Wednesday 24 April 2024

Three Body Problem

The three-body problem is a fascinating conundrum that defies total explanation. And no, I'm not referring to the time when I got drunk and was discovered in bed by the missus with a barmaid and the next-door neighbour. Ooo, missis. 

The dynamics of two orbiting celestial bodies acting under their direct gravitational attraction can be readily described by Kepler's laws of planetary motion and Newton's laws of gravitational attraction; all is well with the binary world. However, things become a lot more challenging when we decide to add a third body of similar mass. In theory, the motion and position of each body at any precise time, given their initial conditions and acting under their mutual gravitational tugs, should be eminently describable by classical mechanics. Nonetheless, this seemingly simple problem lacks a precise solution and has captivated the minds of great scientists for several hundred years. And yet, as complex as this problem manifestly is, the real cosmos, as opposed to the sterile, theoretical three-body system, is decidedly more challenging and, of course, unsolvable (no shit Flaxen).  

How can we discuss physics without mentioning the greatest physicist of all time, Isaac Newton? His role in the formulation of classical mechanics, as well as his stellar contribution to other fields of physics and mathematics, is unsurpassed. Of note, this profound genius and polymath was also a very odd man. In his monumental work: 'Philosophiae Naturalis Principia Mathematical' (pub. 1687), his elucidation of the laws of motion and gravitation provided the backbone for the study of celestial mechanics. His work made the description of the interaction between two celestial bodies comprehensible; however, the application of his work involving three bodies became a daunting prospect, even for the great man. This challenge taxed his mind greatly. In his great work, he famously stated: "The problem of determining the motion of three bodies moving under no other force than that of their mutual gravitation is unsolvable, and it is not possible to find a general solution for their motion."  Although Newton was not able to provide an absolute solution, he did provide insights by utilising the concept of 'Perturbations'. The concept takes account of the perturbation of motion caused by the gravitational 'pull' of one body upon other bodies within the system. By applying a series of successive approximations, Newton was able to obtain an increasingly accurate estimation of the effects of perturbations on the orbits of celestial bodies.  

Newton's initial work and mathematics provided a solid base for further developments in the field. During the 18th and 19th centuries, savants such as Lagrange and Laplace made significant progress on the problem by introducing sophisticated mathematical techniques. With the advent of computing power in the latter part of the 20th century, scientists were now able to solve the equations for the motions of three body systems.  Modelling using slight adjustments of the orbital mechanics revealed an inherent instability of the celestial systems. Changes as small as a millimetre involving the orbit of one body within the system could result in chaos.

Active research continues; however, many scientists acknowledge that the problem defies an ultimate solution. Whilst mathematical tools are effective in providing dynamic solutions given initial starting conditions and parameters, slight variations in any one parameter can result in a dramatically altered outcome. This, of course, is highly reminiscent of the 'Butterfly Effect'. A term coined by an American meteorologist way back in the 1960s at a conference. He specifically referred to the problem of predicting the weather, even in the short term. Subsequently, his utterance has become a metaphor for a host of circumstances unrelated to whether/weather it's going to piss down or not. As my astute readers are well aware, predicting the weather has never been an exact science. Anyway, the concept of the 'Butterfly Effect' is eminently worthy of a post-, but only if I remember to take my prescribed medication. Too many variables and too much unpredictability.    

While we expect nay demand chaos and uncertainty at the quantum level, we can comfort ourselves in the absoluteness of the macro world in which we bathe. But our illusions have been shattered, at least at the cosmic level. If we struggle to understand the complexity of the gravitational interaction of three entities, what are we to do when the number of interactions is numerous, as is the case with the solar system. Here, we have nine planets, eight if you are pedant, in addition to planetary moons, asteroids, and accumulations of dust and ice, various. Each body will have a gravitational effect dependent upon its mass and distance from other bodies. The gravitational effect each has upon others is subject to Newton's inverse square law. I've always thought that the gravity of any particular body is infinite in scope. Therefore, the gravitational force of a body should still be felt, albeit extremely weak, by an object at the 'other' extent of the universe. At this stage, I'm still within the scope of Newtonian mechanics. Of course, 'Infinite Gravity' may be a mathematical concept that is untenable when applied to unfeasibly vast distances. Nevertheless, Einstein's insight into special relativity enables us to grasp, although loosely, the concept of gravitational force as an artefact and consequence of mass warping space/time. In this scenario, we envisage gravitational fields radiating out at the speed of light. And so, ultimately, we are left with a universe enveloped with grids of overlapping and interacting gravitational fields or perturbations. When considered in this way, it is hard to fathom how there can be any form of orbital stability at all. With so many gravitational perturbations, how can we achieve the orbital cohesion and dependability that we actually observe? I would like to petition the views of any of my readership, who are blessed with a better understanding of these physical conundrums, in order to throw a little light onto the dark regions of space between my ears. Nuff said.    


