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Mechanics of love and happiness

Mike Ciavarella's picture

An interesting application of non linear dynamics, including happiness with a simple 1D damped oscillator!

Stemming from a model in the well known book by Strogatz on non/linear dynamics, where Romeo and Juliet's model was proposed as a simple two-dimensional linear continuous-time dynamical model of the love/hate relationship, JC Sprott has further described the rich dynamics of this simple model and suggest some nonlinear extensions and models of love triangles with chaotic solutions (a talk as a PowerPoint Presentation and in HTML format)
dR/dt = aR + bJ
dJ/dt = cR + dJ
where R is Romeo’s love for Juliet, J is Juliet’s love for Romeo (or hate if negative); a, b, c, d are constants that determine the “Romantic styles”.  a>0, b>0 (eager beaver). a>0, b<0 (narcissistic nerd). a<0, b>0 (cautious lover) it seems it is the best and finally a<0, b<0 (hermit).

Simplifying to only two parameters, due cautious lovers, b > 0, c > 0 leads to big love or war, whereas mixed cases b > 0, c < 0 to never-ending cycle, and two hermits unrequited love.

With Self-Awareness, and bc < 0 (nerd +lover), instead of never/ending cycle, the only equilibrium is apathy, or large volatility!  I did not get why Fire and Water, I thought it was included. Peas in a Pod, lead to no oscillations.  Romeo the Robot, can be either love or hate, but her feelings never die, and no oscillations are possible.

More interesting the Love Triangle, Anything can happen!

Combining the two equations, a related linear model for the time evolution of one's happiness in response to external stimuli (hedonics) results as a simple 1DOF linear damped oscillator. Happiness is defined as the derivative of love.

He discusses some implications for psychotherapy and for a personal philosophy of life. An important implication of the happiness model is that one cannot expect to be either exclusively happy or exclusively unhappy over long periods.  A similar response can occur with love/hate.  Also, one's subjective feelings are more volatile and often opposite to those perceived by others.

Did you expect all this mechanics applied to love and happiness?  Be more creative!


Mike Ciavarella's picture

the code can be found at


Mike Ciavarella's picture

Consider a love triangle in which Romeo and Juliet love each other, but Romeo also loves his mistress Guinevere and she loves him. See 


Mike Ciavarella's picture

From the love/hate model of R&J, the derivation of Dynamical Models of Happiness, is in short this.

dR/dt = aR + bJ
dJ/dt = cR + dJ

one can eliminate an eqt for Romeo's love alone
d2R/dt2 + beta dR/dt + omega2 R = 0

beta = -a - d  (damping)
omega2 = ad - bc  (frequency)

so notice that damping, which clearly is one of the key parameter,
depends only on the self/awareness factors, and not sensitivity to
partner. Manic depression for example may correspond to beta ~ 0.

H (your happiness)=dR/dt must average to zero (with positive damping)

But remember beta=-a-d, so generally speaking damping is high
with cautious lover and hermits, but not so for eager beavers
or narcissistic nerds which may have negative damping, i.e. have
either happiness very high or very low, forever!


1. The simplest model supposes that over a sufficiently long time,
people tend to acclimate to their circumstances and thus experience
equal amounts of happiness and unhappiness. Therefore, constant
happi-ness is an unrealistic and unobtainable goal.
2. Suicide is an irrational response to unhappiness, similar to
bailing out of the stock market at the bottom. The model predicts that
if you wait long enough, happiness will return, even if only by virtue
of acclimating whatever is causing the unhappiness.
3. Others tend to perceive less volatility in you than you feel since
they observe primarily your responses to external events (R) rather
than your true feelings (H), and therefore they often wrongly conclude
how happy you are.
4. Since individuals will tend to acclimate to their circumstances,
long prison terms may be ineffective if the goal is punishment rather
than deterrence or protection of society.
5. Individuals may be characterized by two parameters β (how rapidly
they return to equilibrium after a perturbing event) and ω (the
frequency with which their feelings change). These parameters may be
of diagnostic use in describing psychological health.
6. Healthy individuals may have β ~ 2ω, corresponding to critical
damping, while people with β < 0 exhibit bipolar behavior. A strongly
overdamped person (β » 2 ω) may be largely devoid of emotions and
unaffected by events, whether good or bad.
7. A reasonable goal of psychotherapy might be to alter the parameters
β and ω so that the patient responds to external events in a more
healthy manner. It is an open question the extent to which such
therapy is effective, since the parameters may be largely fixed by
one’s personality.

How to do this is left to the psychotherapists, I guess mechanics cannot do much more!  :) 

Mike Ciavarella's picture

The Mathematics of Marriage: Dynamic Nonlinear Models (Bradford Books)

The book does not rely on metaphors, but develops and applies a mathematical model using difference equations. The work is the fulfillment of the goal to build a mathematical framework for the general system theory of families first suggested by Ludwig Von Bertalanffy in the 1960s.The book also presents a complete introduction to the mathematics involved in theory building and testing, and details the development of experiments and models. In one "marriage experiment," for example, the authors explored the effects of lowering or raising a couple?s heart rates. Armed with their mathematical model, they were able to do real experiments to determine which processes were affected by their interventions.Applying ideas such as phase space, null clines, influence functions, inertia, and uninfluenced and influenced stable steady states (attractors), the authors show how other researchers can use the methods to weigh their own data with positive and negative weights. While the focus is on modeling marriage, the techniques can be applied to other types of psychological phenomena as well.

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