Syzygy sequences of the -center problem

2016 ◽  
Vol 38 (2) ◽  
pp. 566-582 ◽  
Author(s):  
KUO-CHANG CHEN ◽  
GUOWEI YU
Keyword(s):  
Periodic Solutions ◽  
Fibonacci Sequence ◽  
Reflection Symmetry ◽  
Center Problem ◽  
Action Functional ◽  
Fixed Length ◽  
Lagrangian Action

The purpose of this paper is to consider the$N$-center problem with collinear centers, to identify its syzygy sequences that can be realized by minimizers of the Lagrangian action functional and to count the number of such syzygy sequences. In particular, we show that the number of such realizable syzygy sequences of length$\ell$greater than or equal to two for the 3-center problem is at least$F_{\ell +2}-2$, where$\{F_{n}\}$is the Fibonacci sequence. Moreover, with fixed length$\ell$, the density of such realizable syzygy sequences of length$\ell$for the$N$-center problem approaches one as$N$increases to infinity. Using reflection symmetry, the minimizers that we found can be extended to periodic solutions.

scholarly journals A Smooth Pseudo-Gradient for the Lagrangian Action Functional

2009 ◽  
Vol 9 (4) ◽  
Author(s):  
Alberto Abbondandolo ◽  
Matthias Schwarzy
Keyword(s):  
Tangent Bundle ◽  
Lagrangian Function ◽  
Quadratic Growth ◽  
Action Functional ◽  
Hilbert Manifold ◽  
Lagrangian Action

AbstractWe study the action functional associated to a smooth Lagrangian function on the tangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H

scholarly journals Hadamard’s variational formula in terms of stress and strain tensors

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Björn Gustafsson ◽  
Ahmed Sebbar
Keyword(s):  
Green Function ◽  
Stress Tensor ◽  
Strain Tensor ◽  
Variational Formula ◽  
Scalar Fields ◽  
Action Functional ◽  
Lagrangian Action ◽  
The Green Function ◽  
Strain Tensors ◽  
Hadamard Variational Formula

AbstractStarting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

Stability of periodic solutions near a collision of eigenvalues of opposite signature

1991 ◽  
Vol 109 (2) ◽  
pp. 375-403 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Periodic Orbit ◽  
Periodic Solutions ◽  
Floquet Theory ◽  
Stability Index ◽  
Surprising Result ◽  
Action Functional ◽  
Stability And Instability ◽  
The Stability ◽  
Pure Imaginary Eigenvalues ◽  
Stability Results

AbstractSome general observations about stability of periodic solutions of Hamiltonian systems are presented as well as stability results for the periodic solutions that exist near a collision of pure imaginary eigenvalues. Let I = ∮ p dq be the action functional for a periodic orbit. The stability theory is based on the surprising result that changes in stability are associated with changes in the sign of dI / dw, where w is the frequency of the periodic orbit. A stability index based on dI / dw is defined and rigorously justified using Floquet theory and complete results for the stability (and instability) of periodic solutions near a collision of pure imaginary eigenvalues of opposite signature (the 1: – 1 resonance) are obtained.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

A Fixed Field/Fixed Length Automated Case Record Using an IBM 1232 Optically Readable Document

1968 ◽  
Vol 07 (03) ◽  
pp. 156-158
Author(s):  
Th. R. Taylor
Keyword(s):  
Clinical Data ◽  
Case Record ◽  
Fixed Field ◽  
Fixed Length

The technique, scope and limitations of a fixed field/fixed length case record utilising the IBM 1232 system is described. The principal problems lie with personnel rather than machinery and with programmes for analysis rather than clinical data.

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