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.
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":
I. Dynamics - HISTORY II. Dynamics - Mathematics - forthcoming, see Calculus IV in the meantime III. Linear Dynamics - Applications in Physics IV. Nonlinear Dynamics - Applications in Physics V. Nonlinear Dynamics - Applications in Other Fields VI. Personal History
III. Linear Dynamics - Applications in Physics A. N= 1 variable (Growth, Decay, or Equilibrium) 1. Exponential Growth 2. RC Circuit 3. Radioactive decay
B. N=2 variables (Oscillations)
1. Linear oscillator 2. Mass and spring 3. RLC Circuit 4. 2-body problem (Kepler, Newton)
C. N>=3 variables
1. Civil engineering, structures 2. Electrical Engineering D. N >>> 1 variables (Collective phenomena)
1. Nonlinear waves (shocks, solitons) 2. Plasmas 3. Earthquakes 4. General Relativity (Einstein) 5. Quantum field theory 6. Reaction-diffusion, biological and chemical waves 7. Fibrillation 8. Epilepsy 9. Turbulent fluids (Navier-Stokes) 42. Life
V. Nonlinear Dynamics - Applications in Other Fields - forthcoming
Click here to see a highly interactive version of Brian Castellani's Complexity Map, as shown below:
VI. Personal History
I'm as happy as a Philosophy Grad Student at the beginning of his first lecture of his first teaching job teaching "Introduction to Plato". *
Why?
Because, I've finally found my specialty, thanks to United Parcel Service delivering the following book from Amazon to my doorstep yesterday:
By the way, I really hate, loathe and despise the term "Chaos Theory." The proper description would be "Structure-in-Chaos" Theory. It's young, it's happening, it's taking off in many different fields, and the high-speed computers of Computer Scientists and their wonderful Algorithms are its very best friend. We have miles to go before we sleep. Time to get cracking! :-)
From Wikipedia, at which an input of "Nonlinear Dynamics" directs to "Nonlinear differential equations" under "Nonlinear systems." Under that section it states:
A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics, the Lotka–Volterra equations in biology, and the Black–Scholes PDE in finance.
One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.
Going to the top of the page:
In mathematics, a nonlinear system is a system which is not linear, that is, a system which does not satisfy the superposition principle, or whose output is not proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system of multiple variables.
Nonlinear problems are of interest to physicists and mathematicians because most physical systems are inherently nonlinear in nature. Nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos. The weather is famously nonlinear, where simple changes in one part of the system produce complex effects throughout.
And that's it for today. For all my regular readers (all 4 of you ... it would be 5 but Mom passed away in 2008) I'm afraid I will spend less time on-line and at this blog, as I have much to read. I won't go away completely, but for the most part I'll be incognito. Cheers and farewell, and here's hoping I do Mom proud, wherever she is, when I publish my first paper in 6 months to 3 years, or so.
* - most of whom start off with: "I am SO ENVIOUS of you people! You are about to hear about Plato for the FIRST time!" They have their point.
Sincerely,
S'Colyer
P.S. For your viewing and listening pleasure (subjective), a VERY non-linear song:
Here is the Number One most popular song in America today, Empire State of Mind by Jay-Z and Alicia Keys. Keys' bits are beautifully linear, Jay-Z's nonlinear. Somehow, they blend well:
"Hammock Physicist" Johannes Koelman has made a bold statement at his blog regarding the title of this blarticle, which I strongly recommend we read then examine by clicking here. Read the replies as well, as they're also interesting and no less important.
Essentially, the bulk of Koelman's work is as follows, and in his words from his article:
I previously posed the question “can dark energy, just like gravity, be understood as an entropic effect?”.
To my astonishment, it appears that a quick 10 minute exercise in determining the entropic force exerted on the entire observable universe indeed yields an effect with the right order of magnitude to explain the cosmic dark energy (or cosmic acceleration). It seems that a dark energy effect emerges from Verlinde's holographic description. All 123 orders of magnitude of the dark energy mismatch evaporate when considering the cosmic acceleration as a result of a holographic entropic force.
