The Three-Body Problem. Context Motivation and History Periodic solutions to the three-body problem...
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Transcript of The Three-Body Problem. Context Motivation and History Periodic solutions to the three-body problem...
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