The Three-Body Problem

Context Motivation and History Periodic solutions to the three-body problem The restricted three-body problem Runge-Kutta method Numerical simulation

Motivations and History

People who formulated the problem and made great contributions: Newton Kepler Euler Poincaré

Newton told us that two masses attract each other under the law that gives us the nonlinear system of second-order differential equations: Motivations and History The two-body problem was analyzed by Johannes Kepler in 1609 and solved by Isaac Newton in 1687.

Motivations and History There are many systems we would like to calculate. For instance a flight of a spacecraft from the Earth to Moon, or flight path of a meteorite. So we need to solve few bodies problem of interactions.

In the mid-1890s Henri Poincaré showed that there could be no such quantities analytic in positions, velocities and mass ratios for N>2. Motivations and History

In 1912 Karl Sundman found an infinite series that could in principle be summed to give the solution - but which converges exceptionally slowly. Henri Poincaré identified very sensitive dependence on initial conditions. And developed topology to provide a simpler overall description. Motivations and History

Periodic Solutions

Newton solved the two-body problem. The difference vector x = x 1 - x 2 satisfies Kepler’s problem: All solutions are conics with one focus at the origin. The Kepler constant k is m 1 +m 2. Periodic Solutions

Filling of a ring is everywhere dense Periodic Solutions

The simplest periodic solutions for the three- body problem were discovered by Euler [1765] and by Lagrange [1772]. Built out of Keplerian ellipses, they are the only explicit solutions. Periodic Solutions

The Lagrange solutions are x i (t) = λ(t)x i 0, λ(t) C is any solution to the planar Kepler problem. To form the Lagrange solution, start by placing the three masses at the vertices x 1 0,x 2 0, x 3 0 of an equilateral triangle whose center of mass m 1 x 1 0 +m 2 x 2 0 +m 3 x 3 0 is the origin. Periodic Solutions

Lagrange’s solution in the equal mass case Periodic Solutions

Lagrange’s solution in the equal mass case Periodic Solutions

The Euler solutions are x i (t) = λ(t)x i 0, λ(t) C is any solution to the planar Kepler problem. To form the Euler solution, start by placing the three masses on the same line with their positions x i 0 such that the ratios r ij =r ik of their distances are the roots of a certain polynomial whose coefficients depend on the masses. Periodic Solutions

Euler’s solution in the equal mass case Periodic Solutions

Most important to astronomy are Hill’s periodic solutions, also called tight binaries. These model the earth-moon-sun system. Two masses are close to each other while the third remains far away. Periodic Solutions

New periodic solution “figure eight”. The eight was discovered numerically by Chris Moore [1993]. A.Chenciner and R.Montgomery [2001] rediscovered it and proved its existence. Periodic Solutions

The figure eight solution Periodic Solutions

Some examples the figure eight 6 bodies, non-symmetric 19 on an 8

Some examples 21 bodies7 bodies on a flower

Some examples 8 bodies on daisy4 bodies on a flower

The restricted three-body problem.

Formulation of Problem The restricted three-body problem. The restricted problem is said to be a limit of the three-body problem as one of the masses tends to zero. Hamilton’s equations:

Runge-Kutta Method

Abstract: First developed by the German mathematicians C.D.T. Runge and M.W. Kutta in the latter half of the nineteenth century. It is based on difference schemes.

2 nd order Runge-Kutta method : Cauchy problem: Let’s take Taylor of the solution :

If u(x i ) solution, then u ’ (x i )=f(x i,u i )

If we substitute derivatives for the difference derivatives, We get: 0<β<1, y j+1 is approximated solution.

Now if we take β=1/2, we obtain classical Runge- Kutta scheme of 2 nd order. If we continue we obtain scheme of 4 th order:

2 nd order Runge-Kutta method :

Method for the system of differential equations: Let’s denote u ’ =v,.

The system takes on form:

If is a vector of approximations of the solution, at point x j, and are vectors of design factors, then:

Th.(error approximation in the RK method): ε h (t 1 )=|y h (t 1 )-y(t 1 )|≈ 16/15∙|y h (t 1 )-y h / 2 (t 1 )| where ε h is the error of calculations at the point t 1 with mesh width h.

Numerical simulation

Numerical simulation is based on: 4 th order Runge-Kutta method Adaptive stepsize control for Runge- Kutta Program is developed in Delphi. Numerical simulation

Some obtained orbits

“A New Solution to the Three-Body Problem”, R. Montgomery “Numerical methods”, E. Shmidt. “Lekcii po nebesnoj mehanike”, V.M. Alekseev. “Chislennie Metodi”, V.A. Buslov, S.L. Yakovlev. “From the restricted to the full three-body problem”, Kenneth R., Meyer and Dieter S. Schmidt. Charlie McDowell. References