  1. Dear Flax, fckn complex questions you are putting... but... never surrender.
    First: mind Ludwig Wittgenstein! "If a question could be put at all (logically nothing but the human mind could have done that) the human mind is always competent enough too to find an answer on it." A question can´t be unanswerable, no problem unsolveable.
    Second: Don´t mistake expecting that Physicians can explain any essence. Try for example "what´s the meaning of life?". For natural sciences impossible to answer, for philosophers/epistemologists easy going: "The meaning of life means to give life a meaning." Do so and you´ll see, how well it´s working. Doubtless great Isaac Newton knew very well why he dedicated his "Principia" to René Descartes. Yes, he observed gravitation but never knew how/by what it is caused. Nobody of todays knows. Einstein said it´s electro-magnetism, but not made of iron we stick on earths surface in anyway, ignoring centrifugal powers... and hell... so much more.
    Third: Temet nosce... it´s all in your mind. You are creating the whole bullshit. (100%proof you can - in any dream you do) Gongrats man. Reality is only as real as you can really realize. If not to your liking, do better. Cheers.

    1. Just been listening to a neuroscientist called, Julia Mossbridge. She is of the belief that our consciousness is really the only thing we can take as real (Descartes rules!). Consciousness exists and everything has to be extrapolated from this, including our physical world construct. She infers that consciousnes is not a phenomenum dependant upon the brain. I understand that her view is a minority one amongst neuroscientists.

  2. Newton, greatest physicist of all time. Difficult to argue with that. His theoretical basis lasted about two hundred years before it started to struggle.

    I’ve been fascinated with Newton since I first encountered calculus almost fifty years ago now. Supreme genius indeed!

    His basic model was that the physical world was composed of point masses obeying precise laws (and in celestial mechanics of course, planets, stars and anything else can be considered as points of the requisite mass). A revolution in celestial mechanics and so much else that was to follow.

    He was aware (not the only one to be aware of course) of the fundamental question of action at a distance – point to point forces acting between particles in a straight line, something must be transmitting the force – but given that his elegant mathematical construct brought order to the heavens and so much else, it was a question that could be left in limbo.

    It wasn’t the only such omission. Huygens’ construct, every point on a wavefront as a source of a spherical wave the interference of which created a new wavefront going forward. It was an elegant construct and could explain many optical effects – refraction, reflection etc – but the question, “don’t these wavelets propagate backwards as well?” similarly could be brushed under the carpet.

    Coming back to gravity, ff the transmitting medium was some sort of fluid, Laplace calculated that gravitational forces were being transmitted at least 7 million times faster than light!

    Come the 19th century and electricity & magnetism – phenomena Newton never studied - started to raise questions.

    It had been known for the previous hundred years that electric and magnetic forces followed the same inverse square law as gravity. But they were also different.

    There are positive and negative charges, north and south magnetic poles, lines of force between them. Faraday coined the term field in this context - energy associated with physical phenomena lies in a continuous medium surrounding bodies, rather than in the bodies themselves (you can talk of a gravitational field of course, but that is a subtly different use of the term).

    Faraday was a genius of the first water but he had little mathematics. Maxwell, another genius of the first water did, and he created the mathematical construct in his famous equations.

    In doing this, Maxwell imagined all sorts of intricate models using wheels and gears with balls between them to explain the various phenomena (which he stressed were just constructs although there were others who interpreted his models literally) but it was only when he abandoned these models and looked at the mathematics that it all came together.

    It was a major shift in physical thinking. There was a physical phenomena but we can’t model it, we can’t imagine it as a mechanism. The trusted methods of the previous two centuries (from a mid nineteenth century perspective) no longer worked.

    This is where the great French savants you mention come to the fore. Laplace and Lagrange. They developed generally applicable functions, which could treat a system as a “black box”, relating outputs to inputs, using general characteristics of the system. The details of any actual mechanisms of the system in question were not really important. applicable to generally to physical systems.

    Maxwell used a Lagrangian formulation to produce his field as a coherent, connected system and there was no real need to speculate as to any “internal” mechanisms.

    This was a major shift in perspective, and physics going forward for this point has never really looked back.

    Quantum mechanics, relativity - special and general – came along and today we have various field theories in particle physics and there are – as yet still speculative – things like Penrose’s “twistors” and things like strings, “branes”.

    What is the physical reality underlying all these things?

    Who knows. Can we ever understand? Likely not.

    It’s the mathematical models and techniques which are the “reality” we can try to relate to.

    Where will this all lead.

    Who knows!

