Sunday, September 28, 2008

The Variational Principle

Lawns in public places all suffer from the same problem: people don’t like detours. In cities throughout the world people search for the fastest route to the workplace, the shortest way to the restroom, the least pricey airline, the most convenient parking spot - depending on resources and personal preferences we optimize our day with regard to criteria we regard important. Cooks experiment with recipes to create the most delicious meals, politicians argue about taxation to score in polls, you aim to find the most comfortable position on your couch. In most cases, these are optimizations through incremental modifications and evaluation of the change, little steps of trial and learning, and eventual selection of the optimal solution.

Not only our daily lives reflect our aim to optimize under variation, but Nature itself shows the selection of optimal configurations. A soap bubble minimizes surface area [1]. Electric currents prefer the way of least resistance, water runs downhills around obstacles in its way.

In all cases we have a system with a quantity which is optimized for one of many possible configurations, and the configuration optimal in this regard is the one realized in Nature. Optimization can mean either lowering a quantity to a minimal value, or obtaining a maximal value. Might that be you slouching on the couch with your feet on the table because it’s the most comfortable way to spend your evening, or dozens of students trampling their traces in the campus’ lawn because it’s the fastest way to coffee.

The same idea underlies theoretical physics. For every system we want to describe we have a quantity whose value has to be optimized. The way we find the optimal configuration is to make small changes and to take the configuration which would get less optimal under any change. This is essentially the same procedure one does for finding the extrema of a function by requiring the first derivative to vanish: These small changes are called ‘variations’, are denoted with a small delta δ, and the process is called the ‘variational principle’. For the optimal configuration, the variation has to vanish. In physics, in most cases the quantity optimized is called the ‘action’, and is usually denoted with a capital S. The requirement then reads

The Best of all Possible Worlds

One can’t write about the variational principle without some name dropping. The first to drop is Gottfried Wilhelm Leibniz, German scientist, philosopher, mathematician, and the one with the cookie. Leibniz already in 1710 wrote the “Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal ” (Engl: Essays of theodicy on the goodness of God, the freedom of man and the origin of evil, writing in French was apparently chique in the 18th century), in which he develops the thought that our world is the ‘best of all possible worlds’. Though his argumentation is heavily theological, he made fairly clear that with the ‘best’ world he meant one that is optimal in some sense, possibly in a sense not immediately obvious for us, and that the optimal world has to be among the indeed possible worlds. Certainly, we can imagine a world with less evil, less hunger, less poverty and less spam in my inbox, but is this a possible world? The question touches on the trouble with all utopias that build upon idealized caricatures of men, and remain wishful thinking because they fail to describe reality.

Voltaire famously made fun of the idea the world could as it is be optimal in his satire “Candide, ou l'Optimisme,” in which he attacks the Leibnizian optimism. This always seemed odd to me since I find Leibniz' conviction the world we live in is the best possible one rather pessimistic.

Leibnitz however inspired Maupertuis, a French mathematician and philosopher, to put the linguistic argument on a more rigorous base. Maupertius came up with the concept that light travels on the shortest path, known as Maupertius' principle, which was a successor of Fermat's principle and the predecessor to the principle of least action. The calculus of the general variational principle was only shortly after this developed by Leonard Euler and Joseph-Louis Lagrange. Both of their names are today intimately connected with the formalism: the equations of motions one derives from the variation are known as the ‘Euler-Lagrange equations’, and the action is an integral over a function called the ‘Lagrangian’ (though certain people insist on calling it ‘Lagrangean’).

Examples

The Lagrangian for Newtonian mechanics for example is
A nicer example is however the Lagrangian for a point particle of mass m in a possibly curved background described by the metric tensor gμν, where the action is just the length of the curve

The equation of motion is then the so-called geodesic equation, the shortest curve in an arbitrary background, which is not usually – as in flat space, e.g. a lawn – just a straight line:

The routes of airplanes for example are to a good approximation geodesics of a sphere (neglecting wind and countries who have issues with their airspace). Though it might not look like it on the routemap you find in the in-flight magazine, these are indeed the shortest connections possible.

For fields instead of single particles, the Lagrangian is a function over space-time and the action is a volume integral. The Lagrangian of General Relativity for example is just the curvature scalar R, the variation yields Einstein’s Field equations. The Lagrangian of free Electrodynamics is F2, where F is the field-strength tensor, the variation yields the free Maxwell equations. One can couple these to matter in a straight-forward way to also get the source terms.

One thing that is important to notice here is that these variations are not actual changes of the real system. Unlike you trying several non-optimal ways to your new workplace before you find one that you like best, here the variation is not performed in reality. It is a variation in the space of could-be configurations, a procedure to find the one realized. The world wasn’t created crappy and then became better, it was just optimal all along. Systems solve the equations one obtains from the variational principle, they don’t learn how to do so as time goes by.

