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.