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北票市电动工具维修培训班文章内容：

Post-publication activity

Curator: Alain Chenciner

Contributors:

0.14 -

Alessandra Celletti

0.10 -

Nai-Chia Chen

0.10 -

Eugene M. Izhikevich

0.10 -

Serguei A. Mokhov

0.10 -

Nick Orbeck

0.05 -

Richard Montgomery

0.05 -

Jacques Féjoz

0.05 -

Benjamin Bronner

Richard Moeckel

The problem is to determine the possible motions of three point masses \(m_1\ ,\) \(m_2\ ,\) and \(m_3\ ,\) which attract each other according to Newton's law of inverse squares. It started with the perturbative studies of Newton himself on the inequalities of the lunar motion[1]. In the 1740s it was constituted as the search for solutions (or at least approximate solutions) of a system of ordinary differential equations by the works of Euler, Clairaut and d'Alembert (with in particular the explanation by Clairaut of the motion of the lunar apogee). Much developed by Lagrange, Laplace and their followers, the mathematical theory entered a new era at the end of the 19th century with the works of Poincaré and since the 1950s with the development of computers. While the two-body problem is integrable and its solutions completely understood (see [2],[AKN],[Al],[BP]), solutions of the three-body problem may be of an arbitrary complexity and are very far from being completely understood.

Contents

Equations

The following form of the equations of motion, using a force function \(U\) (opposite of potential energy), goes back to Lagrange, who initiated the general study of the problem: if \(\vec r_i\) is the position of body \(i\) in the Euclidean space \(E\equiv\R^p\) (scalar product \(\langle,\rangle\ ,\) norm \(||.||\)), \[m_i{d^2\vec r_i\over dt^2}=\sum_{j\not=i}{m_im_j}\frac{\vec r_j-\vec r_i}{||\vec r_j-\vec r_i||^3}=\frac{\partial U}{\partial\vec r_i},\; i=1,2,3,\;\hbox{where}\; U=\sum_{i<j}\frac{m_im_j}{||\vec r_j-\vec r_i||}\cdot\] Endowing the configuration space \(\hat{\mathcal X}=\{x=(\vec r_1,\vec r_2,\vec r_3)\in E^3,\; \vec r_i\not=\vec r_j\;\hbox{if}\; i\not=j\}\) (or rather its closure \({\mathcal X}\)) with the mass scalar product \[x'\cdot x''=\sum_{i=1}^3{m_i\langle\vec r'_i,\vec r''_i\rangle}\] we can write them \[{d^2 x\over dt^2}=\nabla U(x),\] where the gradient is taken with respect to this scalar product. In the phase space \(T^*\hat{\mathcal X}\equiv\hat{\mathcal X}\times{\mathcal X}\ ,\) that is the set of pairs \((x,y)\) representing the positions and velocities (or momenta) of the three bodies, the equations take the Hamiltonian form (where \(|y|^2=y\cdot y\)): \[{dx\over dt}={\partial H\over\partial y},\quad {dy\over dt}=-{\partial H\over \partial x}, \quad\hbox{where}\quad H(x,y)={1\over 2}|y|^2-U(x).\]

Symmetries, first integrals

The equations are invariant under time translations, Galilean boosts and space isometries. This implies the conservation of

the total energy \(H\ ,\)

the linear momentum \(P=\sum_{i=1}^3m_i\frac{d\vec r_i}{dt}\) (by an appropriate choice of a Galilean frame one can suppose that \(P=0\) and that the center of mass is at the origin),

the angular momentum bivector \(C=\sum_{i=1}^3{m_i\vec r_i\wedge\frac{d\vec r_i}{dt}}\) (identified with a real number if \(p=2\) and with a vector if \(p=3\)).

If the motion takes place on a fixed line, \(C=0\ ;\) on the other hand, if \(C=0\ ,\) the motion takes place in a fixed plane (Dziobek).

The reduction of symmetries was first accomplished by Lagrange in his great 1772 Essai sur le problème des trois corps, where the evolution of mutual distances in the spatial problem is seen to be ruled by a system of order 7.

Finally, the homogeneity of the potential implies a scaling invariance:

if \(x(t)\) is a solution, so is \(x_\lambda(t)=\lambda^{-\frac{2}{3}}x(\lambda t)\) for any \(\lambda>0\ .\) Moreover \(H(x_\lambda(t))=\lambda^{\frac{2}{3}}H(x(t))\) and

\[C(x_\lambda(t))=\lambda^{-\frac{1}{3}}C(x(t))\ ;\] it follows that \(\sqrt{|H|}C\) is invariant under scaling:

\[\sqrt{|H|}C(x_\lambda(t))=\sqrt{|H|}C(x(t))\ .\] Homographic solutions

A configuration \(x=(\vec r_1,\vec r_2,\vec r _3)\) is called a central configuration if it collapses homothetically on its center of mass \(\vec r_G\ ,\) defined by \(\vec r_G={1\over M}\sum{m_i\vec r_i}\) (here \(M=\sum m_i\)) when released without initial velocities (such a motion is called homothetic)

This means that there exists \(\lambda <0\) such that \(\sum_jm_j\frac{\vec r_j-\vec r_i}{||\vec r_j-\vec r_i||^3}=\lambda(\vec r_i-\vec r_G)\) for \(i=1,2,3\ ,\) which is equivalent to \(\sum_jm_j\left(\frac{1}{r_{ij}^3}+\frac{\lambda}{M}\right)(\vec r_j-\vec r_i)=0\ ,\) where \(r_{ij}=||\vec r_j-\vec r_i||\ .\) For non collinear configurations, the two vectors \(\vec r_j-\vec r_i,\, j\ne i\) are linearly independent and so the coefficients in the last sum must vanish. It follows that \(x^0\) must be equilateral, a result first proved by Lagrange in his 1772 memoir. Collinear central configurations of three bodies were already known to Euler in 1763: the ratio of the distances of the midpoint to the extremes is the unique real solution of an equation of the fifth degree whose coefficients depend on the masses.

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