So, now that we've talked about the ‘best’ world, let us replace ‘best’ with ‘fittest’...

Cosmological Natural Selection

Cosmological Natural Selection (CNS) is Lee Smolin’s suggestion for a testable alternative to the anthropic principle (see hep-th/0612185). You find my thoughts on the anthropic principle here, in brief: The (weak) anthropic principle is a nothing but a constraint on the parameters of our theories. It is trivially correct our universe allows the existence of life (we can discuss whether the life we know is intelligent or not), and thus whatever model we use better were not in disagreement with that. This can indeed provide constraints on the parameters of your theory. However, if your theory can’t accommodate the requirement you’d not throw out the assumption life is possible in our universe, but rather your theory.

The most severe problem with the anthropic principle is that a mathematically useful definition of life is absent, and the presence of life is not something that anybody has yet managed to quantify. Thus, it is not a scientific argument, but a rhetoric one and it’s for practical purposes useless. However, as Lee points out correctly (in his paper and also in his recent talk at the Multiverse conference, see PIRSA 08090050) anthropic arguments typically have nothing to do with life in the first place, but with some more general preconditions such as formation of galaxies or existence of carbon molecules, which can then indeed be scientifically evaluated.

CNS tackles the question why our universe is as it is essentially by suggesting a quantity that is optimized for our universe with the parameters that we observe. The specific quantity in this case is the number of black holes and Lee’s argument goes that the number of black holes would drop whenever one turns the parameters of our theories - one can apparently study a lot of different scenarios where that holds. However, as it stands this quantity, the number of black holes, unfortunately is also ill-defined. The obvious questions lying at hand are: If N black holes merge does that count as one or as N? What if a black hole evaporates? Do virtual black holes count?

But forget for a moment the number of black holes specifically and take any quantifiable function of the parameters in the Standard Model and ΛCDM. Then the idea is simply our universe with the parameters as we have measured them optimizes this quantity. Turning the parameters to see whether the number increases or decreases is a poor man’s way to finding a maximum, so pretty much a variational principle.

That doesn’t quite explain though why that scenario is called something-with-natural-selection. The reason for this I suspect is a severe case of Santafeism, and that’s where the idea stops making sense to me. See, the black holes, they supposedly don’t have singularities inside but form new ‘baby universes’ with slightly modified parameters, and in such a way the ‘fittest’ universe, i.e. the one with most black hole offspring, is the one we’d likely find ourselves in. That baby-universe story sounds cute, but is as far as I am concerned wishful thinking.

But again, forget for a moment also the story with the baby universes.

What remains is the idea that we live in the “best of all possible worlds” in some sense. Might that be the “best” world to produce black holes or something else. The question here is simply whether there exists a quantity that is optimized for exactly the parameters of the Standard Model + ΛCDM that we observe in our universe. It lies at hand to think of complexity as a possible alternative, but here again one runs into the problem that it is not a well-defined quantity (what’s the complexity of planet Earth?) and thus useless for practical purposes. It remains the question however, whether the non-optimal worlds “really” exists.

I’m not a big fan of CNS because of the problems mentioned above (and some others), but I like the general idea that the function to be optimized might be a macroscopic quantity that is not easily derivable from the fundamental laws.

A Principle of Everything?

The variational principle has proved to be enormously useful and successful. In addition to that it is also a compelling, simple and elegant formulation. It has everything a theoretical physicist desires. Nevertheless, I can’t but occasionally wonder what if this principle does not indeed hold for the yet to be discovered unified fundamental laws that govern our universe?

Neither Einstein nor Maxwell initially formulated their theories starting from a Lagrangian, they started with the field equations. Yet at some point during the last century, it has become a standard procedure to start from the Lagrangian [2], which reduces the space of possible theories as there are indeed equations of motions that do not follow from any Lagrangian. Given that the examples I know are not examples I’d consider particularly interesting, this might not be a big loss. But the existence of a Lagrangian is nevertheless an implicit assumption that not typically is much discussed.

Sensemaking

The principle of least action appeared on our curriculum in my second year at College, and it has to me always been the most beautiful explanation of the world around us. Not only is the idea of optimization compelling, but the same principle can be used for completely different systems, and for different theories. The only thing one needs to change is the function to be optimized. With that function, you perform the variation according to a well-defined mathematical procedure and get the equations of motions. What a relieve that was to eventually have a clear procedure to arrive at the relevant equations after we had spend years in physics assembling equations on a case-by-case basis! Textbooks frequently offered explanations that only made sense if you already knew the result, it typically involved a lot of guessing and hand-waving, or knowing where to find the solution to the exercise. Now suddenly that all made sense.