The line of reasoning is as follows (for ease of notation I will work in natural units and leave out factors c, G and h-bar):
1. Consider the cosmic horizon (the edge of the observable universe: a sphere with radius R approximately equal to 2.7 10^61 in natural units)
2. According to the holographic principle the observable universe is encoded in N = pi R^2 bits located at the cosmic horizon,
3. A finite temperature T is associated with this horizon. This temperature is determined by an equipartition of the energy Mc^2 contained within the horizon over the bits (degrees of freedom) associated with the horizon. Here M = 1.4 10^60 is the observable mass of the universe. Using the equipartition expression ½kT = M/piR^2, it follows that kT = 3 10^-64.
4. An entropic force F = kT grad(N) = kT grad(pi R^2) =2 pi kT R is associated with the horizon. This results in a cosmic expansion (1/R) d^2 R/dt^2 = F/MR = 2 pi kT / M = 1.3 10^-123. Presto!
--- Postscript ---
Two critical notes to the above speculative derivation need to be mentioned here:
1. As I set out to explain more than a hundred orders of magnitude mismatch, I have not bothered myself with numerical factors of order unity. As a result,the end result could be off by a factor of two or so.
2. More importantly, a full evaluation of the above simple expansion model yields an expansion (1/R)d^2R/dt^2 = (2c/R)^2. Whilst this expression yields the right order of magnitude for the expansion, it is not constant and therefore not in line with the full Lambda-CDM model cosmological model. Is it a coincidence that the current value of the dark energy density comes out right? Or does the Lambda-CDM model need a modification?
You can read about it in the Science News article I came across today: here.
DARK MATTER has yet to be proven, and the competing theory of MOND/TeVeS in which Newton's equations (particularly F=ma) are modified and "Dark Matter" "particles" are unnecessary, has yet to be falsified, yet it gets little to no attention, in comparison.
Mordehai Milgrom is THE guy in MOND. Here is his picture:
Read all about Mordehai Milgrom and the MOND/TeVeS vs Dark Matter controversy: here.
"Historically, the greatest difficulty in scientific revolutions is usually not the missing piece but the extraneous one - the assumption that we've all taken for granted but is actually unnecessary. Philosophers are trained to smoke out these mental interlopers. Many of the problems that scientists now face are simply the latest guise of deep questions that have troubled thinkers for thousands of years. Philosophers bring this depth of experience with them. Many have backgrounds in physics as well."
.....George Musser, Scientific American Senior Editor
I. Time
II. Gravity
III. In Conclusion
I. TIME
Time. It's a Dimension. It's the 4th of the 4 dimensions we know of. It's also the strangest of them all, due to its apparent uni-directionality. Entropy and The Second Law of Thermodynamics seem to be involved.
First up, Richard Feynman's lecture at Cornell:
And then there's this, by Science Comedian Brian Malow :
"If we are considering the fundamental level of reality, and asking the most fundamental questions about dynamics, we come up against the question “What decides how things change?” At this fundamental level, the physical laws can seem somewhat arbitrary (for example, the amount of charge on an electron). In fact, at this most fundamental level, the only principle which seems likely to describe dynamics seems to come from mathematics not physics: a system will have many more possible disordered states than ordered states, so a system which changes state randomly will most likely move to a more disordered state.
"While the second “law” of thermodynamics is “just” a statistical principle, it is a mightily powerful statistical principle! This is because the basis of the second law – that “disorder will increase” – seems so obvious, and seems to appeal to a fundamental, platonic principle of mathematics. For this reason, the second law manages to appear even more fundamental and unbreakable than the other physical laws, which seem rather arbitrary in comparison. Hence Arthur Eddington’s famous quote: “If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can offer you no hope; there is nothing for it but to collapse in deepest humiliation.”
"At the most fundamental level, I would just imagine physical dynamics are described by change of entropy – I can’t imagine any more fundamental principle which could possibly describe change." ..... Andrew Thomas of "What is Reality?" fame, on "The Arrow of Time"
II. GRAVITY
Gravity. It's a Force. It's the force we've known about the longest, yet, the one we seem to know the least about. It's the strangest of them all due to its weakness, and its range.
