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

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 = x1 - x2 satisfies Kepler’s problem:

• All solutions are conics with one focus at the origin.

• The Kepler constant k is m1+m2 .

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 xi (t) = λ(t)xi0,

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 x1

0,x20, x3

0 of an

equilateral triangle whose center of mass m1x1

0+m2x20+m3x3

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 xi (t) = λ(t)xi0,

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 xi

0 such that the ratios rij=rik 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 bodies 7 bodies on a flower

Some examples

8 bodies on daisy 4 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

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.

2nd order Runge-Kutta method :

Cauchy problem:

Let’s take Taylor of the solution :

If u(xi) solution, then u’(xi)=f(xi ,ui)

If we substitute derivatives for the difference derivatives,

We get:

0<β<1, yj+1 is approximated solution.

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

If we continue we obtain scheme of 4th order:

2nd 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 xj, and

are vectors of design factors, then:

Th.(error approximation in the RK method):

εh(t1)=|yh(t1)-y(t1)|≈ 16/15∙|yh(t1)-yh/2(t1)|

where εh is the error of calculations at the

point t1 with mesh width h.

Numerical simulation

Numerical simulation is based on:

• 4th order Runge-Kutta method

• Adaptive stepsize control for Runge-

Kutta

Program is developed in Delphi.

Numerical simulation

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.

• http://www.cse.ucsc.edu/~charlie/3body/, Charlie McDowell.

References