More about the history and applications of the variational principle in this nicely illustrated book:

The Parsimonious Universe: Shape and Form in the Natural World
By Stefan Hildebrandt and Anthony Tromba

[1] An optimization that heavily inspired Frei Otto's architecture, who for example designed the Olympia Stadium in Munich. See e.g. The lightweight champion of the world - How soap bubbles and cobwebs helped Frei Otto win architecture's greatest prize, by Jonathan Glancey.
[2] Rspt. some decades later the path integral.

Lee Smolin (2006). The status of cosmological natural selection arXiv

Robert said...

When discussing the variational principle I think it is important to mention path integrals as well as those "explain" the magic of why the extrema of some functional should be relevant to nature (unless you would like to attach values like beauty or effectiveness to nature): In the quantum theory, all possibilities contribute and they are added with a phase given by e^iS. It's only in the hbar->0 limit that the method of steepest descent localises the integral to stationary points of the action.

Re equations of motion that do not come from a Lagrangian: What you think is relevant of course depends on your personal opinion. However, things like self-dual fields (more specifically: in 2n dimensions, consider theories with (n-1)-forms A where you have the constraint that F=dA is self-dual) tend to be difficult to write in terms of a Lagrangian that has all the symmetries (especially Lorentz invariance). Type IIB supergravity is an example of this (although there are some tricks you can play to write down an action).

Phil Warnell said...

Hi Bee,

An excellent post and obviously one you spent a lot of time in composing and considering. It is also interesting to find that action principle is so central to your thinking, even though you propose perhaps a little wiggle room away from it. It has always fascinated me that so much of our world can either be explained or perhaps more importantly restricted with just this one principle and was something I chose to write one of my own few blog entries (awkward noodlings) about a few years back.

In our world it places tight parameters on almost everything from something as common as Humpty Dumpty’s path to his demise to the path taken of light from galaxies at the extreme limit of our observable universe. Also, it is interesting to find that Smolin thinks that this action also places limits on what is possible as to what reality might be. I have for some time thought that this principle is to often ignored when many are considering what is possible in terms of what can be real.

Best,

Phil

Arun said...

Robert beat me to it.

Marcus said...

Bee, the link for "in his _paper_ "
in second paragraph of section on CNS may be broken. Is it supposed to point to the same paper
arXiv:hep-th/0612185
that you mentioned in preceding paragraph?

Anonymous said...

"Optimization can mean either lowering a quantity to a minimal value, or obtaining a maximal"

I would like to show example of optimization in Nature
Mp/Me=1836
1+8=3+6=9
9 maximal 1- digit number
in decimal system.
in binary 1001 nice mirror symmetry....

Anonymous said...

another example of optimization see my thread:

changcho said...

Well Bee, I wish I had your explanation of the variational principle when I was making my way through Goldstein!.

"writing in French was apparently chique in the 18th century". Just like it is fashionable in the 21st to write in English.

BTW, where can I find the cosmological movil for my baby's crib?

;-)

Uncle Al said...

Theory arises from deep symmetries but observables arise from symmetry breakings. How does the variational principle accommmodate symmetry breakings?

Bee said...

Hi Marcus,

Thanks, I've fixed the link, it's the wysiwyg editor that always messes up the html, I wish there was a way to just turn it off. I think Lee has several other papers on the topic (not to mention the book), but they are probably all cited in the above reference. Best,

B.

Bee said...

Hi Robert,

I've mentioned the path integral, see footnote [2], just so the reader knows I've heard of it. It should be obvious, but I've tried to keep this post as lean as possible and didn't want to make it more complicated by going into quantization. If you want to write a post on the path integral, just go ahead, I will add a link.

Thanks for the example with the self-duality, that's interesting. Best,

B.

Bee said...

Hi Changcho,

Funny you're mentioning Goldstein. I was just some days ago talking to Stefan about the book. Seems we both found that as a beginner's book it is quite hard to get through, but becomes really useful if one already knows most of the basics. In fact, the way I learned the variational principle best was not from any physics book, but from a maths book (that doesn't have an English translation however). It didn't come with all the mysteries and philosophy around the δ, but just with a proof and period. Best,

B.

Bee said...

Hi Phil,

I'm not sure my interpretation of CNS matches much with Lee's intention. At least I can't recall he ever put it this way. It's just that I like the idea if one strips away the details, and I think it hasn't quite gotten the attention it deserves. Best,

B.

Phil Warnell said...

Hi Bee,

In viewing the lecture of Smolin you referred to I think your understanding as to what underlies his proposal is correct. It’s strange however that though in fact this what he means in essence he doesn’t actually come straight out to state it. I would say if you wanted to put it another way one could say it’s what you might end up with when you consider the action limit of chaos in the context of set theory. In essence he extends Darwinian concepts to the cosmological realm while giving it a more firm scientific backbone by rendering it falsifiable, while dismissing other approaches for lacking this. I really enjoyed how he summed this up at the end of his lecture with a quote of Daniel Dennet.

