IntroductionThe study of the n-body problem—the problem of determining the motion of n massive particles interacting through gravitational attraction—is essentially as old as the idea of gravity. Beginning with the publication of the Principia Mathematica in 1687, itsstudents include many of the great mathematical thinkers of the 17th, 18th, and 19th centuries: Newton, Bernoulli, Euler, Lagrange, Poincaré, and Jacobi, to name a few. Because most celestial bodies can be approximated as point particles,1 the n-body problem is also essentially the foundation of celestial astronomy. Twenty-three years after Newton formalized his theory of gravity, Johann Bernoulli proved that for the n = 2 case the orbits of the two bodies always describe a conic section.2For cases where n > 2, however, the motion of the bodies becomes incredibly complex and the system exhibits chaotic properties. In fact, Poincaré’s work on the three-body problem in the late 19th century led him to the discovery of stable and unstable manifolds,homoclinic points, and the beginnings of chaos theory and the theory of dynamical systems. For an overview of the n-body problem’s history, see the appended timeline.State Space and Equations of Motion1 Most of the significant bodies in our solar system are spherical, and the distances between them are relatively great.2 The left figure shows the four possible conic sections. http://www.answers.com/topic/conic-section?cat=technology. The right figure is a trajectory of two bodies simulated in MATLAB.The exact state of any body can be expressed in two vectors—a position vector describing the location of the particle and a momentum vector describing its velocity and mass. Each of these vectors has three spatial dimensions, which yields a state space of 6n. For the 3-body problem, the special case where n = 3, the state space is 18-dimensional and is expressed in a single vector:QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.Equating Newton’s expression for gravity with his second law, F = ma, we can describe the motion of the 3 bodies. Let Pi represent particles with masses mi for i = 1,2,3.Define rij to be the distance between Pi and Pj. Also define qi to be the jth component of the position vector for the ith body. Then the equations of motion where G is the gravitational constant are:QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.Applying Poincaré Maps and Hamiltonian Analysis to the 3-Body ProblemEuler proposed a set of simplifications to the 3-body problem which later came to be known as the restricted 3-body problem. In this subset of the n = 3 case it is assumed that one of the bodies, known as the planetoid, is so much less massive than the other twoobjects that its gravitational force on them is negligible. The orbits of the two massive objects, then, describe a conic section, and the problem is reduced to determining the movement of the planetoid. Also assuming that the z-component of the planetoid's initial velocity is zero, we are left with only four unknown variables: the position and 2momentum of the planetoid in the x and y dimensions. In this way, the original 18-dimensional state space of the 3-body problem is reduced to a system of two second orderdifferential equations, which can be further simplified to four coupled first order equations.QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.At this juncture in his analysis, Poincaré realized that if he imposed a rotating coordinate system and set y = 0, the Hamiltonian for the restricted 3-body problem would simplify to a palatableQuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.where C is a constant of the system. Using this knowledge, he began to examine the states of the system as it pierced the surface of section y = 0. Recording the x-positions and momentum in the x-dimension every time the planetoid intersected the plane y = 0 and the momentum in the y-dimension was greater than zero, he developed a map which transformed the continuous dynamical system into a discrete one. He soon realized that his new map, (later known as a Poincaré map) preserved many properties of the continuous system's periodic and quasiperiodic orbits, despite the fact that its state space was one dimension smaller than the original system's.QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.Libration Points3Lagrange first discovered the Libration or Lagrangian points in 1772, during his studies of the restricted 3-body problem. They are the five locations in space for which the force imparted by the two massive bodies will provide the planetoid with “precisely the centripetal force required to rotate with them.”3 Because of the triangular configuration of the bodies at L4 and L5, the above conclusion holds even if the third object has a non-negligible mass. For this reason we often used the L4 initial conditions for our simulations.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.SimulationsWe used MATLAB to create an environment to simulate the 3-body problem for various initial conditions. In order to view these trajectories, our first task was to break down the 9 second order differential equations into 18 coupled first order equations. We then used MATLAB’s ode45 function to evaluate them at different time intervals. The second challenge was determining the initial conditions that would yield the few known orbits of the 3-body problem. We were able to accomplish this through a reading of published research, intuition, and trial and error.3 Hobbs, David. "Lund Observatory: David Hobbs." Lund Observatory: Space Astronomy. 1 June 2007. Lund University. 5 Dec. 2007 <http://www.astro.lu.se/~david/Gaia1.html>. 4The figure to the left is a contour plot of the gravitational potential felt by the planetoid. The red and blue arrows indicate the slop of the contour plot, red meaning an upward slope and blue meaning downward. L4 and L5, then, are maximums, and L1, L2, and L3 are saddle points.Image courtesy of http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.htmlThe above orbits are two known solutions of the Libration points L4 and L5. In both orbits all three particles have equal masses. For both orbits the magnitudes of the velocities of all three particles are equal. In the orbit to the left, the three particles alwaysdescribe an