Sunday, January 24, 2010

LOVE AFFAIRS and DIFFERENTIAL EQUATIONS

From Albert Einstein's notebooks, on a sweet spring day in the early days of The Institute for Advanced Study at Fine Hall in Princeton once upon a time, on his attempt to solve The Three-body Problem:*


* - 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

Rdot=0

Jdot=0

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":




34 comments:

Plato said...

Good one Steven,

The restricted three-body problem assumes that the mass of one of the bodies is negligible; the circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun-Earth-Moon system and many others). For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe; for another stable system, see Lagrangian point.


It's that third party that can definitely throw a tumultuous variable when the third party was present in thought, but, I have to warn you about something Einstein said.

In mathematics we call this a three-body problem. It’s notoriously intractable, especially in the astronomical context where it first arose. After Newton solved the differential equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun), he turned his attention to the three-body problem for the sun, earth and moon. He couldn’t solve it, and neither could anyone else. It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.

----------------------

Einstein:Gravity cannot be held responsible for people falling in love." :)

Best,

Plato said...

These quasi-periodic Lissajous orbits are what most of Lagrangian point missions to date have used. Although they are not perfectly stable, a relatively modest effort at station keeping can allow a spacecraft to stay in a desired Lissajous orbit for an extended period of time. It also turns out that, at least in the case of Sun–Earth L1 missions, it is actually preferable to place the spacecraft in a large amplitude (100,000–200,000 km/62,000–120,000 mi) Lissajous orbit, instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct Sun–Earth line, thereby reducing the impact of solar interference on the Earth–spacecraft communications links. Another interesting and useful property of the collinear Lagrangian points and their associated Lissajous orbits is that they serve as "gateways" to control the chaotic trajectories of the Interplanetary Transport Network.

Steven Colyer said...

Yes, the 3-body problem is fascinating, and the source of our understanding of Chaos in the first place. Good old Henri Poincaré the Polymath, what DIDn't that man do?

Of course then there's the 4-body problem. If we call the bodies in question Hermann Weyl, Anny Schrödinger, Erwin Schrödinger, and Anonymous Woman, then what gets spit out is ... the Wavefunction! lol

Seriously, I am expanding the outline of Nonlinear Dynamics on the page previous. This whole "Love" stuff, while fascinating, is an example of Linear Dynamics, which is why I put it up first AND because ... it's so amusing. :-)

Andrew Thomas said...

I think the behaviour of a woman is more accurately plotted as a chaotic system rather than a system of linear differential equations.

Steven Colyer said...

I think the behaviour of a woman is more accurately plotted as a chaotic system rather than a system of linear differential equations.

Indeed you are quite correct, except the subject is Linear, not NONlinear ... Dynamics.

Before one understands NONlinear one must start with that which NONlinear is actually the "NON", of.

As a fellow Engineer, I suspect you know the difference between powers in a differential equation, that is to say if the highest power is one we are talking about Linear equations; if higher, the Nonlinear.

I just mention that because not everyone knows what we're talking about. I'll get into RC vs RLC circuits soon enough, trust me mate.

Neil B said...

Some people make fun of reducing "love" to equations as many social scientists do, but economists trying to model our complex market behavior as almost as overconfident in their ability to represent thereby.

BTW all, Orzel finally admitted my claimed formal result was correct, but says it doesn't matter to QM measurement issue. Oh yes it does, since the common claim is that decoherence produces a "mixture", and there's "no way to tell the difference" between that and real superpositions. Uh, way! That was my whole point ...

Steven Colyer said...

OK, Neil, that will be the first and LAST time you bring up your particular theory of Interpretation of Quantum Mechanics "Falsification of Decoherence" HERE.

You're in MY house now, man.

I'm open-minded, but I have my limits, and I reserve the right to delete any further expression of your theory ala Woit/Motl in the future, which I would strongly prefer NOT to do so don't tempt me.

In short, you have your own website, and Plato's and Chad Orzel's et. al., on which to pontificate. Please don't do so here. Respect me man that I even have tyrannogenius up as a site I explore ... please don't abuse that.

Having said that ... you're more than welcome to stay and comment as long as your comments are constrained to the topic at hand, which in this case is ...

... Love as a Linear v Nonlinear dynamic.

Neil B said...

OK, Steven, but remember that you and Phil were asking for popcorn and such about it, at my blog - so it's like a business contact calling you back, not a "telemarketer" out of the blue. But sure, not in an unrelated thread.

Steven Colyer said...

I should really delete that comment, as it added nothing to what I'm about, and probably will.

In the meantime ... latitude. What do you have to say about this blog article, in question?

Neil B said...