“If you were to give an award for the single best idea anyone ever had, I’d give it to Darwin, ahead of even Newton or Einstein and everyone else. In a single stroke, the idea of evolution by natural selection unifies the realm of life, meaning and purpose with the realm of space and time , cause and effect, mechanism and physical law.

Darwin’s idea has been born as an answer to questions in biology, but it threatened to leak out, offering answers--welcome or not—to questions of cosmology.”

-Daniel Dennet, Darwinn’s Dangerous Idea (1995) p. 21

Thanks Bee once again for giving me much more to ponder and consider.

Best,

Phil

Tom said...

I have very often wondered over the years WHY nature should be so kind(?) as to satisfy a variational principle. In finding the 'next' step in fundamental physics are we being fooled that this condition is always true? Having had Goldstein FROM Goldstein himself many years ago(!), I remember asking him this question as a grad student at Columbia to which he had no good answer. As we all know, revolutions in the sciences often occur when we realize that one of our cherished assumptions about the Universe is not as generally true as we had believed. But the trick is to know
which ones these are...at this point we need some guidance from experiment. Will
this come from the LHC or someplace else?

Kris Krogh said...

Certainly, we can imagine a world with less evil, less hunger, less poverty and less spam in my inbox, but is this a possible world? The question touches on the trouble with all utopias that build upon idealized caricatures of men, and remain wishful thinking because they fail to describe reality.

Wow! These two sentences put utopia in perspective as well as I've seen it done. I think they also relate to problems with modern physics, which to me seems built on mathematical caricatures of the real world.

Neil' said...

Observations:

1. Our universe seems to combine some basic logical concepts such as the VP combined with arbitrary details such as the value of alpha, which is not "mathematically sensible" (i.e., not equal to one, etc.)

2. The whole point (IMHO, and as put forth by Barrow and Tipler in the classic TACP) of noticing anthropic design features of our universe is not the silly tautology that our being here must be consistent with those parameters being life-friendly. It is rather, the very fact that so many such parameters are life-friendly to begin with (originally argued as being within narrow margins, maybe or maybe not), and why should they be "picked" such as to be like that? The circular argument is dumb: if you're a realist, then you can imagine the universe not having life-friendly features, and thus no life. So the question is, why are the laws and constants one way, versus another way, "to begin with" (not about time or initial moments per se, but that properties logically precede their effects) versus it having to be X-friendly if X comes about later (that's Bee's "so what" moment.)

3. As I have explained and argued before, there is no logical way to pick out some "possible worlds" (as in, sets of laws and constants) to "exist", but not others. It's like the number 23 being made in brass numerals alone among all the other numbers, those having none other than airy platonic existence. Indeed, modal realists cogently argue that "existence" can't be logically defined in any way other than "as numbers" (it from bit), and ironically then all possible worlds must exist. That's what Max Tegmark implies he believes, but I don't think he appreciates that it involves accepting cartoon universes too - since they involve descriptions of things in space and time, etc.

I don't agree with MR, for three main reasons. One is the strange nature of quantum reality, in which the wave function isn't really representable (has to "collapse" all over the universe when particle is observed.) The second is, the nature of conscious experience, which I don't think is mere computation. The third is (unexpectedly to most people), that "time" can't be defined mathematically. Sure, you can draw the coordinates and the world-lines, but there's no way to represent "flowing time" as we experience it. Really, dy/dx is the same as dy/dt in math. Time, real time, has to be put in by intuition. Some deny it's existence, with the "block universe" idea, but what the hell are we experiencing? It ironically makes our minds even more magical, to construct a new form of reality.

Oddly, strong AI means there's no way to think "I am real" in a non-platonic way, so all SAI believers must admit we can't know if we have "real [material] brains" or are just the program representing our mental activity, in conceptual form (as in "possible chess games.") It's an odd expansion of the "brain in a vat" problem, here it's the "brain in fact" problem.

So, either MR is true and there's no point in asking "why is the world like this" because all of them "exist" somewhere, or: MR is not true, and some special "virtus" decides what is real and what isn't. Whether "God" or some principle, there has to be some controlling authority to make some descriptions ("worlds") real and not others. Logic by itself doesn't allow a distinction, for such a basic non-predicate claim, to be made. ("Predicate" is like the difference between a circle and a square. However, the difference between a "real" circle (as in "material"), and one that is just "the math" is not accessible to logic.) Think about that "23" example if you need to.

(PS: As you can imagine, I studied and got into "philosophy" a lot - well it isn't just "metaphysics" because it gets to the heart of what you are *saying* when you try to make a point about *anything*.)

Dr Who said...

