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The Newtonian 3-Body Problem

A system of $N$ point masses, all of which mutually exert attractive forces on each other, will evolve in time according to Newton's second law:

\begin{displaymath}\ddot{{\bf x}_i}(t) = \sum_{i \neq j=1}^N \frac{{\bf F}_{ij}}{m_i} \end{displaymath}

where ${\bf x}_i : {\cal O} \subset {\bf R} \rightarrow {\bf R}^3$ is the position of body $i$ at time $t$, ${\bf F}_{ij}$ is the force exerted on body $i$ by body $j$, and $m_i$ is the mass of body $i$.

The $N$-body problem refers to the study of the dynamical properties of the second-order system of equations (of which there are $3N$). One needs only to specify the force law ${\bf F}$ and the mass of each body to specify the system.

A common force law in the study of the $N$-body problem is gravitation, the force of which is given by Newton' law of universal gravitation:

\begin{displaymath}{\bf F}_{ij} = \frac{Gm_im_j}{r_{ij}^2} \hat{\bf r}_{ij} = \frac{Gm_im_j}{r_{ij}^3} {\bf r}_{ij} \end{displaymath}

where $G$ is Newton's constant of universal gravitation and ${\bf r}_{ij} = {\bf x}_j - {\bf x}_i$ is a vector pointing from the position of body $i$ to that of body $j$. Strong-force laws (those whose potential energies satisfy $U({\bf r}) = o(r^{-1})$) have also been used in studies of the $N$-body problem.

This simulator applies the gravitational force law to the $N=3$-body problem. All three bodies have equal masses $m_1 = m_2 = m_3 = 1$, and Newton's gravitational constant is normalized to $G=1$.


next up previous
Next: The Shape Sphere Up: 3-Body Problem Configuration Simulator Previous: 3-Body Problem Configuration Simulator
Matthew Salomone 2003-07-24