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 problemmay 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}=http://www.scholarpedia.org/article/Three_body_problem//sum_{j/not=i}{m_im_j}/frac{/vec r_j-/vec r_i}{||/vec r_j-/vec r_i||^3}=http://www.scholarpedia.org/article/Three_body_problem//frac{/partial U}{/partial/vec r_i},/; i=1,2,3,/;/hbox{where}/;U=http://www.scholarpedia.org/article/Three_body_problem//sum_{i<j}/frac{m_im_j}{||/vec r_j-/vec r_i||}/cdot/]Endowing the configuration space /(/hat{/mathcal X}=http://www.scholarpedia.org/article/Three_body_problem//{x=(/vec r_1,/vec r_2,/vec r_3)/in E^3,/; /vecr_i/not=http://www.scholarpedia.org/article/Three_body_problem//vec r_j/;/hbox{if}/; i/not=j/}/) (or rather itsclosure /({/mathcal X}/)) with the mass scalar product /[x"/cdot x""=http://www.scholarpedia.org/article/Three_body_problem//sum_{i=1}^3{m_i/langle/vec r"_i,/vecr""_i/rangle}/] we can write them/[{d^2 x/over dt^2}=http://www.scholarpedia.org/article/Three_body_problem//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 thethree 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 integralsThe equations are invariant under time translations, Galilean boosts and space isometries. This implies the conservation of
the total energy /(H/ ,/)
the linear momentum /(P=http://www.scholarpedia.org/article/Three_body_problem//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=http://www.scholarpedia.org/article/Three_body_problem//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/ ,/) themotion 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)=http://www.scholarpedia.org/article/Three_body_problem//lambda^{-/frac{2}{3}}x(/lambda t)/) for any /(/lambda>0/ ./) Moreover /(H(x_/lambda(t))=http://www.scholarpedia.org/article/Three_body_problem//lambda^{/frac{2}{3}}H(x(t))/) and
/[C(x_/lambda(t))=http://www.scholarpedia.org/article/Three_body_problem//lambda^{-/frac{1}{3}}C(x(t))/ ;/] it follows that /(/sqrt{|H|}C/) is invariant under scaling:
/[/sqrt{|H|}C(x_/lambda(t))=http://www.scholarpedia.org/article/Three_body_problem//sqrt{|H|}C(x(t))/ ./]Homographic solutions
