Is the 3 body problem in physics solved or not?

Have physicists solved the 3 body (or n-body) problem ? another question : is this problem pure classical mechanics or something else with classical mechanics ?

The folklore is that the three-body and higher n-body solution is not only unsolved but unsolvable, and that Poincare and Bruns proved this in 1888.

This folklore is wrong. Or at least, it’s astonishingly misleading, even if you start adding the caveats of “closed form solution” or “analytic solution”. For Poincare and Bruns proved nothing of the kind. You have to tailor your criteria for what counts as a “solution” suspiciously closely in order to hold that there’s a sense in which the two body problem is solvable, but the three and higher n-body problem is not.

Let’s start with the two-body problem. Here it is.

m_{1} {\ddot{x}}_{1} = \frac{G m_{1} m_{2}}{r_{12}^{3}} (x_{2} - x_{1})

m_{2} {\ddot{x}}_{2} = \frac{G m_{1} m_{2}}{r_{12}^{3}} (x_{1} - x_{2})

(where

r_{12} = | x_{1} - x_{2} |

). There is a well-known (and rather neat) approach to solving this that is taught in all intermediate courses in Newtonian physics. The steps are to substitute in an “effective” mass term and transform to a centre of mass frame to reduce it to a problem of central potential, move to polar co-ordinates, eliminate the time variable, and then make the simple but but effective substitution of

u = 1 / r

, whereupon the whole problem collapses into a familiar form of differential equation that can easily be shown to be satisfied by an ellipse, a hyperbola, or a parabola — the selection depending on the initial conditions.

This approach works by quickly searching out and exploiting the constants of motion (the linear momentum of the centre of mass, the various angular momenta, and the total energy of the system) to simplify the problem down.

This process allows you to write down the equation of this curve as a function of the radius

r

and angle

θ

, and you can do this in so-called “closed form”. That is, it can be written in a finite number of functions and terms from a general accepted set of “allowed” ones. This “allowed” set includes algebraic functions, exponentials and indefinite integrals, and not usually much more than that.

This definition of “closed form” looks pretty arbitrary. Why choose those set of functions, why specify a finite set of them, and so on? What it tries to do is capture a sense in which the orbit can be calculated in some “manageable” form — at least manageable in the pre-computer age when all calculations had to be done by hand.

Before we leave the two-body case we should note one more interesting thing. There is a reasonable sense in which the two-body case is not “solvable” either.

The procedure outlined above gives you an equation for the shape of the orbital curve a body produces, but considering where we started (two second order differential equations in terms of

r

and

t

) you might have been expecting a solution of

r

and

t

. It is not usually mentioned, but it is not generally possible to write down a closed form expression for

r

as a function of

t

for the two body problem. And we might be troubled by this. What if we want to wind time

t

forward and solve where the body will be relative to another at a specific time? It turns out that we cannot write down such an expression in closed form (though a variety of parametric forms can be produced for as general cases as you could wish).

It’s worth stopping to consider why this lack of a closed form of the

r (t)

isn’t thought important. It’s not important because we can derive any property that we find interesting from other forms of equation (e.g., the elliptical solution, or the parametric forms), and if we really want it in precisely the

r (t)

form, there are plenty of tractable numerical approaches we can solve with a computer. But it’s interesting that we can make the two body problem “unsolvable” in a certain sense, so long as we put enough restrictions on what counts as a “solution”.

Three-body problem:

It looks like this:

m_{1} {\ddot{x}}_{1} = \frac{G m_{1} m_{2}}{r_{12}^{3}} (x_{2} - x_{1}) + \frac{G m_{1} m_{3}}{r_{13}^{3}} (x_{3} - x_{1})

m_{2} {\ddot{x}}_{2} = \frac{G m_{2} m_{3}}{r_{23}^{3}} (x_{3} - x_{2}) + \frac{G m_{2} m_{1}}{r_{21}^{3}} (x_{1} - x_{2})

m_{3} {\ddot{x}}_{3} = \frac{G m_{1} m_{3}}{r_{13}^{3}} (x_{1} - x_{3}) + \frac{G m_{2} m_{3}}{r_{23}^{3}} (x_{2} - x_{3})

That is, very, very similar to the two-body problem. But it’s much harder to attack.

