“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.”

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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.

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The model can be made more realistic in various ways. For instance, Romeo might react to his own feelings as well as to Juliet’s. He might be the type of guy who is so worried about throwing himself at her that he slows himself down as his love for her grows. Or he might be the other type, one who loves feeling in love so much that he loves her all the more for it.

Add to those possibilities the two ways Romeo could react to Juliet’s affections — either increasing or decreasing his own — and you see that there are four personality types, each corresponding to a different romantic style.

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My students and those in Peter Christopher’s class at Worcester Polytechnic Institute have suggested such descriptive names as Hermit and Malevolent Misanthrope for the particular kind of Romeo who damps out his own love and also recoils from Juliet’s. Whereas the sort of Romeo who gets pumped by his own ardor but turned 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 whimsical, the equations that arise in them are of the far-reaching kind known as differential equations. They represent the most powerful tool humanity has ever created for making sense of the material world. Sir Isaac Newton used them to solve the ancient mystery of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion applied to both.

In the 300 years since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. This is true for the equations governing the flow of heat, air and water; for the laws of electricity and magnetism; even for the unfamiliar and often counterintuitive atomic realm where quantum mechanics reigns.

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In all cases, the business of theoretical physics boils down to finding the right differential equations and solving them. When Newton discovered this key to the secrets of the universe, he felt it was so precious that he published it only as an anagram in Latin. Loosely translated, it reads: “It is useful to solve differential equations.”

The silly idea that love affairs might progress in a similar way occurred to me when I was in love for the first time, trying to understand my girlfriend’s baffling behavior. 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 even more like the first Juliet.

The cycling of our relationship was driving me crazy until I realized that we were both acting mechanically, following simple rules of push and pull. But by the end of the summer my equations started to break down, and I was even more mystified than ever. As it turned out, the explanation was simple. There was an important variable that I’d left out of the equations — her old boyfriend wanted her back.

In mathematics we call this a three-body problem. It’s notoriously intractable, especially in the astronomical context where it first arose. After Newton solved the differential equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun), he turned his attention to the three-body problem for the sun, earth and moon. He couldn’t solve it, and neither could anyone else. It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.

Newton knew nothing about chaotic dynamics, but he did tell his friend Edmund Halley that the three-body problem had “made his head ache, and kept him awake so often, that he would think of it no more.”
I’m with you there, Sir Isaac.
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NOTES:
For models of love affairs based on differential equations, see Section 5.3 in Strogatz, S. H. (1994) “Nonlinear Dynamics and Chaos.” Perseus, Cambridge, MA.
For Newton’s anagram, see page vii in Arnol’d, V. I. (1988) “Geometrical Methods in the Theory of Ordinary Differential Equations.” Springer, New York.
Chaos in the three-body problem is discussed in Peterson, I. (1993) “Newton’s Clock: Chaos in the Solar System.” W.H. Freeman, San Francisco.
For the quote about how the three-body problem made Newton’s head ache, see page 158 in Volume II of Brewster, D. (1855) “Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton.” Thomas Constable and Company, Edinburgh.
For readers who are curious about the math used here:
In the first story above, Romeo’s behavior was modeled by the differential equation dR/dt = aJ. This equation describes how Romeo’s love (represented by R) changes in the next instant (represented by dt). The amount of change (dR) is just a multiple (a) of Juliet’s current love (J) for him. This equation idealizes what we already know – that Romeo’s love goes up when Juliet loves him – by assuming something much stronger. It says that Romeo’s love increases in direct linear proportion to how much Juliet loves him. This assumption of linearity is not emotionally realistic, but it makes the subsequent analysis much easier. Juliet’s behavior, on the other hand, was modeled by the equation dJ/dt = -bR. The negative sign in front of the constant b reflects her tendency to cool off when Romeo is hot for her. Given these equations and an assumption about how the lovers felt about each other initially (R and J at time t = 0), one can use calculus to inch R and J forward, instant by instant. In this way, we can figure out how much Romeo and Juliet love (or hate) each other at any future time. For this elementary model, the equations should be familiar to students of math and physics: Romeo and Juliet behave like simple harmonic oscillators.

