Brief history and Brief history and development of thedevelopment of the

n n body problembody problem

Jeff Kaplan

Lance Min

Tom Yohannan

Newton’s laws and deterministic Newton’s laws and deterministic universeuniverse

Newtonian mechanics Newtonian mechanics argues that the physical argues that the physical matter of the universe matter of the universe operates according to operates according to Newton’s lawsNewton’s laws

With the atomic nature With the atomic nature of the matter, predicting of the matter, predicting every event in the every event in the universe should be as universe should be as possible as predicting possible as predicting how the billiard balls with how the billiard balls with move on a table, only move on a table, only with more number of with more number of balls.balls.

Deterministic universe and Deterministic universe and the n-body problemthe n-body problem

Given the initial positions and Given the initial positions and velocities of N number of objects that velocities of N number of objects that attract one another by Newton’s law attract one another by Newton’s law of gravitation, determine their of gravitation, determine their configuration at any time in the configuration at any time in the futurefuture

The 3-body problem before The 3-body problem before PoincaréPoincaré

During the 18During the 18thth century, century, mathematicians began to mathematicians began to realize the impossibility of realize the impossibility of finding a closed solution in finding a closed solution in terms of elementary terms of elementary functions. functions.

Clairaut succeeded in Clairaut succeeded in predicting the date of the predicting the date of the perihelion passage of Halley’s perihelion passage of Halley’s comet within 1 month, which comet within 1 month, which was exactly the margin was exactly the margin allowed by the approximation allowed by the approximation method. method.

Poincaré and the chaosPoincaré and the chaos

competition honoring the 60competition honoring the 60thth birthday birthday of Oscar II, King of Sweden and Norway of Oscar II, King of Sweden and Norway featured the 3-body problemfeatured the 3-body problem

Poincaré did not succeed in solving the Poincaré did not succeed in solving the problem, but the ideas he presented in problem, but the ideas he presented in his work were so revolutionary that he his work were so revolutionary that he was awarded the prize for the was awarded the prize for the competition anywayscompetition anyways

The 3-body problem in the 20The 3-body problem in the 20thth centurycentury

Karl Sundham Karl Sundham Stephen Smale Stephen Smale Richard FeynmanRichard FeynmanWang Qiu Dong Wang Qiu Dong

Solutions and Methods

If we have N masses, then the force on any one is the sum of the forces exerted on it by all the others. This gives us the nonlinear system of second-order differential equations

To solve, write in form of Keplar’s equation

In N=2 case, x=x1-x2 with k constant m1+m2

Two body motions are similar conic sections, and in periodic case, ellipses.

Two Body Problem

Found simple periodic solution in 1772 start by placing the three masses at the vertices

x01,x02, x03 of an equilateral triangle whose center of mass, m1*x01 +m2*x02+m3*x03 is the origin. Identify the plane of the triangle with the complex

plane, so that x0i belongs in C. Take any solution L(t) to the planar Kepler equation where the Kepler constant k is a certain rational expression in the three masses mi

The Lagrange solutions are xi(t) = L(t)*x0i

Langrange’s 3-body solution

Each mass moves in an ellipse in such a way that the triangle formed by the three masses evolves by a composition of instantaneous dilations and rotations and hence is equilateral for all time.

Euler’s 3-body solution For the Euler solutions start by placing the three

masses on the same line with their positions x0i such that the ratios rij=rik of their distances are the roots of a certain polynomial whose coefficients depend on the masses

Take any solution L(t) to Kepler’s equation (2) where the Kepler constant is a certain rational expression in the masses mi.

The Euler solutions are xi(t) = L(t)x0i

At every instant the masses are collinear, and the ratios of their distances remain constant.

Figure Eight Solution and Calculus of Variations

S is called the action. It is a number with the dimensions of (Energy) * (Time). S depends on L, and L in turn depends on the function x(t). Given any function x(t), we can produce the number S.

