Have you ever wondered if the swirling emotions of a love affair could be understood, or even predicted, with something as precise as mathematics? It may sound like a strange idea, but what if I tell you that the passionate tango between Romeo and Juliet can be described by lines and curves, using the very same math we learn in calculus class?
Imagine our star-crossed lovers, Romeo and Juliet. Their feelings for each other are constantly changing. Sometimes Romeo loves Juliet intensely, sometimes his feelings might cool. The same goes for Juliet. How can we possibly put numbers to something so famously unpredictable? This is where the brilliant ideas of mathematician, Steven Strogatz come in. In his fascinating paper, “Love Affairs and Differential Equations,” he shows us how a powerful tool called differential equations can help us model this ever-changing relationship.
What follows is a highly simplified version of Strogatz’s paper. It may be suitable as an introduction to differential equations for senior school (Grades 11/12) students.
The Tragic Love Story: A Simple Mathematical Model
- Let’s say Romeo’s love for Juliet is represented by a number, call it ‘R’. If R is positive, he loves her. If R is negative, he dislikes her (gasp!).
- Similarly, let’s use ‘J’ for Juliet’s love for Romeo.
How does Romeo’s love change over time? And how does Juliet’s love change over time? This “change” is what calculus is all about!
Strogatz cleverly suggests that the way Romeo’s love changes don’t just depend on his own feelings, but also on Juliet’s current feelings. And vice versa for Juliet.
The “Follower” Affair
Imagine Romeo is a bit of a follower. His love for Juliet grows or shrinks depending on how much she loves him. If she’s head-over-heels, he catches feelings. If she’s indifferent or dislikes him, his love fades.
At the same time, perhaps Juliet is also a follower. Her love for Romeo changes based on his current feelings. This sounds like a recipe for a rather unstable relationship, doesn’t it? If they both only react to each other, their love might swing wildly.
We can mathematically model this ever-fluctuating relationship as follows:
This can also lead to a discussion in the classroom – where the teacher asks the students to imagine a smooth continuous curve / function that might represent the above data (i.e. the variation of R and J with time)
In turn, this may lead to the insight that a very simple model might be that of two sine waves that are out of phase.
A student’s activity with simple pendulums can also be conducted if there is sufficient time.
The teacher can mention that at different points of time, Romeo may like, or dislike Juliet and Juliet may like or dislike Romeo.
This naturally leads to a crucial question – what percentage (%) of the time would both Romeo & Juliet like each other at the same time?
Enter the Villain: Differential Equations
Using very simple differential equations, we can write something like:
- How Romeo’s love changes = (some constant) * How Juliet’s current love changes
- How Juliet’s love changes = (some constant) * How Romeo’s current love changes
Consistent with the above assumptions, let’s frame them in mathematical terms:
Solving the Equations of Love
The teacher can then lead the class through the following solution steps:
Equation (4) can be further simplified with the substitutions below. This stage is ripe for a class discussion where the teacher asks the students to guess what R vs J looks like and hence if they can now guess what % of time Romeo and Juliet, both like each other?
The answer is of course an ellipse and hence it naturally follows that Romeo and Juliet simultaneously like each other 25% of the time!
The Courtship Period
Here are some potential questions that may spark conversations to help students realize both the power limitations of modelling real life situations.
- What are some of the defects with this model? Which real life relationship scenarios do it not take into account?
- If p or q have larger values, what does that tell you about how emotional Romeo and Juliet are?
- If p or q are zero, what does it say about Romeo and Juliet’s relationship?
- What if q was also +ve? Can you plot the graph of R vs J and conclude if this situation is realistic?
Happily, Ever After
This idea is fantastic for a calculus classroom because it makes abstract concepts like “rates of change” and “modeling” incredibly relatable. Instead of just solving equations about water flowing out of a tank, students can see how math can describe something as complex and humane as love.
It shows that:
- Calculus is about change: How do things evolve over time?
- Differential equations are powerful: They let us predict what might happen in the future based on current conditions and rules.
- Math is everywhere: Even in the most poetic and emotional aspects of life, mathematical principles can be found.
So, the next time you hear about Romeo and Juliet, perhaps you’ll think not just of balconies and poisoned potions, but also of the elegant curves of differential equations, quietly mapping out the rise and fall of their legendary love. Who knew calculus could be so romantic?
By Nilanjan P Choudhury, STEM Leader, CuriousEd
