Convex Central Configurations of the n-Body Problem Which are not Strictly Convex

It is well-known that if a planar central configuration for the Newtonian 4-body problem is convex, then it must be strictly convex. In some literature, same conclusion was believed to hold for the case of five or even more bodies but rigorous treatments are absent. With the help of some numerical calculations, in this paper we provide concrete examples of central configurations which are convex but not strictly convex. Our examples include planar central configurations with five bodies and spatial central configurations with seven bodies.     

Convex Four-Body Central Configurations with Some Equal Masses

We prove that there is an unique convex noncollinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such a central configuration possesses a symmetry line and it is a kite-shaped quadrilateral. We also show that there is exactly one convex noncollinear central configuration when the opposite masses are equal. Such a central configuration also possesses a symmetry line and it is a rhombus.     

The Existence Problem for Steiner Networks in Strictly Convex Domains

We consider the existence problem for ‘Steiner networks’ (trivalent graphs with 2π/3 angles at each junction) in strictly convex domains, with ‘Neumann’ boundary conditions. For each of the three possible combinatorial possibilities, sufficient conditions on the domain are derived for existence. In addition, in each case explicit examples of nonexistence are given.     

Four-Body Central Configurations¶with some Equal Masses

We prove firstly that any convex non-collinear central configuration of the planar 4-body problem with equal opposite masses β >α > 0, such that the diagonal corresponding to the mass α is not shorter than that corresponding to the mass β, must possess a symmetry and therefore must be a kite. Then by a recent result of Bernat, Llibre and Perez-Chavela, this kite is actually a rhombus. Secondly we prove that a convex non-collinear planar 4-body central configuration with three equal masses must be a kite too. We also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center, or an isosceles triangle with the fourth mass on the symmetry axis.     

Morse Theory and Central Configurations in the Spatial N-body Problem

In the spirit of Palmore and Pacella, Morse Theory is used to obtain a lower bound for the number of central configurations in the spatial N-body problem. The homology of the configuration ellipsoid with the collision and collinear manifolds removed and the SO(3) symmetry quotiented out is calculated. As intermediate steps, homology calculations are carried out for several additional manifolds naturally arising in the N-body problem.     

Finiteness and Bifurcations of some Symmetrical Classes of Central Configurations

We consider here the classical question of finiteness (see Smale [13] and Wintner [15]) – given n point masses, is the corresponding number of central configurations finite? We prove finiteness for a particular family of d-dimensional symmetrical configurations of d+2 point masses. Also, we study the bifurcations of these configurations and provide the exact number of central configurations when d=2, 3. All our results stem from the application of a new method for studying symmetrical classes of central configurations, which is presented in this work. (Accepted September 23, 2002) Published online February 14, 2003 Communicated by P. Rabinowitz     

Collinear Central Configurations and Singular Surfaces in the Mass Space

For a given m=(m₁,...,m_n)(R⁺)ⁿ, let p and q(R³)ⁿ be two central configurations for m. Then we call p and q equivalent and write pq if they differ by an SO(3) rotation followed by a scalar multiplication as well as by a permutation of bodies. Denote by L(n,m) the set of equivalent classes of n-body collinear central configurations in R³ for any given mass vector m=(m₁,...,m_n)(R⁺)ⁿ. The main discovery in this paper is the existence of a union H₃ of three non-empty algebraic surfaces in the mass half space (m₁,m₂–m₁,m₃–m₂)R⁺×R² besides the planes generated by equal masses, which decreases the number of collinear central configurations. The union H₃ in R⁺×R ² is explicitly constructed by three 6-degree homogeneous polynomials in three variables such that, for any mass vector m=(m₁,m₂,m₃)(R⁺)³, ^# L(3,m)=3, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if two of m₁, m₂, and m₃ are equal but not the third, ^# L(3,m)=1, if m₁=m₂=m₃. We give also a sharp upper bound on ^#L(n,m) for any positive mass vector m(R⁺)ⁿ.     

On Co-Circular Central Configurations in the Four and Five Body-Problems for Homogeneous Force Law

We study the central configurations (cc for short) for four masses arranged on a common circle (called co-circular cc) in two different situations, namely with no mass inside and later adding a fifth mass at the center of the circle. In the former, we focus the kite shape configurations by proving the existence of a one-parameter family of cc which goes from the kite containing an equilateral triangle up to the square shape. After, by putting a fifth mass at the center, we feature the planar cc of five bodies as a tensor of corange two see, “Albouy and Chenciner (Invent Math 131:151–184, 1998)” and we prove that cc is stacked see, “Hampton (Nonlinearity 18:2299–2304, 2005b)” in a such way that the center of mass of the four bodies should be the center of the circle. We emphasize that our approach includes not only the Newtonian force law, but the homogeneous ones with exponent $a\le -1$ a ≤ ? 1 .     

