Rose solutions with three petals for planar 4-body problems

For planar Newtonian 4-body problems with equal masses, we use variational methods to prove the existence of a non-collision periodic choreography solution such that all bodies move on a rose-type curve with three petals.     

Numerical Computation of Figure-eight Solution for 3-body Problems

The main goal of this paper is to compute the Figure-eight solutions for the planar Newtonian 3-body problem with equal masses by finding the critical points of the functional associated with the motion equations of 3-body in plane R2. The algorithm adopted here is the steepest descent method, which is simple but very valid for our problem.     

三体问题8字形解的数值计算

The main goal of this paper is to compute the Figure-eight solutions for the planar Newtonian 3-body problem with equal masses by finding the critical points of the functional associated with the motion equations of 3-body in plane R2. The algorithm adopted here is the steepest descent method, which is simple but very valid for our problem.     

Homographic solutions of the curved 3-body problem

In the 2-dimensional curved 3-body problem, we prove the existence of Lagrangian and Eulerian homographic orbits, and provide their complete classification in the case of equal masses. We also show that the only non-homothetic hyperbolic Eulerian solutions are the hyperbolic Eulerian relative equilibria, a result that proves their instability.     

Geometric characterizations for variational minimizing solutions of charged 3-body problems

We study the charged 3-body problem with the potential function being (-α)-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the π2-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.     

一类非牛顿流体流动问题的变分原理和广义变分原理

本文将钱伟长教授[1]的不可压缩粘性流的最大功率消耗原理推广到一类特殊的非牛顿流体─-广义牛顿流体的流动问题,并采用识别的拉氏乘子法来解除变分约束条件,导出其广义变分原理。     

关于4-体问题中心构型的一点研究

N-体问题的中心构型是应用数学领域广泛研究的问题．关于N-体问题的中心构型已有许多研究结果．但是对于n≥4,其中心构型解的计算是比较困难的．作者运用Wu-Ritt零点分解方法和子结式序列研究了一般的平面4体中心构型问题,给出了这类4体中心构型问题的解析解,从而证明了一类平面牛顿4-体问题的中心构型个数是有限的．     

Variational minimizing parabolic and hyperbolic orbits for the restricted 3-body problems

Using variational minimizing methods, we prove the existence of the odd symmetric parabolic or hyperbolic orbit for the restricted 3-body problems with weak forces.     

Homographic Solutions of the N -body Generalized Lennard-Jones System

In this paper,we obtain the existence of non-planar circular homographic solutions and non-circular homographic solutions of the(2+N)-and(3+N)-body problems of the Lennard-Jones system.These results show the essential difference between the Lennard-Jones potential and the Newton's potential of universal gravitation.     

Some counterexamples to a generalized Saari's conjecture

For the Newtonian -body problem, Saari's conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.

     

On the minimizing properties of the 8-shaped solution of the 3-body problem

The starting point of our study was the recent results of Alain Chenciner and Richard Montgomery concerning the discovery of the 8-shaped orbit of the planar 3-body problem with equal masses (in the sequel, we will call it just “the Eight,” [4]). Geometrically this orbit consists of 12 pieces such that each of them minimizes the Lagrangian action between Euler and isosceles configurations of the bodies. Our aim was to understand whether the larger pieces of the Eight are still solutions of some minimizing problem. The paper presents some preliminary analytical and numerical results on the minimizing properties of the Eight. Using the technique of the so-called Jacobi curves, we numerically show that the solution of Chenciner and Montgomery is no longer optimal after 0.52 of its period. Moreover, we find a better solution for the fixed endpoint problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

Variational minimizing parabolic and hyperbolic orbits for the restricted 3-body problems Dedicated to my Teacher Professor Yang Wannian on the Occasion of his 75th Birthday

Using variational minimizing methods,we prove the existence of the odd symmetric parabolic or hyperbolic orbit for the restricted 3-body problems with weak forces.     

General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.     

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.     

Fractional Variational Approach for Dissipative Mechanical Systems

More recently, a variational approach has been proposed by Lin and Wang for damping motion with a Lagrangian holding the energy term dissipated by a friction force. However, the modified Euler-Lagrange equation obtained within their for- malism leads to an incorrect Newtonian equation of motion due to the nonlocality of the Lagrangian. In this communication, we generalize this approach based on the fractional actionlike variational approach and we show that under some simple restric- tions connected to the fractional parameters introduced in the fractional formalism, this problem may be solved.     

带约束广义变分不等式问题的一般分解算法的收敛性分析

本文考虑如下带约束广义变分不等式问题的增广Lagrangian对偶理论:寻找一点x∈Γ使满足,〈F(x),y-x〉 φ(x,y)-φ(x,x)≥0,y∈Γ,其中,Γ={y∈X｜Θ(y)∈-C}.对于求解这类一般变分不等式问题的基于增广Lagrangian对偶理论分解算法,本文给出了算法的收敛性分析.     

A stochastic variational approach to viscous Burgers equations

We consider viscous Burgers equations in one dimension of space and derive their solutions from stochastic variational principles on the corresponding group of homeomorphisms. The metrics considered on this group are L ^p metrics. The velocity corresponds to the drift of some stochastic Lagrangian processes. Existence of minima is proved in some cases by direct methods. We also give a representation of the solutions of viscous Burgers equations in terms of stochastic forward-backward systems.     

Optimality in Infinite-Horizon Variational Problems under Sign Conditions

We consider infinite-horizon variational problems on several spaces of curves. We establish relations between these problems and the properties of their solutions. Notably, we exhibit situations where optimality in a given space of curves implies optimality in a bigger space of curves. We work with a domain of definition of the Lagrangian which has a very general form and we provide assumptions to ensure a satisfactory theory of the necessary conditions of optimality. We apply these results to actualized Lagrangians.     

The <Emphasis Type="Italic">n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.     

Global symplectic regularization for some restricted 3-body problems on

We develop a regularization for binary collisions in some restricted 3-body problems moving in planar one-dimensional spaces with constant curvature. The main characteristic of the regularization is that it preserves the Hamiltonian structure of the equations and it regularizes all the binary collisions with just one transformation. We apply this global symplectic regularization to the 2-body problem on the unit circle and we show the global dynamics. Also, we tackle the restricted 3-body problem with one fixed center in the unit circle and we give the global dynamics for the case when it has two fixed centers.