Relative equilibria in the 3-dimensional curved n-body problem

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Stability of motion is one of the oldest problems in mathematical physics, with important contributions dating back to the eighteenth century. In this article we discuss stability of equilibria of Hamiltonian flows. Stability of periodic orbits , by Floquet theory , and of quasiperiodic orbits, by the powerful Kolmogorov-Arnold-Moser Theory , are treated elsewhere in this encyclopedia. An historical account of progress in stability analysis for Hamiltonian systems reads like a Who's Who in Mathematics and Physics.

For the computation of linear stability, and in particular spectral stability, important early contributions, dealing with properties of the zeros of polynomials, were made by Cauchy, Lagrange, Fourier, and Hermite, culminating in the great theorems of Sturm, which provided necessary and sufficient conditions that a given real polynomial have all its zeros in a specified real interval.

The relevance of these seminal studies in the theory of equations to orbital stability was recognized and developed further by Airy and Clifford, and subsequently carried to completion by Maxwell and Routh.

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Maxwell, in his celebrated Adams Prize Essay of on the stability of the rings of Saturn obtained linear stability criteria in two and three degrees of freedom. Routh devised two different methods, the first based on Clifford's idea of forming the polynomial whose zeros are the twofold sums of the zeros of the characteristic polynomial, the second based on the Cauchy Index Theorem , coupled with Sturm's Theorem. Routh's investigations on the stability of governors evolved into the field now known as control theory.

His second method is still widely used for dissipative systems, together with the equivalent method of Hurwitz. Stability considerations for equilibria of Hamiltonian flows differ markedly from those encountered in dissipative systems , where the existence of attractors and repellors results in asymptotic stability. In contrast, stability of Hamiltonian systems is at best neutral rather than hyperbolic.

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For example, Routh-Hurwitz criterion cannot be used since it explicitly assumes that no eigenvalues lie on the imaginary axis. Other techniques, such as the powerful Lyapunov function method are widely used in dissipative systems and find occasional use in Hamiltonian systems Rumyantsev and Sosnitskii, The familiar Lagrangian method Goldstein, is applicable only when the kinetic energy is positive definite , excluding such important examples as the restricted three-body problem Broucke, Recently a new method was devised which overcomes all the above problems, yielding explicit linear stability bounds for Hamiltonian equilibria in arbitrary dimension.

The method is based on two innovations: i introduction of a "reduced characteristic equation" of degree half that of the usual characteristic equation, ii use of Sturm's theorem to relate the eigenvalues of the reduced characteristic equation to spectral stability. Significant progress has also been made on converse stability , where conditions are obtained for instability of non-natural systems.

See Rumyantsev and Sosnitskii for a recent review. There are many notions of stability of equilibria, but only three of these are of primary importance for Hamiltonian systems.

Note however, that spectral stability does not imply linear stability. Nevertheless, it can be shown that the boundaries of linear and spectral stability coincide.

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The precise relation is that an equilibrium is linearly stable if and only if it is spectrally stable and all its Jordan blocks are one-dimensional Abraham and Marsden, ; Arnold We shall return to this example below. Other types of stability encountered in Hamiltonian flows are orbital stability , which describes the divergence of two neighboring orbits, regarded as point sets, and structural stability , which describes the sensitivity or insensitivity of the qualitative features of a flow to changes in parameters.

See Stability for details. A general method for analyzing stability in Hamiltonian systems is the energy momentum method which uses a combination of energy and momentum to provide a Lyapunov function. A special case of this is the energy Casimir method that uses Casimir functions, functions that commute with every other function under the Poisson bracket.

These ideas were developed by Arnold who originally applied them to fluids and Marsden and his collaborators see Marsden and Ratiu for references. The role that dissipation plays in either stabilizing or destabilizing equilibria is discussed in Bloch, Krishnaprasad, Marsden and Ratiu , See also references therein including to the work of Chetaev.

Hence, eigenvalues occur in the following configurations:. It follows that an equilibrium can loose spectral and therefore linear stability in only two ways:. Of course, combinations of these configurations or eigenvalues of multiplicity greater than one are possible.

The essential fact is that every equilibrium on the boundary of spectral stability must have a multiple eigenvalue which is either zero or nonzero imaginary. However, not every equilibrium satisfying one of these conditions is actually on the boundary of spectral stability! This is a consequence of the existence of additional invariants for Hamiltonian flows which may prevent an eigenvalue pair from leaving the imaginary axis.

These are the Krein Signatures , which we now describe. Which eventually actually occurs depends on a special invariant peculiar to Hamiltonian flows, the Krein signature. For more information on the mixed signature case see Mackay, Volume 10 Issue 4 Nov , pp.

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Chapter 1. Introduction

Our goal is to define and study relative equilibria, which are orbits whose non- periodic orbits ever found in a 3-dimensional n-body problem. The author considers the 3-dimensional gravitational n-body problem, n≥2, in spaces of constant Gaussian curvature κ≠0, i.e. on spheres S3κ.

Next Article. Abstract In this paper, we study possible shapes of 4-body non-collinear relative equilibria for any positive masses, and give estimates on their geometric quantities. About the article Received : Published Online : Published in Print : Citing Articles Here you can find all Crossref-listed publications in which this article is cited. Classifying four-body convex central configurations.

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