If you mean the current topic, I could repeat here again since it needed editing anyway:
Some people make fun of reducing "love" to equations as many social scientists do, but economists trying to model our complex market behavior with math are almost as overconfident in their ability to represent it thereby.

Visit my blog anytime and I appreciate your interest as Follower.

Plato said...

I am expanding the outline of Nonlinear Dynamics on the page previous. This whole "Love" stuff, while fascinating, is an example of Linear Dynamics, which is why I put it up first AND because ... it's so amusing

Sign of a engineer's meta view?:)

Engineers often use the term nonlinear to refer to irrational or erratic behavior, with the implication that the person who "goes nonlinear" is on the edge of losing control or even having a nervous breakdown.

So love can seem like that?:)Just kidding...as I see you are trying to get a handle on it.

Best,

Steven Colyer said...

Engineers often use the term nonlinear to refer to irrational or erratic behavior, with the implication that the person who "goes nonlinear" is on the edge of losing control or even having a nervous breakdown."

Wow, where did THAT expression come from, Plato? I certainly never said that, besides the fact that it is completely untrue.

Please enlighten.

Plato said...
This comment has been removed by the author.
Plato said...

Hi Steven,

Sorry I couldn't get back to you sooner.

I got that quotation from the nonlinear link at wikipedia in above post to yours.

When I said meta I was referring to that link and the subject title of this section of nonlinear "metaphorically explained" as.

Nothing personal to you other then a large statement taken overall to what must have been written by a engineer?

Untrue then it is Steven.

Best,

Andrew Thomas said...

Hi Steven, referring to your answer to my previous comment, it's possible to have higher-order (2, 3, etc) derivatives in a linear differential equation - it's how those terms are combined decides if its nonlinear or not.

Wikipedia describes this at the start here.

"The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y."

(T'Pau?? Really??)

Steven Colyer said...

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and nonlinear otherwise from here

@Neil - we're cool, but wavefunction collapse isn't my thing. I question the What? moreso than the How? or Why? Meaning, I question wavefunctions collapsing as a valid question. The How? and Why? are fascinating and important, yes, but the What? comes first, and I don't think the proper one has been asked, and no I don't have a clue as to what the proper What? actually is. Just not interested, BUT, I'm glad to see others are on the job. Seems more like Philosophy than Physics to me.

Plato, thanks, yeah slang term. It's kind of funny now that I think of it, but I've never heard it used, personally.

Andrew, yes, T'Pau. I just got hooked on that video back in the day, 1987 I think it was. Seemed like the perfect song to me. And the girls!

Andrew Thomas said...
This comment has been removed by the author.
Andrew Thomas said...

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed)

Hi Steven, yes, I think that means you cannot have squares or cubics or high powers of the derivatives,
for example you couldn't have:

(dy/dt)^2

or:

(d^2y/dt^2)^2

so you couldn't have anything squared. However, that's not the same as a higher-order derivative, so in a linear equation you can have:

dy/dt

or:

d^2y/dt^2

There's nothing squared in either of those - the second one is just a second-order derivative. As it says here: "The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y." Yes?

Andrew Thomas said...

I don't think it's really necessary to get bogged-down in the maths, though. I think the general principles of all this is what is really important.

Plato said...

Andrew,

Can you give classical examples of Linear and Nonlinear. Steven mentions a derivation of, as the three body problem, while communication methods in computerization might all be thought of as such?

This might help toward general principles and it's applications.

Best,

Andrew Thomas said...

If Steven will excuse me, this is my take on it.

Well, a linear system is usually a very simple system where the output varies linearly (obviously) with the input. For example, the output of an audio amplifier should increase linearly as the input increases linearly. There's a few examples here.

The nice thing about linear systems is that there is plenty of mathematical systems theory which allows us to analyse them - like the impulse response described in that link (whereas there is no such maths available for nonlinear systems). The definition of a linear function f() is:

f(a+b) = f(a) + f(b)

So we can simply find the output of the system as the function applied to the sum of the inputs. If the weather was linear, for example, instead of nonlinear it would be very easy to predict it: we could just model the behaviour of a single raindrop and then find the behaviour of a storm consisting of a million raindrops as:

one million x one raindrop.

So that would be a really easy way to describe a storm. But in reality the storm system cannot be described by a simple nonlinear formula - the millions of interactions are too complex.

So to sum up, systems designers (of amplifiers, for example) love linear systems as they have loads of math techniques to describe them. However, most of the universe is nonlinear, and we don't have the maths to analyse it (in fact, it's probably not possible to produce a simple analysis). The only solution is to get a super-computer and consider each one of the millions of interactions separately, which is basically what they have to do to predict the weather.

