This claim might seem incongruous with the fact that they obviously analyze three-body systems and "discover" interesting things about their dynamics.
The subtlety rests on the definition of "solvable", which here means (loosely) to express the trajectories of the gravitational bodies in terms of "known functions", thus to have an "exact solution" that one can write down in mathematical notation, say x = sin(w t), instead of an approximate (numerical) solution that is a stream of numbers.
The chase of exact solutions was a holy grail for 18th/19th century physics but has lost much of its mystique in the 20th century. On the one hand the systems of interest become much more complex (thus in general "unsolvable") and on the other hand the invention of digital computers enabled new forms of analysis.
But the irony is that in the broad scheme of things "interesting" exact solutions are very rare, almost freakish. The only systems we can reliably solve are linear :-). So while numerical approaches have weaknesses, they are the de-facto available tool in most cases.
What constitutes a closed-form expression depends on context/is up for debate:
- https://en.wikipedia.org/wiki/Closed-form_expression#Alterna...:
“Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well known since numerical implementations are widely available.”
- https://mathworld.wolfram.com/Closed-FormSolution.html:
“For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.”
so that’s not a given.
It is of practical use often with spacecrafts, where the spacecraft is much much smaller than the Sun-Planet or Planet-Moon it orbits around. Jokingly one could say this is why you don't need to worry about the Sun-Earth system blowing up every time we boost the international space station to a higher orbit.
The variable is not ignored. The effect of that 3rd light mass on the other two is ignored. The effect of the two massive bodies on the 3rd is not ignored.
As an example when you are planning orbital manoeuvres for the JWST space telescope you assume that the movements of the telescope won't affect the Moon or the Earth. But the Moon and the Earth will effect the position of the JWST. While this is not "true", it is accurate enough to work in practice.
And of course the significance is that the 3rd light object is the only one the engineer controls. We can't change what the Earth or the Moon does, but we can make our spacecraft burn engines in a direction of our choosing. That gives a changed velocity vector. And then we can use the restricted 3-body problem to efficiently calculate where it will end up as a function of time.
I always suspected that Lebensraum discourse.
If a bunch of puppets can fly stars around the galaxy in a Klemperer rosette, How Hard Can It Be?!
The paper itself merely claims to have done a bunch of work to further characterize and statistically analyze 3-body situations.
For anyone interested in the history: https://en.wikipedia.org/wiki/Three-body_problem
For a fun animation of a bunch of solutions, see: https://en.wikipedia.org/wiki/File:5_4_800_36_downscaled.gif
Three-body problems can be chaotic and can be ugly. But in the simplest case you can make them look like two-body problems and things will be just fine.
