# Romeo & Juliet phase portrait

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In 1988, Steven Strogatz proposed using two people’s changing feelings for each other over time as an example for teaching ordinary differential equations. Read his eloquent one-page note.

Each person has two personality parameters determining whether they respond positively or negatively to their own feelings for someone else, and whether they respond positively or negatively to another person’s feelings for them. They also have some initial conditions here (first impressions of each other, you could say), though in a larger model you might want everyone to start at apathy and ignorance.

So if `t` is time; `R` is Romeo’s feelings for Juliet; `J` is Juliet’s feelings for Romeo; and `a`, `b`, `c`, and `d` are personality parameters; you have:

``````dR/dt = aR + bJ
dJ/dt = cR + dJ``````

I think I first came across this in Prof. Pietraho and Prof. Zeeman’s classes at Bowdoin. I think it was on a final exam. J. C. Sprott has a good exposition of Strogatz’s model here.

Of course this is all very silly. But I like exercising a more dynamic vocabulary for relationship state than the static “being together” or “not being together”. I like thinking about orbits. There is something romantic about constant freefall under centripedal acceleration toward your mutual baryocenter. And I like puns about three-body systems.

But if there is one way in which math classes shaped my mind in a way unfit for relationships with people, it is the focus on the limiting behavior of systems. There is a lot of fatalism in math: either something will converge as `t` approaches infinity, or it will diverge. But sometimes things converge very slowly! I guess this is an embarrassing error, but I think I internalized the asymptotic outlook to such an extent that hearing someone in my life on Earth say “timing is everything” is somehow jarring.

Math is necessary, universal, and eternal. Love is contingent, personal, and fleeting. In math, anything that isn’t always true is dishonest. In love, anything that would be true for anyone else is dishonest. In math, your proof should hold true for anyone, anywhere, at any time. In love, you are at your most honest when your words would be lies, or devoid of meaning, if they were said to anyone else, or at any other time.

Anyway, I know very little of either. There are probably bugs in here. Good chance something’s entirely upside-down.

—14 February 02016