Mathematicians get annoyed by nonlinearity as you can't analyse it with a pencil and paper - you need a supercomputer. So nonlinearity puts them out of a job. But like I say, the universe is basically one huge nonlinear computer, so nonlinearity is really the only maths which counts.

Andrew Thomas said...

Oops, should have said "the storm system cannot be described by a simple *linear* formula.

Andrew Thomas said...

With all the fuss about this Verlinde paper recently, it has been attacked for having "high school maths". But, of course, as we get closer to describing the fundamentals of the universe we should indeed expect the maths to get simpler - not more difficult. Because at the most fundamental level we are really dealing with the simplest of intereactions - and this idea of entropic gravity is a wonderfully simple idea. The complexity of the universe is produced by the emergent effect of all these billions of simple interations which are operating in a nonlinear fashion. So it's the magic of nonlinearity which produces our wonderful universe.

Thanks, Steven.

Neil B said...
This comment has been removed by the author.
Neil B said...

Wow, I'm glad someone might be providing new insight with "high school maths." But even as Andrew thinks math should get simpler as we get closer to fundamentals, many do not agree - and are in fact quite prejudiced against simple math. I get such flak about my paradoxes and such (OK, Steven? ;-) QM does use linear equations which are simple at heart, or I'd have trouble arguing about it.

However, if string theorists are right, very hard math is needed at the level of getting particles and fields off the ground. Maybe it's a mixed bag.

Heh, so "going nonlinear" is the nerd's version of "going postal"?

Steven Colyer said...

Yeah lol. I guess I'm not nerdy enough to have used it, or just out of the loop, I guess.

Yes, Verlinde's paper and his theory, re-christened Jacobson/Verlinde, is under savage attack. This is nothing new in Physics. These guys are ultra-competitive weenies at times. Every day is their own personal Super Bowl (go Colts!) or World Cup finals. It can get quite annoying unless one chooses to look at it as entertainment. That's my choice ... one that's kept me relatively sane about this stuff.

Take Lubos Motl, for example. I knew it was too good to be true that he would have a moderate opinion and be open-minded for a change. His attack on Verlinde is typical Lubosian pit-bullery. Essentially, if verified then Verlinde's work would negate huge tracts of String Theory which is Motl's holy grail.

Lubos must have originally been thrown for a loop (no pun intended) as Verlinde was a noted String Theorist. Day-am, people are leaving ST in droves these days. Get out of Dodge while the getting is good? Shrug.

Gratuitous Linear v Nonlinear section, 2 parts

Phew, it's nice to return to Calculus again, rusty though I may be atm. Calculus 1 is the single most important course on Earth. ALL of Science uses Mathematics, and Calc 1 is the Fulcrum of the seesaw of Mathematical knowledge. I loved it.

Then along came Calc II and Calc III, which bored me to tears. I was about to throw in the towel, when along came ...

Calculus IV ! Differential Equations! Great! Good stuff. My strongest memory of those years was when the professor said (in 1976) ".. but you know, solving second-order nonlinear differential equations is quite difficult, requiring great intuition. Many feel computers are required, although their answers are only approximate."

Wow, I thought, do you mean there's a "war" in Mathematics-land, between Math and CompSci ? Awesome. Give me some algorithms, or give me death! lol

End part 1

Steven Colyer said...

Andrew, let's not forget that sin x and cos x etc. are also nonlinear. For most functions one can just keep taking derivatives to make them linear, but we must be careful to know when to stop, otherwise we get a constant and then 0. :-)

For example, and from Strogatz' book:

The swinging of a pendulum is governed by the equation:

x(double-dot) + (g/L)sin x = 0

where x is the angle of the pendulum from vertical, x(double-dot) is the second derivative of x, g is the acceleration due to gravity, and L is the length of the pendulum. The equivalent system is nonlinear:

x(sub=1)dot = x(sub=2)

x(sub=2)dot = - (g/L) sin x(sub=1)

Non-linearity makes the pendulum equation very difficult to solve analytically. The usual way around this is to fudge, by invoking the small angle approximation sin x is approximately equal to x for x << 1. This converts the problem to a linear one, which can then be solved easily. But by restricting to small x, we're throwing out some of the physics, like motions where the pendulum whirls over the top. Is it really necessary to make such drastic approximations?

It turns out the pendulum equation can be solved analytically in terms of elliptic functions. But there ought to be an easier way. After all, the motion of the pendulum is simple: at low energy, it swings back and forth, and at high energy it whirls over the top. There should be some way of extracting this information from the system directly. This is the sort of problem we'll learn how to solve, using geometric methods.

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