The Dynamics of Love

Some physicists like to believe that physics is the science of everything, even hoping to find the ultimate reason – (a theory called, rather arrogantly, the “Theory of Everything”) which attempts to describe why nature is the way it is.

Right from the astronomical scale of galactic superclusters, to the miniscule fundamental constituents of atoms, physics can be extended even beyond the limits of matter to complex and unpredictable systems such as the stock market, biology, and even the weather.

The maths used to describe the laws of nature are differential equations. Here at thefullapple, we show you how they can be used to analyse the most confusing and unpredictable thing of all, love – and some of our results are shockingly apt, showing how powerful results and intricate patterns can emerge from even the simplest equations.

First things first – simplifying assumptions

The ancient Greeks had more than 10 words for experiences that we might call love. There’s no denying that relationships are complicated and difficult to describe, and in order to start quantifying things like affection, love, lust or just plain infatuation, we need to make a couple of simplifying assumptions.

As a common first step in mathematical modelling while unravelling the mysteries of complex systems, physicists use simplified models to capture the most important features of the behaviour. The point is that even such highly simplified models can often give correct predictions of the qualitative behaviour, so even if they don’t tell us the exact numbers, we get a feel for approximately how a system behaves, which is often the really interesting part.

Part I: A never ending chase

In order to try and model different scenarios, we have chosen a “test couple”, Romeo and Juliet. In our first scenario, Romeo is a simple man. His feelings exactly reflect those of Juliet, he’ll do anything for her when she shows that she’s into him but gets discouraged and feels like giving up when she’s cold or mean.

Juliet, on the other hand, is a lot more complicated – she enjoys men who play hard to get.  The more Romeo shows his love for her, the more she retracts herself. But when Romeo gets discouraged and backs away, it suddenly makes him all the more attractive for Juliet.

We can put everything we’ve just said into mathematical equations, which will then tell us what happens to our couple:

romeopics.004

In our formulae, R is Romeo’s love for Juliet and J is Juliet’s love for him. When R is positive, Romeo loves Juliet, but when it turns negative, Romeo’s love has turned into negative love… or hate. If it’s zero, then he just doesn’t care. The same goes for Juliet.

The human mind, however is known to be fickle. We need to incorporate the fact that our protagonists can change their minds about each other in our system. The way to express changes mathematically is by calculating the rate of change of a quantity. For example, speed is the rate of change of position.

In our equations, the rate of change of Romeo’s love (denoted by R  with a dot on top) depends on Juliet’s love for him – like we said, Romeo’s feelings reflect those of Juliet. Juliet’s feelings however always evolve oppositely to Romeo’s, which is why there is a minus sign in the second equation.

These equations can be solved using calculus to predict the course of the love affair. We’ve done this for you guys and summarised all the results in some simple pictures:

romeopics.002

When Romeo and Juliet first meet, they will start of at a specific point in the diagram, depending on how they initially feel about each other. The black arrowed lines then show how their feelings will evolve from that point onwards.

Our forecast in this case is a tragic infinite cycle of love and hate. No matter what their initial impression of each other, they will invariably end up on a love-carousel going round and round, and only 25% of the time they will be happily in love with each other (whenever their feelings are in the green part of the diagram).

This makes a lot of sense, if Romeo will only love when he’s loved, and Juliet will only love when she feels she’s not loved, we land ourselves in a bit of a conundrum. So far so good, our equations are holding up.

Part II: Cautious lovers

We can make our model more realistic by also letting Romeo and Juliet respond to their own feelings. For cautious lovers, this means they try to avoid throwing themselves at the other. In maths speak this means that when their love for the other increases, their rate of falling in love decreases, after all nobody wants to appear too eager!

romeopics.005

In these equations, we have quantified how cautious they are using the cautiousness c, and also how excited they get by the other’s advances by their responsiveness r.

As you can imagine, solving this takes even more scrap paper than the simple ones above, so we’ll again stick to nice pictures. This time, we get two different pictures depending on the relative size of their cautiousness and their responsiveness:

romeopics.003

The diagrams we get are slightly more complicated, but their message is simple: If the lovers are too cautious about their feelings, they will end up indifferent about each other – 100% of the time. Again, an astonishingly insightful result from equations with just four variables. This is shown in the left diagram, where all the arrows go to the point R = J = 0, and there is no hope for any form of love story … leaving Shakespeare with nothing to write about.

If however, the lovers’ responsiveness to the other person dominates over their cautiousness, we find some exciting movie material. Depending on the chemistry of their first encounter, the whole thing either turns into an explosive love relationship or into hateful war.

Let’s look more closely at what happens when they first meet in this scenario. If their initial feelings for each other lie anywhere in the green area of the diagram, they will invariably fall in love. This includes parts of the diagram where Romeo already likes Juliet, but she is just not interested in him, even dislikes him. But, as our mathematical solutions show, she just won’t be able to resist his charms in the long run. Interestingly, starting off in this area, they will not only fall in love, but their feelings tend to the line of exactly equal affection at 45°, where R = J

If however at least one of the two has strong resentment for the other (red area), the whole thing will tend the other way into wild hatred.

Love triangles lead to chaos

Despite the ludicrous simplicity of our Romeo-and-Juliet model, we’ve seen some pretty interesting behaviour, some of which you may identify with. If we want to make this even closer to real life, we could introduce non-linear terms (preventing unbounded love and hate) or external and unpredictable factors (such as some bad drunk decisions, a new job in a different country, parent related drama and so on…)

The most dramatic change would be to introduce more than two protagonists to the game. If for example we add Juliet’s ex-boyfriend into the equation, we are left with the infamous three-body problem. Funnily, this problem remains unsolved to this day. A few centuries ago, Newton complained to his friend that it “made his head ache, and kept him awake so often, that he would think of it no more.” Newton was not the first and nor the last guy to have sleepless nights thanks to a pesky ex. In fact, nowadays we know that any dynamical problem with more that two protagonists will lead to chaotic dynamics – meaning that the outcome is uncertain and different every time. Ladies and Gents, let exes stay exes.

Such models can be immensely powerful and offer incredible insights. We shouldn’t underestimate the power of mathematical equations – physicists are attempting to use models to predict climate change, modelling cell suicide or even the dynamics of olympic success.

In the end, love will forever remain a problem of infinite complexity and not even the most complicated physics model will be able to help you – like Einstein said: “Gravity cannot be held responsible for people falling in love.”

Resources

  1. This article is based an idea proposed by Steven Strogatz in this paper.
  2. More calculations can be found here.
  3. More about the solution of linear systems using phase-space diagrams can be found in chapter 5 of the book “Nonlinear Dynamics and Chaos” by Steven Strogatz.

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