I agree with Bee that CNS deserves more attention. Bee rightly emphasises that the variational principle makes a variation over situations that *do not* occur ["in reality"] and selects out one that does occur ["in reality"]. So it really cuts to the core of the role of physical laws as things that distinguish "reality" from "non-reality". Smolin would probably say that there is a precedent for this: natural selection decides what kinds of animals we really have, as opposed to all of the kinds that we don't have. It's a very clever idea indeed. But the problem is that it is *very* hard to see how to make a baby universe *that grows up to look like ours*. In view of our extreme lack of understanding as to what happens inside black holes [like Bee, I don't think that we are anywhere near a solution of the information paradox etc, as we discussed a couple of months back], I don't think it is unreasonable to imagine that the singularity is replaced by a bounce into some new world [though personally I don't believe it....]. That is understandable. What I cannot understand is how that baby can possibly have extremely low entropy, as we know our universe did when it was born. So something major is being missed here. I have not seen LS discuss this point, but perhaps I just missed it? How does he address this problem?

g said...

"A soap bubble minimizes surface area."

Sort of. Soap bubbles are a good example of optimization to some local minimum, not necessarily to the global minimum.

http://www.scottaaronson.com/papers/npcomplete.pdf

Bee said...

Hi G,

Yes, good you mention that. It seems I forgot to mention local vs global optimization. Well. Some other time...

Best,

B.

Bee said...

Hi Tom,

Sure, one can always ask for a further why. As I mentioned in the post, it is of course a good question whether one can maybe throw out the variational principle, as one should generally question all principles from time to time. But to convince anybody of that being a way forward it needs a good reason. I couldn't come up with one (at least not so far). Best,

B.

island said...

Dr Who said:
Smolin would probably say that there is a precedent for this: natural selection decides what kinds of animals we really have, as opposed to all of the kinds that we don't have. It's a very clever idea indeed. But the problem is that it is *very* hard to see how to make a baby universe *that grows up to look like ours*.

Man, I just want to shout out the answer... 'The "baby" evolves, of course, to higher orders of the same basic configuration'... and all of your problems are resolved... duh.

http://www.edge.org/q2005/q05_6.html
Richard Dawkins thinks that all intelligence, all creativity and all 'design' anywhere in the universe, is the direct or indirect product of Darwinian natural selection.

I would suggest that the universe evolves by this mechanism, as well.

Bee said:
The most severe problem with the anthropic principle is that a mathematically useful definition of life is absent, and the presence of life is not something that anybody has yet managed to quantify.

One way of mathematically modeling life as a dissipative system is given in the article on wandering sets:
http://en.wikipedia.org/wiki/Wandering_set

I could be wrong, but I think that master equation in the Lindblad form is capable of quantifying life AND the universe as far from equilibrium dissipative structures.

As quantum mechanics, and any classical dynamical system, relies heavily on Hamiltonian mechanics for which time is reversible, these approximations are not intrinsically able to describe dissipative systems. It has been proposed that in principle, one can couple weakly the system – say, an oscillator – to a bath, i.e., an assembly of many oscillators in thermal equilibrium with a broad band spectrum, and trace (average) over the bath. This yields a master equation which is a special case of a more general setting called the Lindblad equation that is the Quantum equivalent of the classical Liouville equation. The well known form of this equation and its quantum counterpart takes time as a reversible variable over which to integrate but the very foundations of dissipative structures, imposes an irreversible and constructive role for time.

island said...

Just to cover all bases above, before somebody asks, while hoping to stay on topic, Bee:)

This is how it works:
http://pespmc1.vub.ac.be/ASYMTRANS.html

This is how it is illustrated to work:
http://pespmc1.vub.ac.be/ASYMILL.html

And this defines the physics and our role in it:
http://www.lns.cornell.edu/spr/2006-02/msg0073320.html

*The* anthropic principle, as a thermodynamic energy conservation law that enables the universe to periodically "leap"/bang to higher orders of the same basic system in order to preserve the arrow of time, the second law, and causality... indefinitely... ... ...

Dick said...

A few years ago, when I was developing what I now call “predictive metaphysics,” I realized that I was sorely lacking in predictability on the question, “Why this universe and not some other possible universe?” Then I read Lee Smolin’s “The Life of the Cosmos,” in which he discusses his Cosmological Natural Selection (CNS) idea. Bingo! The lights went on! I was saved! I stole the idea immediately, modifying it to fit my theory (the black holes went away and were replaced by another, more metaphysical universe replication mechanism). I will be forever grateful to Lee, who continues to impress with his amazing ability to think outside the box.

Dick Dolan

Arun said...

I didn't want to write this comment, but it really, really came about because of the variational principle.

All the neuro-muscular activity in typing this up, the electrons jiggling about in my computer's transistors, the packets down the Internet Tubes, the magnetic domains changed on your server's hard drives - everything, everything - happened to extremize some action functional. Amazing but true!

Dr Who said...

Island said: Man, I just want to shout out the answer... 'The "baby" evolves, of course, to higher orders of the same basic configuration'... and all of your problems are resolved... duh."

