To what extent are unstable the maxima of the potential?
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To what extent are unstable the maxima of the potential? Antonio J. Ureña1 Received: 22 August 2018 / Accepted: 11 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The classical Lagrange–Dirichlet stability theorem states that, for natural mechanical systems, the strict minima of the potential are dynamically stable. Its converse, i.e., the instability of the maxima of the potential, has been proved by several authors including Liapunov (The general problem of stability of motion, 1892), Hagedorn (Arch Ration Mech Anal 42:281–316, 1971) or Taliaferro (Arch Ration Mech Anal 73(2):183–190, 1980), in various degrees of generality. We complement their theorems by presenting an example of a smooth potential on the plane having an isolated maximum and such that the associated dynamical system has a converging sequence of periodic orbits. This implies that the maximum is not unstable in a stronger sense considered by Siegel and Moser. Keywords Converses of the Lagrange–Dirichlet stability theorem · Maxima of the potential · Periodic solutions Mathematics Subject Classification Primary 37C75 · 37J25; Secondary 37J45 · 37J50
1 Introduction Consider the Newtonian system of equations
q̈ = −∇V(q),
(1)
q ∈ ℝd ,
where V ∶ ℝd → ℝ is a given potential having a local maximum at some point q∗ ∈ ℝd . In his 1892 PhD dissertation [6], Liapunov showed that if this maximum is nondegenerate, then the corresponding equilibrium is dynamically unstable. Liapunov’s instability theorem was extended by Hagedorn [4], who used variational methods to prove the instability of all isolated maxima of the potential, and Taliaferro [11], who removed the isolatedness assumption from Hagedorn’s theorem. Results of this type have been named as converses
Partially supported by Spanish MICINN Grant with FEDER funds MTM2014- 5223. * Antonio J. Ureña [email protected] 1
Departamento de Matematica Aplicada, Universidad de Granada, 18071 Granada, Spain
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of the Lagrange–Dirichlet stability theorem, and the associated literature is very ample, see, for instance, [5, 7] and the references therein. In the above-mentioned papers, the word instability is understood as the logical negation of Liapunov stability. In particular, for equations of the form (1) and more generally in the Hamiltonian framework, it means both past and future instability; namely, the two concepts are equivalent. See [9, p. 150] and [8, p. 114–115]. (In this latter reference, the statement is made in dimension 2, but the proof works for all dimensions.) However, a stronger notion of instability was considered by Siegel and Moser in [10, §25 ]. According to their definition, the equilibrium q∗ ∈ ℝd is unstable if there is a neighborhood N of (q∗ , 0) in the phase space such that every globally-defined solution ̇ q )) ∉ N for some tq ∈ ℝ . In other words, q ∶ ℝ → ℝd of (1), q(t) ≢ q∗ , satisfies (q(tq ), q(t {(q∗ , 0)} is the maximal subset of N which is invaria
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