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* Text by Steven Strogatz, May 26, 2009

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chica milkshake


Some Comments:

* I never could understand differential equations and I never could understand women. Oh well.
It’s easy to see the ebb & flow in practice but the trick is to see where you both are on the curve and adjust for it at the right time. Each woman is different so there’s probably some other underlying wave form that dampens or reinforces the peaks and valleys.
— Johnny E

* Rather wonderful isn’t it that God has given us the ability to see the very cosmic principles that influence our lives? Being aware of them can only lead to more freedom for the human mind and spirit.
— Steve

* OK, math unpredictable is fun. Still, there are only a small set of actions that are possible given the real world constraints of actions. The same stories occur again and again with only small varions from story to story where we fill in the names and the details.
I can appreciate math and life but I sense there is still more to grasp even as I try to run variables in my mind.
Keep us thinking. I am just frustrated at the prospect there must be many variables that we can not phantom … . Somehow, if we ever where to meet other intelligent life we would better appreciate the role of math in the differences and similarities reflected in the possible.
Hard to think it all through, but I am tempted to try… .
— Mark

* There are four factors necessary for having the best chance at a love that will last:
1. Mutual attraction.
2. Mutual good-heartedness.
3. Compatibility.
4. Maturity.
Maybe that’s why lasting love is so hard to find.
— Ronald E. Maxson

* Here is a mathematician who twigged in time: “There was an important variable that I’d left out of the equations.” Mathematicians often analyze models of reality, which first require that they determine what variables to consider and what to leave out. History, especially in economics and the biosciences, shows they can sometimes get this profoundly wrong. In economics there is, from time to time, a “meltdown,” that brings economists (or at least those who trust economists) to there senses. In the biosciences the meltdowns are less obvious. Nevertheless – Bravo differential equations!
Donald R. Forsdyke, Kingston, Canada
— Donald R. Forsdyke

* Although the concept might be beyond comprehension of the ‘present day scientists/ mathematicians’, in the ‘recent times’ it is accepted that the mysterious moon – that in the three-body problem made Newton’s head ache – evolved from earth-moon itself. And, reading between lines the ‘Hindu mythology’, one would find the same phenomenon is indicated with the help of cryptic clues in the story related with Androgynous Shiva losing His ‘better half’, or consort, in a ‘sacred fire’ that was arranged by her father and, therefore, at a later stage Shiva remarrying her as the daughter of Himalaya – as she was yet another form of His original wife itself!
Thus moon virtually is indicated as a reflection/ projection in space of the energy at the centre of earth that apparently acts as a protective layer, like a bullet-proof jacket as its model, to help sustain eternally not only earth but also the entire solar system…The ancient astronomers resorted to astrology and Palmistry etc. to deal with expected human behaviour, and not simple mathematics…
— JC Joshi

* Nice introduction to differential equations, though a small error might confuse people – it should be ” The amount of change (dR) that occurs every moment (dt) is just a multiple (a) of Juliet’s current love (J) for him . “
— LJ Graham

* Attending la creme de la creme instiution of higher education Shimer College, my astute classmates came to the conclusion that, “Most people say two plus two are four. By the time we leave here, we’ll say, ‘Oh yeah, want to make a bet?’”
And so an interesting love square/pentacle of the century was predicted.
Call them Romeo and Juliet, or Hermit and Malevolent Misanthrope.
Or you could call them MICKEY and MICKEY or DONALD and DONALD. Let me regress.
I had an interesting, influential experience as a child. Likening myself as a princess, I played in tower of blue with the tough kids, who kept shouting, “Donald Duck.” We were the elite, with wings we could access the tower complete with slide. At the bottom were the angry peasants, working class, shouting MICKEY mouse. These kids were blocked off from the ladder to the slide and lacked the wings and the latitude to enforce order. They were rioting, crying with disorder, “MICKEY mouse.” Their Furies had been awoken. They felt their day of reckoning had come. I stood atop the tower, smug at my position, watching the contorted angry expressions of the children without access, I wanted to help them. I wanted to say that I like the elitist Donalds, but I like you MICKEYS too. So mass chaos ensued. Tower children shouted, “Donald Duck,” while the children below angrily insisted “MICKEY Mouse.” I was captivated by this scene. I didn’t know what I should be shouting. Even as a child, I had a sense of “Don’t be a racist, be an elitist mentality” while similtaneously wanting to help the rest rise. I suppose that is why I am still single, I am searching for that right MICKEY, Donald combo.
— Eve of Blue Tower

* I have always advised my engineering students not to draw on the equations that we use in classes in order to understand social interactions, if for no other reason than that even though there are some amusing comparisons they will probably not get a useful answer and will waste time more usefully spent on social interactions trying to do math problems.
That said, one can’t help but wonder about the three (or more) body problem that is usual at that age, and their LaGrange points (points of stability between mutually attractive large objects). Perhaps the constant tweeting and texting that occurs these days is the maintenance of some kind of metastable equilibrium in an ages-old problem.
— Scott