Human babies have a low-entropy state which they owe to the extremely delicate operations of earlier systems with still lower entropy. The interior of a black hole, by contrast, is a very violent place where entropy is large and rapidly increasing --- at least this is the standard picture of a [realistic -- not AdS or something like that] black hole. A nice smooth baby universe is about as likely to result from a black hole interior as a human baby is likely to emerge from putting meat, blood etc into a blender and letting it run for a few hours.

I guess LS would have to argue that random fluctuations which miraculously produce a low-entropy baby universe somehow help to maximize the number of black holes.....

island said...

And I think that you should finish reading everything that I wrote in response to Bee's statement, which was written in context with what I said to you, and does not necessitate a singularity nor a family tree of off-spring universes.

Phil Warnell said...

Hi Bee,

With all the side talk here about universe models and all it may be interesting to note that at the PI lecture I attended last evening Roger Penrose discussed his cyclic model. It dispenses with any concept of Multiverses altogether, as well as any crunch to bring about a new beginning. To be truthful I’m still struggling with the proposal as to what all the implication are and what not.

The interesting thing for me about it is that he does propose how it might be tested or supported from data being gathered in the latest background radiation experiments with the signature of gravity waves from a previous phase. It may be something or it may be nothing, yet I like it for nothing other that like Smolin’s proposal it is falsifiable or supportable by observation.

Best,

Phil

oleg said...

Hi Bee,

Apropos the variational principle, it's well worth mentioning the Bernoullis, the founding brothers of variational calculus, so to speak.

In 1696 Johann Bernoulli formulated and solved the brachistochrone problem: find the curve connecting two given points that results in the shortest travel time for a bead constrained to slide along the curve, subject to gravity. His solution cleverly used Fermat's principle of least time for light refraction (1662).

Bernoulli posed the problem in a journal and a year later he, his brother Jakob as well as Leibnitz and Newton printed their solutions. Jakob Bernoulli's solution was the most profound of the four: it was applicable to a broader class of problems and eventually led to the development of variational calculus.

Here is a paper discussing the problem in a historical context: J. Babb and J. Currie, The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem, The Montana Mathematics Enthusiast 5, 169-184 (2008). Paper in PDF.

Bee said...

Hi Dr. Who,

What exactly is the problem you are talking about with the entropy? I don't get what this has to do with bouncing universes, as it's not the whole universe that collapses to a black hole and is 'reborn'. Best,

B.

Bee said...

Btw, since you ask what Lee says about the entropy, this question has come up repeatedly in various places. If I recall correctly, somebody asked it in the end of the above mentioned talk, so check the recording. I think he basically said the BH entropy doesn't count the internal states.

Bee said...

Hi Oleg,

Yes you are of course right. Stefan actually wanted to add some paragraphs about the brachistochrone, but I suggested he does that in a separate post. Best,

B.

stefan said...

Hi Oleg,

thanks for the reference on the brachistochrone problem!

Indeed, Sabine and I had discussed it when she wrote the post, as one of the first examples of a variational problem to be solved analytically.

I have been "collecting" different ways to present the solution of the brachistochrone since some time, but still searching for an optimal discussion ;-). So, I'll definitely have a look at the paper.

Best regards, Stefan

Dr Who said...

Bee said: "What exactly is the problem you are talking about with the entropy? I don't get what this has to do with bouncing universes, as it's not the whole universe that collapses to a black hole and is 'reborn'."

Sorry, what I meant was that according to LS, the *interior* of the black hole avoids a final singularity and re-emerges as a baby universe --- let's avoid the word bounce because it is misleading. Anyway, the point I am making is very simple: the baby's properties are determined by the black hole [plus random variations of parameters], since the black hole was its parent. But a black hole interior is usually a very *high* entropy place, and the entropy density is also very high because of the squashing. The random variations of parameters will just make this worse. So if the second law of thermodynamics holds throughout this process, baby universes should be born with very high entropy, *unlike* our Universe.

"Btw, since you ask what Lee says about the entropy, this question has come up repeatedly in various places."

I'm sure it has; LS is a smart guy. But I could not find it in his papers.

"If I recall correctly, somebody asked it in the end of the above mentioned talk, so check the recording."

I couldn't find it there, but that may be because the sound volume is very low even when I turn everything up to the maximum. Jim Hartle did seem to ask a related [but much deeper] question but I could not hear what LS said in response.

"I think he basically said the BH entropy doesn't count the internal states."

That may be, but it's not what I am worried about. Basically I am worried about the same old Penrose story: our Universe started with extremely smooth spatial geometry. How are you going to get anything like that out of a black hole interior, with all that BKL anisotropy growing like crazy as you approach the singularity [or whatever replaces it]?

PS I'd like to see the brachistochrone story too. I have never seen a good clear modern discussion of it!

Plato said...

