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By M.I. Zelikin, S.A. Vakhrameev

The one monograph at the subject, this e-book matters geometric tools within the concept of differential equations with quadratic right-hand facets, heavily on the topic of the calculus of adaptations and optimum keep watch over conception. in response to the author’s lectures, the ebook is addressed to undergraduate and graduate scholars, and clinical researchers.

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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.

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