t, x) = Hy (t, x, h(x, y)) defines an extremal field. 1 tells us that instead of considering extremal fields we can look at surfaces which are given as the graph of gx given, where g is a solution of the Hamilton-Jacobi equation gt = −H(t, x, gx ) . The can be generalized to n ≥ 1: We look for g ∈ C 2 (Ω) at the manifold Σ := {(t, x, y) ∈ Ω × Rn | yj = gxj }, where gt + H(t, x, gx ) = 0 . 2 a) Σ is invariant under XH . b) The vector field x˙ = ψ(t, x), with ψ(t, x) = Hy (t, x, gx ) defines an extremal field for F .

The existence of an extremal field is equivalent to stability. 1. GLOBAL EXTREMAL FIELDS 43 are invariant under the flow of XH . The surface Σ is an invariant torus in the phase space T2 ×R2 . The question of the existence of invariant tori is subtle and topic the so called KAM theory. We will come back to it again later the last chapter. Definition: An extremal solution x = x(t) is called global minimal, if R ˙ − F (t, x, x) F (t, x + φ, x˙ + φ) ˙ dt ≥ 0 for all φ ∈ Lipcomp (R) = {φ ∈ Lip(R) of with compact support.