Introduction to Mechanics and Symmetry by Marsden J. Ratiu T.

By Marsden J. Ratiu T.

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Modified Step 3. Assume P is a linear space. (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.

Given by H(q, p) = 1 2 1 2 p + q + pq 2 2 has zero as a double eigenvalue so it is spectrally stable. On the other hand, q(t) = (q0 + p0 )t + q0 and p(t) = −(q0 + p0 )t + p0 is the solution of this system with initial condition (q0 , p0 ), which clearly leaves any neighborhood of the origin no matter how close to it (q0 , p0 ) is. Thus spectral stability need not imply even linear stability. An even simpler example of the same phenomenon is given by the free particle Hamiltonian H(q, p) = 12 p2 .

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. For a survey of this method, additional references and numerous examples, see Holm, Marsden, Ratiu, and Weinstein [1985]. There are some situations (such as the stability of elastic rods) in which the above techniques do not apply. The chief reason is that there may be a lack of sufficiently many Casimir functions to even achieve the first step.

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