* - OK, I lied, sorry. That is NOT from Einstein's notebooks. I don't know where it's from, but it's funny. :-)
No need to explore three bodies for the moment though. The two body problem is easier, is linear, and here's a sweet application by Strogatz:
First, click here to see a New York Times guest columnist piece by Stephen Strogatz on Romeo-and-Juliet Mathematics. The replies are pretty funny. :-)
Click here to see Strogatz' original 1988 piece on the topic.
Modeling of Love-Hate relationships (from here)
See: Strogatz S.H., Love Affairs and differential equations, Math. Magazine, 61,35,1988.
Lets imagine a Romeo (R) and Juliet (J) "troubled" romance, where:
R(t)=Romeo's Love/Hate for Juliet at time t
J(t)= Juliet's Love/Hate for Romeo at time t
with positive values signifying love and negative values hate.
A first order system of equations to model the evolution in time of the relationship can be written as (Rdot = dR/dt = rate of change of R, and similarly for Jdot):
Rdot = a R + b J
Jdot = c R + d J
where a,b,c,d are parameters which can be positive, negative or zero, with the following "meanings":
a and d: "cautiousness" (throw towards (if a,d>0) the other or avoid (if a,d<0) the other)
b and c: "responsiveness" (degree at which they react to the other's advances)
For instance a case where Romeo has both a>0 and b>0 can be called an "eager beaver" (he gets excited by Juliet's love and is further excited by his own feelings into a "snowball of affection").
But if a<0 and b>0 ("cautious lover"), it means that the more Romeo loves Juliet (R>0), the more he wants to "run away" from her (Rdot more negative, particularly acute near marriage decisions...); and the more he hates her (R<0) the more he increases his love (Rdot more positive, nothing like distance to inflame his fellings).
If a<0 and b<0 ("cautious and unresponsive") usually not a good chance for romance, "lets just be friends" type...
If a>0 and b<0 ("daring but unresponsive") is more the "narcisist" type...
Typical issues of these "dynamical love systems" is which relationships are "viable"...
Notice that the "fixed points", that is where the system will stabilize would be given by
that is :
a R + b J =0
c R + d J =0
which is a system of two algebraic equations with two unknowns (R and J).
Let's analyze some special cases:
1) Two identical cautious lovers: a=d<0 , b=c>0
Then det=ad-bc=a2 -b2 , and the solutions behave in the following way:
i) If a2>b2 the lovers are more cautious than "responsive" and the relationship "fizzles out " to mutual indifference R=J=0 (caution leads to apathy)
ii) If a2
2) Out of touch with their own feelings: a=0, d=0 (lovers only react to the others feelings)
Then det=ad-bc=-bc, and the equations are:
Rdot = b J
Jdot = c R
Find out what happens!
3) Do opposites attract? Analyze d=-a, c=-b.
4) Do identical lovers make for good couples? d=a, c=b
5) Analyze your own "made up" case of interest!
Here's what interests me, the Eurorock band "T'Pau" performing their 1987 hit, "Heart and Soul":