It's hard to know where to begin when one is working on one's own.:)

It 's how the blackholes contribute to the greater whole? You have to account for what the universe is doing now, and what is "contributing to it" at this time. IN it's past, and in it's future?

Gravitational collapse within locations has it's consequences? Provides for, the current conditions of the universe?

To see The WMAP with a gravitational inclination, one can find the routes within "this place" to say that the photon can move "faster here" or "slower there?" So "lensing" should not be a problem when to see it linked to a Lagrangian view of the universe. It's an accumulative effect of viewing the universe in this way.

Best,

Anonymous said...

Plato said...

Albert Einstein (1879–1955)

One part of the theory of Relativity was inspired when a painter fell off a roof. Einstein found out that while the painter was falling freely, he felt weightless. This led Einstein to realize that gravity was a form of inertia, a result of the way things moved through space - and General Relativity was born.Albert Einstein (1879–1955)

Applicable tendencies in the natural world need some explanation, when in consideration of mathematical constructs? :) Principal of least action

For the layman then and the world beyond with measure those illustrious images of earth..:)

On planet Earth, we tend to think of the gravitational effect as being the same no matter where we are on the planet. We certainly don't see variations anywhere near as dramatic as those between the Earth and the Moon. But the truth is, the Earth's topography is highly variable with mountains, valleys, plains, and deep ocean trenches. As a consequence of this variable topography, the density of Earth's surface varies. These fluctuations in density cause slight variations in the gravity field, which, remarkably, GRACE can detect from space.

What did it take to transfer "the landscape" to views in the cosmos in this gravitational sense?

Of course then, the dynamics of satellite travel in space, for sure.

Best,

Plato said...

It's not much of a "leap of inductive/deductive faith then to consider the value here? Our determination then. How shall one see the world and the cosmos around them?:)

Best,

Bee said...

Hi Dr. Who,

Well, doesn't that depend on the size of the system whether an entropy is considered large or small?

But I have no clue how the 'birth' of the new universe is supposed to work either.

Another thing I never understood is what happens to the total energy. I mean, I know that in GR the total energy isn't a well-defined quantity, but still, it just doesn't make any sense to me. You have a universe. It contains a lot of matter and makes a lot of black hole babies. Yet every single one of these babies is as massive as the parent? If that wasn't the case, the universe family would die out after some iterations due to lack of substance. Alternatively, you'd have to shrink everything, but that would be far off from being a small variation of constants (if possible at all).

Best,

B.

Dr Who said...

Bee said: "Well, doesn't that depend on the size of the system whether an entropy is considered large or small? "

In Sean Carroll's model, the baby universe is born from a big universe which has inflated so that its entropy *density* has become small, so that the baby universe carries off only a small amount of entropy because it is so tiny. But the entropy *density* inside a black hole is not small, so even a tiny baby will carry away a large amount of entropy. True, one should make this more quantitative, but I think it's clear that this is a problem.

"Another thing I never understood is what happens to the total energy. I mean, I know that in GR the total energy isn't a well-defined quantity, but still, it just doesn't make any sense to me. You have a universe. It contains a lot of matter and makes a lot of black hole babies. Yet every single one of these babies is as massive as the parent? If that wasn't the case, the universe family would die out after some iterations due to lack of substance. Alternatively, you'd have to shrink everything, but that would be far off from being a small variation of constants (if possible at all).

Yes, I think you are right. I don't think that the baby would necessarily have *less* mass than the parent [all kinds of weird things are going on inside the black hole, eg "mass inflation"] but there is no reason to think that the baby's mass will be the *same* as that of the parent; so the process is unstable. Perhaps this is where the analogy with natural selection breaks down: in the case of animals there are well-understood laws controlling how big/small etc they can get, but in the black hole case there are no such laws, or at least none that we know.....

bellamy said...

With that function, you perform the variation according to a well-defined mathematical procedure and get the equations of motions. What a relieve that was to eventually have a clear procedure to arrive at the relevant equations after we had spend years in physics assembling equations on a case-by-case basis! Textbooks frequently offered explanations that only made sense if you already knew the result, it typically involved a lot of guessing and hand-waving, or knowing where to find the solution to the exercise. Now suddenly that all made sense."

It would've make sense a lot fuckin sooner if they taught shit top-down. My did school generally suck ass till, in college, I encountered a music teacher (doctorate in theory and comp, aside from his major in piano performance) that did just this.

Arun said...

Though not a variational principle, speaking the truth as one perceives it takes the least action, lying takes effort and causes stresses detectable by the polygraph.

One could imagine a human physiology where deception was more natural.

Bee said...

Gee, I'm not sure about that. It probably depends very much on what you've been taught what comes easy to you. I have the impression that to most people lying comes more easily than telling the truth, both to themselves and others. I guess in a certain way it's an attempt to make the world nicer than it is.

island said...

Barring random quantum effects, if every action increases the entropy of the universe, then how could *any* action possibly violate the least action principle?

