08-23-2026, 08:29 PM
Differential Equations of Love and Love of Differential Equations
Author: Isaac Elishakoff
Journal:Journal of Humanistic Mathematics, Vol. 9, No. 2
Publication: July 2019, pp. 226–246
Summary
Isaac Elishakoff uses the familiar story of Romeo and Juliet to demonstrate how very simple systems of ordinary differential equations can model interactions between two people. The idea follows the classical mathematical treatment introduced by Steven Strogatz: let $R(t)$ represent Romeo's feelings toward Juliet and $J(t)$ Juliet's feelings toward Romeo, with positive values representing love and negative values representing dislike. A general linear model can be written as
$\frac{dR}{dt}=aR+bJ$
$\frac{dJ}{dt}=cR+dJ$
where the coefficients describe how each person's feelings respond both to their own current emotions and to those of the other person. Depending on the signs and magnitudes of these coefficients, the relationship can converge toward mutual affection or indifference, grow without bound, or repeatedly alternate between love and hate. In one particularly instructive configuration, one person's affection increases when loved while the other's decreases, producing oscillatory behaviour—an endless mathematical cycle of attraction and rejection.
Elishakoff's purpose is primarily pedagogical rather than psychological. The romantic metaphor makes concepts from differential equations—coupled systems, equilibrium, oscillations, eigenvalue behaviour, and stability—more intuitive and memorable. He argues that examples of this kind could be incorporated into engineering mathematics courses to increase students' interest in differential equations. The important point is not that equations can genuinely predict romantic relationships, but that the same mathematical structure can appear in situations that look completely unrelated. The article thus illustrates the broader modelling principle that variables may have very different interpretations while obeying mathematically identical dynamical laws.
The most striking part of the paper reverses the metaphor: instead of using mechanics to explain love, Elishakoff uses love and hate to interpret mechanical vibration. A simple undamped one-degree-of-freedom oscillator satisfies
$m\frac{d^2x}{dt^2}+kx=0$,
or
$\frac{d^2x}{dt^2}+\omega^2x=0$,
where $\omega^2=k/m$. Introducing velocity $v=\frac{dx}{dt}$ converts this into two coupled first-order equations,
$\frac{dx}{dt}=v$
$\frac{dv}{dt}=-\omega^2x$.
The variables $x$ and $v$ can therefore be viewed metaphorically as a pair whose responses to one another generate a perpetual oscillation. For a stable mechanical system the solutions are trigonometric, such as $\sin(\omega t)$ and $\cos(\omega t)$. When the effective stiffness changes sign, however, the equation becomes
$\frac{d^2x}{dt^2}-\lambda^2x=0$,
whose solutions involve exponential or hyperbolic functions such as $\cosh(\lambda t)$ and $\sinh(\lambda t)$. The bounded oscillation is replaced by rapidly growing motion—mechanical dynamic instability. Elishakoff memorably describes this as a transition from “trigonometric love” to “hyperbolic love.”
Key takeaways
ARTICLE [PDF]
Author: Isaac Elishakoff
Journal:Journal of Humanistic Mathematics, Vol. 9, No. 2
Publication: July 2019, pp. 226–246
Summary
Isaac Elishakoff uses the familiar story of Romeo and Juliet to demonstrate how very simple systems of ordinary differential equations can model interactions between two people. The idea follows the classical mathematical treatment introduced by Steven Strogatz: let $R(t)$ represent Romeo's feelings toward Juliet and $J(t)$ Juliet's feelings toward Romeo, with positive values representing love and negative values representing dislike. A general linear model can be written as
$\frac{dR}{dt}=aR+bJ$
$\frac{dJ}{dt}=cR+dJ$
where the coefficients describe how each person's feelings respond both to their own current emotions and to those of the other person. Depending on the signs and magnitudes of these coefficients, the relationship can converge toward mutual affection or indifference, grow without bound, or repeatedly alternate between love and hate. In one particularly instructive configuration, one person's affection increases when loved while the other's decreases, producing oscillatory behaviour—an endless mathematical cycle of attraction and rejection.
Elishakoff's purpose is primarily pedagogical rather than psychological. The romantic metaphor makes concepts from differential equations—coupled systems, equilibrium, oscillations, eigenvalue behaviour, and stability—more intuitive and memorable. He argues that examples of this kind could be incorporated into engineering mathematics courses to increase students' interest in differential equations. The important point is not that equations can genuinely predict romantic relationships, but that the same mathematical structure can appear in situations that look completely unrelated. The article thus illustrates the broader modelling principle that variables may have very different interpretations while obeying mathematically identical dynamical laws.
The most striking part of the paper reverses the metaphor: instead of using mechanics to explain love, Elishakoff uses love and hate to interpret mechanical vibration. A simple undamped one-degree-of-freedom oscillator satisfies
$m\frac{d^2x}{dt^2}+kx=0$,
or
$\frac{d^2x}{dt^2}+\omega^2x=0$,
where $\omega^2=k/m$. Introducing velocity $v=\frac{dx}{dt}$ converts this into two coupled first-order equations,
$\frac{dx}{dt}=v$
$\frac{dv}{dt}=-\omega^2x$.
The variables $x$ and $v$ can therefore be viewed metaphorically as a pair whose responses to one another generate a perpetual oscillation. For a stable mechanical system the solutions are trigonometric, such as $\sin(\omega t)$ and $\cos(\omega t)$. When the effective stiffness changes sign, however, the equation becomes
$\frac{d^2x}{dt^2}-\lambda^2x=0$,
whose solutions involve exponential or hyperbolic functions such as $\cosh(\lambda t)$ and $\sinh(\lambda t)$. The bounded oscillation is replaced by rapidly growing motion—mechanical dynamic instability. Elishakoff memorably describes this as a transition from “trigonometric love” to “hyperbolic love.”
Key takeaways
- Love serves as a teaching metaphor for dynamical systems: coupled differential equations can represent how two quantities continuously influence one another.
- Different parameter choices produce equilibrium, growth, decay, or periodic love–hate cycles, illustrating stability and qualitative behaviour of differential equations.
- The mathematics is intentionally playful; the paper's central goal is to make ordinary differential equations more engaging for engineering students, not to claim that human relationships are literally predictable.
- The same mathematical structure describes a mechanical oscillator: stable motion corresponds to trigonometric oscillation, while instability produces hyperbolic/exponential growth, showing how one mathematical model can connect humanistic metaphor with engineering mechanics.
ARTICLE [PDF]
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