Interestingly, Lubos Motl has a blarticle up about Gravity. It's interesting to me because for the first time in a long time, Lubos is actually UNdecided about something ... for a change. The title of the blarticle is "Gravity as a Holographic Entropic Force", and the replies are no less important than Lubos' excellent blarticle. Click here and read it. It won't be "time" wasted, heh.
It refers to U. Amsterdam's Erik Verlinde's January 6, 2010 paper, here, titled On the Origin of Gravity and the Laws of Newton.
UPDATE (Jan.11): Peter Woit notes Verlinde's paper along with Sean Carroll's new book and something called "The Entropic Landscape" at "Not Even Wrong" in the blarticle "The Entropy Decade", here.
UPDATE (Jan. 12): Erik Verlinde has received criticism, and defends himself today: here.
UPDATE ( Jan. 14): Verlinde defends himself at Lubos Motl's The Reference Frame: here. It's worth reading the comments section as Lubos finds this new way of looking at things both interesting and vexing.
UPDATE (Jan. 17) - "The Hammock Physicist's" Johannes Koelman's Blog has a very nice History of Verlinde's work spread over 3 articles in Dec. '09 and Jan. '10, which should be read in the following order, including the replies. They are:
1) Dec. 14 - "Holographic Hot Horizons" - Click here.
2) Dec. 17 - "Holographic Horizons Get Hotter" - Click here.
3) Jan. 7 - "It From Bit: The Case of Gravity" - Click here.
They should be read in order, but the replies to 2) above are very interesting, in which Sunu Engineer (not verified) claims Verlinde's work has already been done by another Scientist named Thanu Panmanabhan. It's a bit messy, but their two approaches are different. I don't get into political sparring among scientists. The gossips love it but I find it messy and embarrassing.
Speculative Physicists are falling all over themselves in trying to describe "Time" and "Gravity", sometimes together, but eventually they fall back into Philosophy in trying to defend (cough) excuse me, I meant describe their own individual takes on this stuff.
So my question is, are there any TRUE Philosophers out there who care to weigh in?
Remember, the purpose of Philosophy is to challenge not the math so much, but the ASSUMPTIONS. George Musser taught me that.
So, Philosophers, I ask you ...
Is Time REALLY a Dimension? Or is it something else? A partial Dimension? An illusion? An absolute value or an ever changing thing?
Is Gravity REALLY a Force? Or is it something else? Simple geometry? An illusion, being the reflection of a true force on a supra-dimensional plane? Since it seems to be tied to mass, what is mass, exactly?
I understand all sorts of mathematics work out splendidly when Time is treated as a Dimension and Gravity as a Force, and it is not my intention to get into semantic arguments. I'm just asking.
IV. MUSIC (Ice for the overheated brain)
Music to contemplate by (from "The Continuing Adventures of Paul on the Floor" by Johnny and the Moondogs, at the first ever outdoor stadium concert way back in 1965):
Finally, Ringo requests more Feynman. Here you go, Ringo:
Finally, in my Philosopher buddy Phil Warnell's (see relies below) honor, here is one of the most haunting songs of the 1960's, from a singing duo that rivaled The Beatles in their day. The music is beautiful, it's the lyrics that haunt. They remind me of Paul Dirac, and David Deustch, and ... me. In my (early) teenage years, anyway.
Art Garfunkel (the guy on the right) got his Masters Degree in Mathematics. He was set to go for his PhD., when destiny (Stardom) called.
Finally, and in great honor to my dear friend Andrew Thomas of Swansea, Wales, UK, who saved me from ditching Mathematical Physics entirely, and at the very last moment before I would have done so, thanks to his GREAT Indroduction to Quantum Mechnics website "What is Reality?", and who furthermore doesn't appreciate The Beatles as much as he should, yet DOES appreciate that great "unifier" of Elvis, Beatles and Motown that is Michael Jackson ... I give my personally favorite video of MJ's, the wonderful "Black or White",
