The Three Body Problem

[SLD]

Is the three body truly unsolvable? Despite centuries of advances in mathematics a direct general solution to the problem of three bodies orbiting each other has yet to be solved. We have found numerous solutions to particular periodic orbits, but not generally in terms of all possible orbits. Of course using computers and numerical methods we can get arbitrarily close and over the last twenty or so years we have gotten far better, and are able to apply these techniques to many other bodies.

But is it unsolved because it is not possible to solve it? Or is it just way too complicated for us to solve, involving way too many variables?

I understand that some formulations involve 18 or so differential equations. While difficult, not necessarily impossible. But some differential equations cannot be analytically solved, only numerically.

I put this in mathematics because I don’t think it is a physics problem per se.

 

[steve_bank]

There are classes of problems that can not be solved analytically, electronics abounds with them.

They are solved with different numerical methods.

Spreadsheets like Excel have a simple one called a goal seeker. Trial and error.

 

[steve_bank]

I have not looked at it before. According to he link the problem chaotic. This means the inertial conditions can not be sufficiently precise to predict long term conditions.

Weather is chaotic. Local temperatures can predicted within a few degrees out a few days but not a year from now. Weather is causal but not predictable in the long term.

Small variations in initial initial conditions lead to large deviations in the long term.

Three-body problem - Wikipedia

en.wikipedia.org

In physics, specifically classical mechanics, the three-body problem involves taking the initial positions and velocities (or momenta) of three point masses and calculating their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.[1] It has no general closed-form solution, unlike two-body problems.[1] When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions, and the only way to predict the motions of the bodies is to calculate them using numerical methods.

The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.

I watched a demonstration on a show.

On a table were three magnets in a triangle. Hanging over the table was a steel ball. The ball was held off to the side by a mechanism and released electrically.

On successive runs the initial trajectories were close but diverged over time. The initial conditions could be made more precise, but they would never be exactly the same.

A similar problem a 3 link pendulum, Chaotic.

 

[Jimmy Higgins]

Is the three body truly unsolvable?

It is solvable, but I thought it was an issue of initial state, and not knowing it exactly. And ridiculously minor variations lead to differing outcomes.

 

[Jarhyn]

Is the three body truly unsolvable?

It is solvable, but I thought it was an issue of initial state, and not knowing it exactly. And ridiculously minor variations lead to differing outcomes.

Well, more it's about knowing a current state, because the further you are from the current state the more chaos evolves into the system.

It's distinctly like the way there are patterns among the primes but the growing complexity forces the viewer to add more and more information, to the point it's impossible to calculate any further with any degree of certainty.

If you stop and recalculate from the current position, it's easy to track, for a while.

Remember that every momentary state is also an "initial state" to the rest of the system. Still, passing through certain moments will also be difficult to parse the outcome of because there will be a "balance on the top of a hill", and knowing which way the "break" will go from there is not always possible.

 

[steve_bank]

Not the same problem, but another that is unlovable by a set of basic Newtonian equations, the solution is a work around.

From mechanics statistically indeterminate structures. More variables than equations.

A statically determinate structure is one that is stable and all unknown reactive forces can be determined from the equations of equilibrium alone. A statically indeterminate structure is one that is stable but contains more unknown forces than available equations of equilibrium.

 

[lpetrich]

 Three-body problem - a series solution does exist, but it has unusably slow convergence.

... in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^8,000,000 terms.

In practice, one does numerical integration or a perturbation expansion. One also does not limit oneself to three bodies or Newtonian gravitational interactions.

Perturbation expansion? That's treating an orbit like a two-body orbit and expanding in powers of small parameters like a mass ratio. One can do it directly in the coordinates or one can use the orbit elements as alternative coordinates.

Using orbit elements often makes it possible to integrate over orbits, thus getting equations of motion for long-term effects.

Orbit elements: a set of parameters for describing an orbit. There are six of them: mean distance or period, eccentricity, three orientation parameters, and position in the orbit.

 

[Swammerdami]

And, for once EVERYONE is correct! And OP himself summarizes the answer in OP: "But some differential equations cannot be analytically solved, only numerically."

An example of a three-body problem is an object traveling with the Earth but at one of the Lagrange points. Depending on which of the Lagrange points, the object might want to describe a small orbit of its own. In practice such spacecraft are equipped with micro-thrusts to correct for any push/pulls. Details anyone?

But OP asks whether an "exact" solution can be constructed for some such equation systems. Proving that NO such solution exists may be impossible even if true, or require time e^N where N is the solution size. Thus we are left to wonder WHICH systems have been solved, and what is known. This is a topic about which I know nothing, nor do I know about near equilibria characterized by a  Lyapunov exponent.

My impression is that only rarely will a complicated set of equations have an exact expressible solution. Even if such a solution exists it is likely to be at an unstable equilibrium, and thus as lpetrich implies, one might still prefer a numeric approach.

One famous example -- or rather trio of examples -- are the  Lagrange, Euler, and Kovalevskaya tops. These three spinning and accelerating rigid bodies have "integrable" equations of motion, each credited to its name-sake. No fourth solvable Top is known. But afaik there is no proof that such a Top doesn't exist.

 

[lpetrich]

Let's now consider the two-body problem, ignoring outside influences. The two bodies have a center of mass, and that center moves at constant velocity. Thus, we can ignore this overall motion and turn to their relative motion.

This has 6 parameters, and of these, energy is 1 and angular momentum is 3, giving 2 left over.

If we consider an inverse-square force, we find an additional conserved quantity, the  Laplace–Runge–Lenz vector or the  Eccentricity vector which differs only by some scaling. Being a vector, it has 3 components, but by interrelationships, it contributes only 1 additional conserved quantity. That leaves exactly 1 left over, and that's for the time reference or epoch.

The conservation of the LRL/E-vector means zero precession.

 

[lpetrich]

For the Euler top, there are 6 variables, the 3 orientation angles and the 3 components of angular velocity. It has 4 conserved quantities: 1 for energy and 3 for the components of the angular momentum. That leaves 2 left over.

There is a closed-form solution for the angular velocity if "closed form" includes  Jacobi elliptic functions - Jacobi Elliptic Functions -- from Wolfram MathWorld

The Lagrange top has 3: energy, angular momentum along vertical axis, and angular momentum along top's axis. That leaves 3.

The Kovalevskaya top has 4: KowaFilm.pdf

 

[steve_bank]

Numerical or analytical the problem in practice for observation of solar scale objects in motion is the initial conditions.

The old saying goes something like your answers are only as good as your data. If your data is only good to 1 or 2 decimal places carrying out arithmetic to 15 decimal places has no meaning.