Steven Strogatz in the New York Times:
“In the spring,” wrote Tennyson, “a young man’s fancy lightly turns to thoughts of love.” And so in keeping with the spirit of the season, this week’s column looks at love affairs — mathematically. The analysis is offered tongue in cheek, but it does touch on a serious point: that the laws of nature are written as differential equations. It also helps explain why, in the words of another poet, “the course of true love never did run smooth.”
To illustrate the approach, suppose Romeo is in love with Juliet, but in our version of the story, Juliet is a fickle lover. The more Romeo loves her, the more she wants to run away and hide. But when he takes the hint and backs off, she begins to find him strangely attractive. He, on the other hand, tends to echo her: he warms up when she loves him and cools down when she hates him.
What happens to our star-crossed lovers? How does their love ebb and flow over time? That’s where the math comes in. By writing equations that summarize how Romeo and Juliet respond to each other’s affections and then solving those equations with calculus, we can predict the course of their affair. The resulting forecast for this couple is, tragically, a never-ending cycle of love and hate. At least they manage to achieve simultaneous love a quarter of the time.