There are certain special cases of orbits that can be solved in a similar way to the two body problem. Euler and Lagrange found some of the most important classes early on. These were usually cases in which the solution has some special property of symmetry, or in which one of the bodies could be considered with negligible mass.

In 1888 King Oscar of Sweden and Norway put forward a prize: the problem was to come up with a particular sort of solution to the three body problem (he set restrictions that made it quite similar to a “closed form”, though also he allowed infinite series, so long as they converged for all reasonable values of the variables).

No-one was able to meet the conditions as written, but Henri Poincare won the prize with a paper that moved forward mechanics in various important ways. It included an impossibility proof, of a type that was also presented by Heinrich Bruns at about the same time. This showed that the n body problem has no integrals, algebraic with respect to time, position and velocities of the n particles, other than the special cases already uncovered (at that point there were 10 types in all).

Why is this important? The whole episode was very important historically for the development of the understanding of differential equations. For Poincare’s work uncovered the strange nature of some of the orbits — they are what we would now call chaotic, and are the first known examples of such systems. (Poincare himself misunderstood these orbits and asserted at first that they were stable, which they are not).

But what Poincare and Bruns’ impossibility proof showed is that there are no closed form solutions producible by a certain method of integration. And this is certainly important: it shows that there are not enough constants of motion of the appropriate type to exploit in the way that we exploited them to produce the solution to the two-body problem. But this is sometimes now paraphrased into saying that general solutions to the three body problem cannot exist. This is wrong.

It soon turned out that we can produce solutions by other methods. In 1907 Karl Sundman developed a series approach for (almost) all initial conditions that actually solves the three body problem. It did not try attacking it via the method that Poincare and Bruns had earlier shown impossible, but went a completely different route and developed a series solution in powers of

t^{\frac{1}{3}} .

This converges just fine (though slowly) for all cases where the angular momentum is non-zero.

Now, you can come up with criteria by which this is not closed form. Most obviously, it gives solutions as infinite, converging series. But I have to say — come on! It’s pretty arbitrary that Sundman’s solution is not allowed, whereas the two-body solution is. You almost have to design your conditions specifically to admit one and not the other. Sundman’s solution certainly fulfils King Oscar’s conditions, since he explicitly admitted infinite series, so long as they converged.

But somehow, Sundman’s solution is not counted by the folklore. The contemporary influence of Brouwers’ Intuitionism in mathematics might have led to the perception that Sundman’s series solution was somehow not a true “solution”. But intuitionism is now almost totally rejected by mathematicians and philosophers, so this is not a good reason any more.

A better reason for objecting to Sundman’s approach is that it’s not very practically useful: it exhibits very, very slow convergence in most cases. So it’s pretty useless for real calculations: you have to calculate thousands of terms to get an accurate answer. But here we can switch horses: we have good numerical methods for the three body problem that — with the aid of computers — give us very accurate solutions to any degree of precision we please (though here, the chaotic nature of many orbits add some spice to the situation.)

Washing up: in most senses you can name, the folklore about the three body problem is simply wrong. The three body problem is solvable. It’s been solved. The sense in which it is “unsolvable” is an arbitrary one. Poincare-Bruns’ impossibility proof shows important things about the system, but doesn’t show what the folklore says it does. And, in passing, the same is true of the general n-body problem. In 1991, Quidong Wang demonstrated a power-series approach to the n-body problem that excludes only collision cases.

But there’s one sense of “solved” which should not be allowed to apply. And that is the sense in which “solved” means “there’s no more to discover”. There is still a great deal to discover in the n body problem. The existence of Sundman and Wang’s series solutions does not reveal much about the character of the orbits that are admitted. New solutions are being discovered all the time (the last I know of was in 2013). In this sense, the problem will be keeping people busy — and producing new insights — for the foreseeable future.

Tuesday, 31 January 2017

Is the 3 body problem in physics solved or not?

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