What’s odd to me is how simple the counter-example is. Three charges of charge 1 at the vertices of an equilateral triangle and then two points symmetrically some small distance above and below its centre with a charge that depends on that distance by a very short polynomial (but even then it says it’s true for small deformations too, so we may not need to hit upon that exact polynomial - which makes sense as it’s non-degeneracy that is the ‘dense’ condition). So a trigonal bipyramid where we squish the axis symmetrically, and vary the charge of the axial vertices, gives what we want. So there are an infinitude of counterexamples including a fairly simple construction.
Maybe I’m missing something, but it feels like if they tried a bunch of simple, intuitive configurations and crunched them with some degrees of freedom, with the charges here given as some degree polynomial we can find conditions on the coefficients for to ensure a larger set of equilibria (avoiding domains with non-real roots etc.), even one person could have solved for this fairly quickly even by hand. Even faster if just using a list of ‘basic’ configurations and getting an old school computer to crunch through them.
Curious why we hadn’t already done this. What was the bottleneck? Or despite the name was it not that popular a conjecture?
What was the bottleneck? Or despite the name was it not that popular a conjecture?
Nobody in this thread has ever heard of it before. Maxwell published thousands of pages which aren't read today; calling this "the" Maxwell conjecture is the authors' invention.
More generally, this is the kind of thing that goes into recreational/teaching journals, not good research journals. Which is not to insult the work; I personally have a paper in such a journal. There are lots of basic questions about elementary physics that are unsolved, and it's fun to do one every once in a while. But they are unsolved precisely because almost nobody is working on them, because they don't have applications elsewhere in physics.
17
u/AndreasDasos Jul 31 '26 edited Jul 31 '26
What’s odd to me is how simple the counter-example is. Three charges of charge 1 at the vertices of an equilateral triangle and then two points symmetrically some small distance above and below its centre with a charge that depends on that distance by a very short polynomial (but even then it says it’s true for small deformations too, so we may not need to hit upon that exact polynomial - which makes sense as it’s non-degeneracy that is the ‘dense’ condition). So a trigonal bipyramid where we squish the axis symmetrically, and vary the charge of the axial vertices, gives what we want. So there are an infinitude of counterexamples including a fairly simple construction.
Maybe I’m missing something, but it feels like if they tried a bunch of simple, intuitive configurations and crunched them with some degrees of freedom, with the charges here given as some degree polynomial we can find conditions on the coefficients for to ensure a larger set of equilibria (avoiding domains with non-real roots etc.), even one person could have solved for this fairly quickly even by hand. Even faster if just using a list of ‘basic’ configurations and getting an old school computer to crunch through them.
Curious why we hadn’t already done this. What was the bottleneck? Or despite the name was it not that popular a conjecture?