* “It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.”
hence: literature, one solution to describing these iterations of love, or love stories –
thanks for a fab article!
— 411bee

* If you enjoyed this piece you will also enjoy Steven Strogatz in programs 12 and 13 of Mathematics Illuminated, a new 13-part multimedia resource that explores major themes in mathematics, from mankind’s earliest study of prime numbers to the cutting edge mathematics used to consider the shape of the universe.
The entire resource is free to all at http://www.learner.org/courses/mathilluminated/.
— Scott Roberts

* I think this theory is the perfect explanation why engineers and mathematicians should feel less confident in explaining love and human relationships:). Applying a clinical equation to human interaction probably works when you calculate how many people might show to a party, but not to predict relationships.
First, you cannot assume a relationship between J&R can exist in isolation from everything else and furthermore you cannot assume you know the number of variables: what do they both want in life, how important are to them the wishes of their family, what are those wishes, new potential love interests, incompatibility and so on. Romeo and Juliet might have experienced absolute love, but they didn’t make it past their teens so there is an inherent limitation of choosing them as examples. And why would you
assume Juliet to be the fickle one?
— Ashleen Hayes

* The problem with modeling real social dynamics is (1) it’s not deterministic, (2) it’s not autonomous, and (3) it’s not as simple as a 2-variable system. But that doesn’t keep us modelers from trying!
One of my favorite social models is by Sergio Rinaldi on cycles in the love poetry of Petrarch. He manages to get rather good fits to the data – and with an autonomous, deterministic differential equation model. But it has three variables ;-)
This article is available free on JStor.
— Sharon Lubkin

* The Romeo/Juliet ambivalent tryst represents about 10% of types of romantic relationships. Since 10% is relatively infrequent, it is probably easier to predict than say romantic relationships that are based on normal attachment processes (70-80%).
There are too many variables in the love equation to try to simplify it with a math model. (Although it is a way to amuse oneself.) People marry for many reasons besides love, for example, they might be pregnant, they wanted to married by a certain age, they receive pressure from family and/or peers, sex guilt, religious influences, fear of not obtaining a better catch, etc. You think the three-body problem is bad, try adding all the other influences!
It might help to read the psychology literature on romantic relationships. Even though I minored in math at Cal, and have tender feelings for the subject, I doubt romance will be explained in math departments.
— Joe

* Interesting article: the example seems to be (inadequate) mathematics to describe social interactions. The gender demographic of readers commenting is also interesting: 7 males (8 including this response), 2 indeterminate and 1 female. Hmmm…
– Mike
— mike mittleman

* This illustrates one problem with blind faith in the power of mathematics–you don’t always know if you have identified all the relevant variables for an equation. Solving problems completely can require an exceedingly large number of variables and exceedingly complex equations. At the end of the day, even supercomputers could turn out to be fallible for lack of the correct variables. As you point out, Isaac Newton recognized this when he tried to understand the movements of only three bodies. Imagine then, trying to understand what influences the behavior of “star-crossed lovers.”
— ttj

*Per Maxson:, there are four independent variables for lasting love:
1. Mutual attraction.
2. Mutual good-heartedness.
3. Compatibility.
4. Maturity.
Maybe that’s why lasting love is so hard to find.
If each variable can only be either there or not there, then the probability of getting all four as positive is 1/16 or about 6%. So lasting love on average only occurs 6% of all couples/marriages. Is this true? Can data support this?
— Manny

* Great correlation evaluated between the hearts and maths. Better if a survey is conducted or the real time analysis is done and results are published. That adds more weight to the theory.
— Harish

* So then now knowing this and having the ability to Meta-think about it, do I then doom myself to a near impossible 2nd order differential equation?
Let me explain. Since I can now know that I might be dooming myself to a SHO off set by pi/2 or something, then I will modify my behavior. This modification, this meta thinking, then would be a 2nd order term. How about solving that one?
Oh sure, if it gets bad enough, I can use a Bessel function, but then again approximating a woman as sphereical might be counter reproductive.
— Rob

* In the arithmetic of love, one plus one equals everything,
and two minus one equals nothing.
— Dru

* Professor Strogatz,
I find it curious that you deemed it necessary to point out the “tongue in cheek” nature of your delightful excursion into the mathematics of love. But I am grateful for your perspective which moves the discussion beyond the plane geometry of “three’s company” to the far more stimulating venue of differential calculus.
I would only add that your observation regarding the collective behaviour of Romeo and Juliet as harmonic oscillators confirms my own observation over the years that families that engage in music strengthen their emotional bonds (i.e., are more harmonious).
Leo Toribio
Pittsburgh, PA