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The key part of the proof is simply the observation that if we add the two inequalities in (b), we get ∆u 2 ≤ H(ue + ∆u) + C(ue + ∆u) − H(ue ) − C(ue ) using the fact that δH(ue ) · ∆u and δC(ue ) · ∆u add up to zero by Step 1. But H and C are constant in time so (∆u)time=t 2 ≤ [H(ue + ∆u) + C(ue + ∆u) − H(ue ) − C(ue )]|time=0 . Now employ the inequalities in (e) to get (∆u)time=t 2 ≤ (C1 + C2 ) (∆u)time=0 α . This estimate bounds the temporal growth of finite perturbations in terms of initial perturbations, which is what is needed for stability.

A) Let ∆u = u − ue denote a finite variation in phase space . (b) Find quadratic functions Q1 and Q2 such that Q1 (∆u) ≤ H(ue + ∆u) − H(ue ) − δH(ue ) · ∆u and Q2 (∆u) ≤ C(ue + ∆u) − C(ue ) − δC(ue ) · ∆u, (c) Require that Q1 (∆u) + Q2 (∆u) > 0 for all ∆u = 0. (d) Introduce the norm ∆u by ∆u 2 = Q1 (∆u) + Q2 (∆u), so ∆u is a measure of the distance from u to ue : d(u, ue ) = ∆u . (e) Require that |H(ue + ∆u) − H(ue )| ≤ C1 ∆u α |C(ue + ∆u) − C(ue )| ≤ C2 ∆u α and for constants α, C1 , C2 > 0, and ∆u sufficiently small.

7) is a solution for every τ > 0 which can be chosen to start arbitrarily close to the origin and which goes to infinity for t → τ . 7-2. 11) is Hamiltonian with p = M q, H(q, p) = 1 1 p · M −1 p + q · V q 2 2 and ∂F ∂K ∂K ∂F ∂F ∂K − i − S ij . 7-3. 11) is p(λ) = det[λ2 M + λS + V ] and that this actually is a polynomial of degree n in λ2 . 7-4. Consider the two-degree of freedom system x ¨ − g y˙ + γ x˙ + αx = 0, y¨ + g x˙ + δ y˙ + βy = 0. 12). (b) For γ = δ = 0 show: (i) it is spectrally stable if α > 0, β > 0; (ii) for αβ < 0, it is spectrally unstable; (iii) for α < 0, β < 0, it is formally unstable (that is, the energy function, which is a quadratic form, is indefinite); and A.