    1. Hi Mark, try philosophy... no discussion how indispensable mathematics are and for sure all natural sciences are super useful too. But - as mentioned above - none of them can ever answer any essential questions (meaning of life, free will or not, who am I, who are the others, what is the "world", where I am comming from/going to, how real is reality....). They are "minor" sciences compared to philosophy and their actual state of art always depends on the temporarly prefered/ruling epistemological concept(s). Example: logic! Try to do natural sciences without that epistemological concept. Don´t let so called astro-physists/cosmologists fool you by making people believe that their business is physics or reasonable true natural sciences. It´s nothing but purest fiction, fantasy, comparable with mythology or religion...(no religion is compatible to logic)... where in fact it´s comming from. The Jesuits started the first systematical campaigns in early 20the cent. , such as bigang-theory, string, dark matter... similar stuff, on purpose to shatter the infinite universe model what no god/creator could have started (infinite in time! no beginning). Have a look at Fred Hoyle, he´s worth. You may know, that Steven Hawking was the most prominent member of the acadamy of the vatican, recruited by incredible tricky Jesuit abbott named George Lemaitre. Have fun.

    2. On Mondays, Wednesdays and Fridays, Neils Bohr would let his imagination run wild. On Tuesdays, Thursdays and Saturdays he would ruthlessly subject these wild speculations to full rigour (this may actually be apocryphal but I have read it in a number of places).

      Genius - that ability to short circuit the plodding that is all most of us can manage - almost by definition defies analysis. Maybe this was Bohr’s attempt to explain (or somebody was explaining on his behalf). Unsatisfactory really, but it does neatly encompass the dichotomy).

      When you can put actual numbers to things, to be able to calculate and accurately predict, you have science, which leads to technology and practical engineering. The fruits of this since the beginning of modern science (when was this? 1642, the year Galileo died and Newton was born; seems as good a place to start as any) have been simply incredible, and there is no need to describe them further.

      It is no longer empirical, finger in the air and it soars

      What can be done with just the empirical is impressive though, but I would posit that it could never lift the masses out of grinding poverty. Look at the great civilizations of the past. Wonder at the architecture, the literature, the art, the philosophy – but never forget the grinding poverty of the masses, a poverty we can scarcely imagine today.

      But the rise of science, technology and engineering soon gets taken for granted and we’ve gotten to the stage where it seems to actually be despised by so many.

      A comedy, which will become a tragedy and could well end up as an apocalypse!

      But I digress. People need wonder and you are right, philosophy can provide that, and I have started on it.

      You need some epistemology, even if you are not directly aware of it, to be a successful scientist or engineer – not necessarily meaning that you will achieve great things - but to at least like and ideally have some passion for it.

      We are awaiting another Newton, another Maxwell, another Einstein to link gravity and quantum theory. At present, it all seems so confused with so many potential paths, speculations, philosophies, and mathematical methodologies.

      But I’m sure it will all become clear and maybe fifty years from now, will all seem so obvious.

      Or maybe not. Are we reaching the point where synthesis by a single mind is no longer likely.

      Is that when the machines take over?

    3. 1642 ain´t bad - René Descartes was in his early 50ies then, so rationalism was in the world.

      Great inventions can be done without epistemological concepts, but they remain exeptions then. Systematical affords did (and can) only happen under the roof of well working epistemological concepts.

      It seems to me that you prefer to measure progress in material stuff. Mind, that no material increase can happen without mental ideas before. That´s why idealism (the junction between rationalism and empirism) is one of the most powerful epistemological concepts.

      A Newton first needs a Bacon and a Descartes, an Einstein needs a Kant and a Husserl.

      We don´t have to wait 50 years - it´s right here now alreeady directly in front of your nose. Clean your glasses, have a look on Ernst von Glasersfeld - Conctructivism. Think that to a logical ending. Out comes Solipsism. It may appear shocking to philosophical beginners, but that´s what the nature of enlightenment always was, is and will be. Watch R. Linklater´s movie "waking life" (kind of arthouse version of "Inception") Very helpful stuff, very close to the truth.

      and no - the machines won´t take over. They are not polyphrenic (enough). Cheers.

  3. Intersting comment. It seems that while we can apply mathematical models to describe the actions of physical processes, we struggle when it comes to conceptual descriptions. Perhaps our brains are not designed to grasp these concepts except with mathematics. My maths ability is not sophisticated, although I did take a calculus course in my first year of uni. Regardless it is facinating to ponder these things and realise that we know very little about the world we exist in.

  4. congrats Flax to this fluffy Schrödinger´s cat. You know few about the world, you exist in and same time anything about it, because the world exits in you/only because of you - as you are it´s only creator. If not to your liking, do better. You are on the right way in just this fckm moment.

    1. What an appropriate comment- did you read my mind Josh? Funny enough, the next post is on the 'Problem of Consciousness'.