Measure of Degenerate Relative Equilibria. I

Palmore, Julian

doi:10.2307/1970964

Cited by 50 publications

(27 citation statements)

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“…Palmore [10]), so the critical set may possibly be infinite. In fact, it is either finite or contains a continuum (cf.…”

Section: Old and Recent Results On Central Configurations (For The Grmentioning

confidence: 99%

On the geometry and behavior of n‐body motions

Straume

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. The kinematic separation of size, shape, and orientation of n-body systems is investigated together with specific issues concerning the dynamics of classical n-body motions. A central topic is the asymptotic behavior of general collisions, extending the early work of Siegel, Wintner, and more recently Saari. In particular, asymptotic formulas for the derivatives of any order of the basic kinematic quantities are included. The kinematic Riemannian metric on the congruence and shape moduli spaces are introduced via O(3)-equivariant geometry. For n = 3, a classical geometrization procedure is explicitly carried out for planary 3-body motions, reducing them to solutions of a rather simple system of geodesic equations in the 3-dimensional congruence space. The paper is largely expository and various known results on classical n-body motions are surveyed in our more geometrical setting.2000 Mathematics Subject Classification. 70F07, 70F10, 70F15, 70F16.1. Introduction. The classical n-body problem studies the motion of n celestial bodies under the mutual influence of gravitational forces. In reality one studies an idealized system consisting of n point masses P i of mass m i in Euclidean 3-space, where the dynamical laws are given by the Newtonian potential function. In the more recent literature, one also finds studies of particle systems whose dynamics are given by various types of potential functions with similar symmetry properties, such as the inverse q force law with q ≠ 2.We start in Section 2 with the kinematics of many particle systems, in the general setting of classical vector algebra. Of particular importance is the decomposition of kinetic energy and the associated kinematic identities and inequalities, including the Sundman inequality which is well known from celestial mechanics. As far as dynamics is concerned, say, with the inverse q force law, 1 < q < 3, a central topic which we will discuss is the asymptotic behavior of motions leading to a general collision (total collapse). This old topic dates back to the pioneering work of Sundman and Siegel on 3-body motions (see [17,18,22,23]) and its partly generalization to n > 3 by Wintner [24], where the time derivatives up to second order of the basic kinematic quantities are investigated. We will extend these results and prove the expected asymptotic formulas for the derivatives of any order. The first part of the proofs appears in Section 2.2; here we establish the case of derivatives up to order 2, largely following Siegel's approach. The proof is completed in Section 6, where we have adapted ideas found in Wintner [24]. We mention that Saari and his collaborators have extended Wintner' ideas and techniques to study (i) collisions involving subsystems of the particles, and (ii) expanding systems and their limiting behavior as t → ∞ (cf. [14]).

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“…Palmore [10]), so the critical set may possibly be infinite. In fact, it is either finite or contains a continuum (cf.…”

Section: Old and Recent Results On Central Configurations (For The Grmentioning

confidence: 99%

On the geometry and behavior of n‐body motions

Straume

2001

International Journal of Mathematics and Mathematical Sciences

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“…By a theorem [3] on the existence of degeneracies of Vm for some (mi) E IR~ and for any n > 4 the minimal classification is not always realized. For any three positive masses ~',n is always a Morse function so that in this case the lower bound is always realized.…”

Section: Forany N >3 and Forany K >0mentioning

confidence: 99%

“…We refer to previous results [3] on the existence of degeneracies of Vm for some (mi)E IR n and for any n > 4. Here there is a one parameter family of relative equilibria classes along which an index change occurs from maximum (index = 2n -4) to saddle (index -2n -6) at a unique ratio of the masses.…”

Section: I N I M a L C L A S S I F I C A T I O N Smentioning

confidence: 99%

Minimally classifying relative equilibria

Palmore

1977

Lett Math Phys

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“…if U is a Morse function. Palmore has shown that this is not always true [17]. The set of masses for which he can give examples of degenerate critical points is of codimension one in the set of all masses.…”

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confidence: 99%

“…The set of masses for which he can give examples of degenerate critical points is of codimension one in the set of all masses. Palmore has also announced that the set of masses which admit degenerate critical points has measure zero in the set of all masses [17]; however, no proof has appeared. Much of the work described above deals with the problem of determining for each fixed m the corresponding relative equilibria.…”

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confidence: 99%

Relative equilibria of the four-body problem

Moeckel

1985

Ergod. Th. Dynam. Sys.

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Abstract. By employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses. Introduction; relative equilibria of the N-body problemThe Newtonian TV-body problem concerns the motion of TV point masses under the influence of their mutual gravitational attraction. We will consider the case when all the particles move in a fixed plane which we take to be the complex plane, C. The position vector zeC N is the vector (z,, ...,z n ) where z,-= x,-+ iy,-£ C is the position of the yth particle. The mass vector m e R n+ is (m l ,..., m N ) where m, is the mass of the jth particle. In complex notation, Newton's laws are:

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Measure of Degenerate Relative Equilibria. I

Cited by 50 publications

References 2 publications

On the geometry and behavior of n‐body motions

On the geometry and behavior of n‐body motions

Minimally classifying relative equilibria

Relative equilibria of the four-body problem

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