Let’s start a new section in physics napkins, called scitoys, for scientific toys. The idea goes as follows: I write down code to illustrate something in physics or maths in an animation. The animation is displayed here, with some explanations and some ideas about further development… So, if you’re a jedi master, you can enjoy the challenges…

This first scitoy is just a running test… the ones I’m preparing are more spectacular. So, in three-body tango, we show three particles interacting through gravitational attraction. I’m pretty sure you all know by now that the two-body gravitational interaction (in Newtonian mechanics) is as regular as a cuckoo clock, while the intrusion of a third body makes it chaotic. You can see in the video how the planets change couple in a chaotic way…

Some technicalities… the program is in C++ for linux with X11. I will send the code to anyone interested and, in due time, I will publish it for all the scitoys. The video capture was done with xvidcap.

A technicality related to this particular program: the gravitational interaction has a short-distance cutoff. I mean: if the planets come closer than a certain minimal distance, I don’t allow the force to accumulate anymore. This avoids some instabilities…

And the challenge: how many perfectly periodic orbits can you find? Are they stable?

With a wink for Miguel Ibáñez Berganza & Daniel Gómez Lendínez

Very nice, Javi! Congratulations for this new section of Physics Napkins, I think it will be of interest to many people. Just a question, what happens with collisions? Are there no collisions in this simulation? or are you “regularizing” them by introducing the short-distance cut off that you mentioned? About your question on periodic orbits, well it seems to me just looking at the simulation that there is no one … how are the masses of the particles, are all of them the same? it would be nice to see the simulation when one of the masses is much smaller than the other two, which move in a Kepler periodic trajectory … is the perturbation of the other mass enough to break the periodic trajectory? ;-)

Hi, Dani! In this case, the masses are equal, as you’ve guessed. And true, there is a regularization in the potential to make collisions “smooth”…

About the periodic orbits, beyond the “trivial” ones (which are also worth discovering), I am not sure, but my guess is that there are non-trivial ones. One might perform a systematic search…

The periodic orbits that you ask for, concern this specific simulation or the three-body problem in general? For the three-body problem there are indeed many configurations with periodic motions (e.g. relative equilibria). Moreover, any two configurations of the masses can be connected by a true solution of the Newton equations … surprising but a theorem ;-)

Seriously? So, given two configurations, one can always find initial velocities such that the trajectory will join them (in unknown time)… This is rather special of this problem, I guess… Or maybe there are more known problems with this property?

Another challenge then would be, given three particles in a line, with equal masses, A-B-C, find the initial velocities to get a permutation B-C-A, for example…

Jajaja this permutation question is nice! … yes, you understood well the claim, the proof is not constructive so one does not know the velocities (maybe time can be fixed, I am not sure) … the proof makes use of the variational characterization of the equations, as well as some symmetries … the point is to show the exisence of minimizers of the action joining two given points in the configurations space … and to be sure that there are no collisions in the process … maybe similar results can be obtained for other variational equations with some “nice” properties …

Wait, I see… the variational principle tells you, given an initial and a final configuration, how to find the trajectory that joins them… if it exists… So, they proved that there is always a stationary point for the action, given the initial and final points… I wonder, a very simple question, what happens when this is not the case… You see, so simple!

Your are right, Javi … well, a difficult technical point in the proof is to show that this stationary point for the action exists (in fact it is a local minimizer) without collisions … maybe one can find some lagrangian of an n-body problem where this result does not hold because some configurations cannot be reached from other ones due to the existence of collisions in the process! … a trivial example is if all the particles are constrained to move on the same line … then of course no permutations are possible! …

Very nice, Javi! Congratulations for this new section of Physics Napkins, I think it will be of interest to many people. Just a question, what happens with collisions? Are there no collisions in this simulation? or are you “regularizing” them by introducing the short-distance cut off that you mentioned? About your question on periodic orbits, well it seems to me just looking at the simulation that there is no one … how are the masses of the particles, are all of them the same? it would be nice to see the simulation when one of the masses is much smaller than the other two, which move in a Kepler periodic trajectory … is the perturbation of the other mass enough to break the periodic trajectory? ;-)

Hi, Dani! In this case, the masses are equal, as you’ve guessed. And true, there is a regularization in the potential to make collisions “smooth”…

About the periodic orbits, beyond the “trivial” ones (which are also worth discovering), I am not sure, but my guess is that there are non-trivial ones. One might perform a systematic search…

The periodic orbits that you ask for, concern this specific simulation or the three-body problem in general? For the three-body problem there are indeed many configurations with periodic motions (e.g. relative equilibria). Moreover, any two configurations of the masses can be connected by a true solution of the Newton equations … surprising but a theorem ;-)

Seriously? So, given two configurations, one can always find initial velocities such that the trajectory will join them (in unknown time)… This is rather special of this problem, I guess… Or maybe there are more known problems with this property?

Another challenge then would be, given three particles in a line, with equal masses, A-B-C, find the initial velocities to get a permutation B-C-A, for example…

Jajaja this permutation question is nice! … yes, you understood well the claim, the proof is not constructive so one does not know the velocities (maybe time can be fixed, I am not sure) … the proof makes use of the variational characterization of the equations, as well as some symmetries … the point is to show the exisence of minimizers of the action joining two given points in the configurations space … and to be sure that there are no collisions in the process … maybe similar results can be obtained for other variational equations with some “nice” properties …

Wait, I see… the variational principle tells you, given an initial and a final configuration, how to find the trajectory that joins them…

if it exists… So, they proved that there is always a stationary point for the action, given the initial and final points… I wonder, a very simple question, what happens when this is not the case… You see, so simple!Your are right, Javi … well, a difficult technical point in the proof is to show that this stationary point for the action exists (in fact it is a local minimizer) without collisions … maybe one can find some lagrangian of an n-body problem where this result does not hold because some configurations cannot be reached from other ones due to the existence of collisions in the process! … a trivial example is if all the particles are constrained to move on the same line … then of course no permutations are possible! …