This article is a for Opinionator by Steven Strogatz.
“In the spring,” wrote Tennyson, “a young man’s lightly to of love.” And so in keeping with the of the , this week’s at love — mathematically. The analysis is offered , but it does a serious point: that the laws of are written as equations. It also helps why, in the words of another , “the of true love never did run .”
To the approach, Romeo is in love with Juliet, but in our of the story, Juliet is a lover. The more Romeo loves her, the more she wants to run away and . But when he takes the and backs off, she begins to find him . He, on the other hand, tends to echo her: he warms up when she loves him and cools down when she him.
What happens to our lovers? How does their love over time? That’s where the math comes in. By writing equations that summarise how Romeo and Juliet respond to each other’s and then those equations with calculus, we can the of their . The for this is, tragically, a never-ending of love and . At they manage to achieve simultaneous love a quarter of the time.
The model can be made more realistic in ways. For , Romeo might react to his own as as to Juliet’s. He might be the type of guy who is so worried about himself at her that he slows himself down as his love for her grows. Or he might be the other type, one who loves in love so much that he loves her all the more for it.
Add to those the two ways Romeo could react to Juliet’s affections — either or his own — and you see that there are four types, each corresponding to a different romantic .
My students and those in Peter Christopher’s class at Worcester Polytechnic Institute have such descriptive names as Hermit and for the kind of Romeo who his own love and also Juliet’s. Whereas the of Romeo who gets by his own but off by Juliet’s has been called a Narcissistic Nerd, Better Latent Than Never, and a Flirting Fink. (Feel free to post your own suggested names for these two types and the other two possibilities.)
Although these examples are , the equations that arise in them are of the kind known as differential equations. They the most has for making of the world. Sir Isaac Newton them to the of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion to both.
In the 300 years since Newton, has come to that the laws of are always in the language of equations. This is true for the equations governing the of , air and water; for the laws of and magnetism; for the and often realm where quantum mechanics .
In all , the business of theoretical finding the right differential equations and them. When Newton discovered this to the of the universe, he felt it was so precious that he published it only as an anagram in Latin. Loosely , it reads: “It is useful to differential equations.”
The idea that love might progress in a way to me when I was in love for the first time, trying to my girlfriend’s behaviour. It was a summer romance at the end of my sophomore year in college. I was a lot like the first Romeo above, and she was more like the first Juliet. The of our was me crazy until I realised that we were both acting mechanically, of push and pull. But by the end of the summer my equations started to break down, and I was more than . As it out, the was . There was an important variable that I’d left out of the equations — her old boyfriend wanted her back.
In we call this a three-body problem. It’s , in the astronomical context where it first arose. After Newton the differential equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun), he his to the three-body problem for the sun, earth and moon. He couldn’t it, and neither could anyone else. It later out that the three-body problem the of chaos, its behaviour unpredictable in the run.
Newton knew nothing about chaotic dynamics, but he did tell his friend Edmund Halley that the three-body problem had “made his head , and kept him so often, that he would think of it no more.”
I’m with you there, Sir Isaac.