Problems: Unless the space of paths used is restricted, the minimum of the action will be achieved where all bodies are infinitely separated with zero velocity. The difficulty is overcome by imposing certain symmetries on the path space to make the action functional coercive, which guarantees to achieve a minimum which will be the desired solution.

A second difficulty arises in trying to ensure the minimizing path avoids collisions. Collisions are instants in the time when the potential energy becomes infinite; yet the action of a path with collisions may in general remain finite, so special care needs to be taken to ensure that the minimizer is a collision-free orbit and hence a genuine solution curve of the system of differential equations

1970, Gordon successfully applies variational calculus to the 2-body problem, showing that Keplerian orbits are action-minimizing;

Figure eight solution 2000, Chenciner and Montgomery use the

principle of least action to prove the existence of a solution to the equal-mass three-body problem in which all three bodies chase each other around a figure-eight. They do this directly by constructing a "test path" of three bodies that has the figure-eight geometry, then showing it has lower action than any possible orbit which includes collisions. (Since their path without collisions has lower action than any collision path, this proves that whatever solution minimizes the action has no collisions.)

The Figure Eight With period T, then x2(t) = x1(t

-T/3) and x3(t) = x1(t-2T/3). This says that the three bodies travel the same planar curve, phase shifted from each other by one-third of a period. This curve has the form of a figure eight.

The double point of the figure-eight curve is at the origin. This is also the center of mass. The figure eight has the reflectional symmetries of the x-y axes.

N-Body Simulation

The Program

Features: Simulates 2-N gravitationally

interacting bodies (tested for 2-4 bodies)

Displays results graphically

Saves initial conditions to file for easy replay of results

Development of The Program

Coded in C++ Initially Developed In Linux

environment Ported to PC for presentation

Graphics Implemented With: OpenGL OpenGL Utility Toolkit (GLUT)

Implementation of The Physics

Vector Class Data type with all the characteristics

of vectors: Vector Addition/Subtraction Multiplication by a Scalar Dot and Cross Product Calculations Magnitude Function

Simplifies coding by allowing us to use vector algebra!

Implementation of The Physics

Star Class Object with the characteristics of a

simple particle for a gravitational simulation:

Position Vector Velocity Vector Acceleration Vector Mass

Allows us to keep track of each body efficiently

Simulation of the Physics

Acceleration calculated on each body j:

aj = Σ ( – G * Mk / rk2 * rk/|rk| )

For all bodies k, where k ≠ j

Simulation of the Physics

Path of Bodies Numerically Integrated with Euler’s Method:

vi+1 = vi + ai * Δt

xi+1 = xi + vi * Δt

Where i is each successive step

Results

When Simulated for Path of Planets: Paths appear correct.

However more extreme conditions and variation of time-step shows a divergence from the solution This Results from Euler’s method Using a Runge-Kutta Integration

method would solve this problem.

Sources Nate Robins – OpenGL – GLUT for Win32

http://www.xmission.com/~nate/glut.html http://www.ams.org/notices/200105/fea-montgomery.pdf http://merganser.math.gvsu.edu/david/reed03/projects/salomne/ http://www.ams.org/new-in-math/cover/orbits1.html “NON-PLANAR MINIMIZERS AND d-ROTATIONAL SYMMETRY IN THE N-BODY

PROBLEM”, MATTHEW SALOMONE AND ZHIHONG XIA (not yet published) Barrow-Green, Poincare and the Three Body Problem, American Mathematical Society, 1997 http://members.fortunecity.com/kokhuitan/nbody.html http://en.wikipedia.org/wiki/Determinism http://www.mathjmendl.org/chaos/ http://www.math.washington.edu/~hampton/Lagrange.html “A Minimizing Property of Keplerian Orbits”, William B. Gordon, American Journal of

Mathematics, Oct. 1977 “A remarkable periodic solution of the three-body problem in the case of

equal masses”, A. Chenciner and R. Montgomery, Annals of Mathematics, 2000.