Deformation of a Convex Hydrogel Shell by Parallel Plate and Central Compression

To characterize the materials parameters and deformation of a convex shell of axial symmetry, a hydrogel contact lens is mechanically deformed by two loading configurations: (a) compression between two parallel plates and (b) central load applied by a shaft with a spherical tip. A universal testing machine with nano-Newton and submicron resolutions is used to measure the applied force, F, as a function of vertical displacement of the plate/shaft, w ₀, while a homemade laser aided topography system records the in-situ deformed shell profile and the contact radius or central dimple, a. A nonlinear shell theory and an iterative finite difference method are used to account for the large elastic deformation, the central buckling for the central load compression, and the interrelationship between the measureable quantities (F, w ₀, a).     

Strictly conjugate stress and strain tensors

a subclass of strictly conjugate tensors, namely, the tensors that satisfy the requirement for transformation by the same law upon rigid motion of the neighborhood of a material particle, is separated into the class of work-conjugate stress and strain tensors. The advantage of the use of strictly conjugate stress and strain tensors in formulating the variational principles for bodies from a hyperelastic material is shown. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 149–154, May–June, 2000.     

Cellular and Tulip Flame Configurations

Flame propagation in a plane channel with the formation of tulip and cellular configurations of the combustion front is simulated. The near-flame flow structure and the thermal flow structure are determined. An analogy is found between the tulip configuration and flame inflections at cell interfaces.     

Memory decay rates of viscoelastic solids: not too slow, but not too fast either

Fading memory is a distinguishing characteristic of viscoelastic solids. Its assessment is often achieved by measuring the stress due to harmonic strain histories at different frequencies: from the experimental point of view, the storage and loss moduli are, hence, introduced. On the other side, the mathematical modeling of viscoelastic materials is usually based on the consideration of a kernel function whose decay rate is sufficiently fast. For several different solid materials, we have collated experimental evidence showing an high sensitivity to frequency variations of both the storage and loss moduli. By contrast, we prove that the commonly employed viscoelastic kernels (Prony series, continuous kernel, etc.) cannot reproduce this experimental behavior, as the resulting frequency sensitivity of the storage modulus is always zero when assessed at low frequency. This leads to identification problems of the material parameters which are strongly ill conditioned. However, we identify the specific kernel property which is responsible for this misbehavior: the long-term material memory must not decrease too fast. Some viscoelastic kernels, showing the correct memory??s rate of decay, are introduced and their improved ability to match the experimental data analyzed.     

Elastic Materials with Two Stress-Free Configurations

It is shown that, if an elastic material exhibits two stress-free configurations, it is dynamically unstable in a definite sense. This revised version was published online in June 2006 with corrections to the Cover Date.     

Non-Uniqueness of Minimizers for Strictly Polyconvex Functionals

In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557–611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals,

Dziobek Equilibrium Configurations on a Sphere

We investigate the n-body problem on a sphere with a general interaction potential that depends on the mutual distances. We focus on the equilibrium configurations, especially on the Dziobek equilibrium configurations, which is an analogy of Dziobek central configurations of the classical n-body problem. We obtain a criterion and then reduce it to two sets of equations. Then we apply these equations to the curved n-body problem in \({\mathbb {S}}^3\). In the end, we find the derivative of the Cayley-Menger determinant.

     

Equilibrium Configurations for a Floating Drop

If a drop of fluid of density ₁ rests on the surface of a fluid of density ₂ below a fluid of density ₀, ₀ < ₁ < ₂, the surface of the drop is made up of a sessile drop and an inverted sessile drop which match an external capillary surface. Solutions of this problem are constructed by matching solutions of the axisymmetric capillary surface equation. For general values of the surface tensions at the common boundaries of the three fluids the surfaces need not be graphs and the profiles of these axisymmetric surfaces are parametrized by their tangent angles. The solutions are obtained by finding the value of the tangent angle for which the three surfaces match. In addition the asymptotic form of the solution is found for small drops.     

Face-Centered Cubic Crystallization of Atomistic Configurations

Archive for Rational Mechanics and Analysis - We address the question of whether three-dimensional crystals are minimizers of classical many-body energies. This problem is of conceptual relevance...     

Convex polynomial yield functions

It is shown that some of the recently proposed orthotropic yield functions obtained through the linear transformation method are homogeneous polynomials. This simple observation has the potential to simplify considerably their implementation into finite element codes. It also leads to a general method for designing convex polynomial yield functions with powerful modeling capabilities. Convex parameterizations are given for the fourth, sixth and eighth order plane stress orthotropic homogeneous polynomials. Illustrations are shown for the modeling of biaxial and directional yielding properties of steel and aluminum alloy sheets. The parametrization method can be easily extended to general, 3D stress states.     

Equilibrium Configurations of an Inflated Cylindrical Membrane

We study the inflation of a cylindrical elastic membrane under conditions where more than one equilibrium radius is possible at the same pressure. In particular, we investigate the existence of solutions connecting two sections of different radii. The existence of such solutions is found to be linked to a Maxwell equal area rule.