Nature is thrifty in all its actions

Where is Laplace's demon when we need (her) the most?... ;)

Bee said...

Hi Island,

Entropy can decrease locally. Best,

B.

island said...

Hi Bee,

That doesn't violate the least action principle.

Bee said...

Nothing in physics, as we know it today, violates the principle of least action.

island said...

Including the observed yet totally unexpected structure of the universe?

I believe that you're right, Bee... for reasons that I gave previously... ;)

Bee said...

I don't understand what you're trying to say. Consider a particle. The principle of least action says it moves on a geodesic. Now consider it doesn't (in some kind of limit we have never directly observed). What does that have to do with the entropy of the universe?

Plato said...

A black hole is an object so massive that even light cannot escape from it. This requires the idea of a gravitational mass for a photon, which then allows the calculation of an escape energy for an object of that mass. When the escape energy is equal to the photon energy, the implication is that the object is a "black hole"

Looking at the universe in this way one would it seem see within the universe that there is a decay from that perfect realm? :) WE know some do not recognize this.

So how does all this come together into a physical theory? It turns out that the proper procedure is to construct every possible diagram allowed by the theory (for a given state of input and output particles and how they're moving) and add up the corresponding complex numbers. The result is essentially the "wave function" for that specific input-output state combination, and by squaring that number you can determine the probability that the given input will result in the given output. Doing that is how theorists at particle accelerators earn their keep.An Introduction to String Theory A Talk by Steuard Jensen, 11 Feb 2004

Best,

island said...

Hi Bee, I am sorry that I didn't get back sooner, but I had to go out.

I had this conversation once before with a female physics blogger from Copenhagen, (that I can no longer find), and I've search everywhere but cannot find it to refresh my memory but it went something like this:

If the Principle of Least Action means that "Nature is thrifty in all its actions", then it is no coincidence that the near-perfectly symmetrical configuration that we observe is also the most energy-efficient means for dissipating energy, because this means that tendency toward “heat-death” is most economically restricted to the most-even distribution of energy possible.

The universe actually expresses a grand scale natural preference toward the most economical form of energy dissipation, in other words, so if the second law of thermodynamics is telling us that the entropy of our expanding universe increases with every action, then the anthropic principle is telling us that this will occur by the most energy efficient means possible, since the flatness of the universe is one of the many coincidentally ecobalanced requirements of the principle.

If the second law of thermodynamics points the arrow of time, then the anthropic principle determines that time is maximized in order to maximize work.

Man, I wish that I could find that conversation!

Plato said...

Glast and LHC are support by evidence contained in calorimetric results.

While one works on earth, it does not discount the way another is put into space to measure evidence from locations in the universe? These results are indicative of perceptions one would appeal too, in the formation of Lagrangian views and support thereof?

Best,

Plato said...

The photon then serves to implicate "an alternate view " in this gravitational measure. Shortest distance, lensing etc?

Best

Bee said...

Hi Island,

I still don't know how you want to derive from the structure of the universe that the principle of least action can not be violated, locally, in some limit or whatever. Also, I have no clue how the anthropic principle and 'ecobalance' made it in here. I've told you at least a dozen times previously that all your talk about ecobalance and conditions for life etc is empty blabla unless you go and define 'life' and prove that some conditions are optimal for some specific function derived from this, e.g. duration of life, number of occurrences of lifeforms over the history of the universe or whatever. Best,

B.

Successful Researcher said...

This is slightly offtopic... but I just learned that the great mathematician Israel Gelfand (inter alia, the author of a classical textbook on the calculus of variations) died yesterday. R.I.P.

Anonymous said...

I am a psychologist who is looking for a theory to help me develop a new tharapy model that works for all disorders. Trying to understand universal laws that may,on some level, govern human behavior may be possible. My question is, does anyone believe the variational principle may be the key to unlocking the door to why people choose the behavior they do to get to where they psychologically need to be to maintain their optimum mental state?

James T Bourland JAMBOURLA@AOL.Com

Andrew Thomas said...

Well, it's true that people take the path of least effort - I know I do! That's interesting. But I don't think mental state comes into it.

j said...

Ill-formed question: my limited understanding is that variational principles can be derived from 'more fundamental' symmetry ideas.

It is also my experience modelers always employ simplified models of reality - these often (almost unavoidably?) have 'symmetries' that are symmetries of the simplified system not the 'real' system.

So perhaps 'variational principles' are really just consequences of the approximation process - if all detail was included would there really be nice minima or symmetries present? These seem to be deduced from the limitations of the modeling process not the basic content itself.

So the question is, is what Ijust said nonsense?

Bee said...

j: I don't know what you're talking about. If you assume certain symmetry requirements you do know what terms can appear in the Lagrangian, but you don't derive the variational principle from the symmetry requirement. Best,

B.