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.    

Wednesday 17 April 2024

Gracchi Brothers

The Brothers Grim

It is time to delve again into Roman history's murky depths. In this post, I'll examine the lives of two highly intriguing Roman brothers from the later Republican period, a time of great political turmoil and unrelenting struggle between the wealthy and the poor. 

Rome of the late republic had gained much wealth through acquisition and conquest. In 146 BC, Carthage (remember Hannibal?), Rome's greatest Mediterranean rival, was destroyed. The Greek city of Corinth was razed to the ground in the same year. The acquisition of such enormous amounts of gelt within a relatively short period was bound to have profound societal consequences. This was especially so as most of the wealth fell into the hands of the already wealthy elite patricians. However, the smug patricians languishing on their lavish estates were soon to experience a political backlash as the poor and landless were not without powerful representation in the Roman Senate. Enter the heroes of the story, stage left.

Tiberius (b. 163 BC) and Gaius Gracchus ( b. 154 BC) were born into privilege during the critical years of the late Roman Republic and would become pivotal figures at this turbulent time in Roman history. Both became champions of the poor and disenfranchised as they attempted to engage the powerful elite. Alas, both brothers lost their lives in their attempt to reorganise the political system. The senatorial patricians were not quite ready yet to devolve their power and, especially, wealth to the common folk. 

The elder of the brothers, Tiberius Sempronius Gracchus, served with distinction in the military before entering politics. Military service was an essential prerequisite for entry into a career in the tempestuous world of Roman politics. 

A Digression is Required

The Roman army of the late Roman Republic was a middle-class militia of small landowners. A destitute Roman could not be enrolled as a soldier at this time, but this would change by necessity. The system worked well when most wars were conducted close to home, and the soldiery was disbanded after the campaigning season to return to plough the land they owned. However, Rome's wars were entering into a phase of unrelenting aggression in lands far away from Itlay. No longer were Rome's wars local and confined to a single season. War was incessant, unrelenting, and now conducted outside Italian soil. Who would plough the field and tend to the crops and harvest?

Back to Tiberius        

Tiberius Sempronius Gracchus, the elder of the two brothers, rose to prominence in the latter half of the second century BC. As a politician, Tiberius became increasingly aware of the dire conditions faced by Rome's rural poor. The problem stemmed from the rapid proliferation of large farms called latifundia. The Roman citizenry could no longer support small farms, resulting in the land being bought at a low price and amalgamated into large farms worked by slaves. As mentioned, the citizenry could no longer till the field as they fought in a foreign field far from home and for many a season. When they returned to the farm, the fields were barren and untilled. This societal disruption resulted in a growing population of poor and landless folk who naturally gravitated to Rome. The rise of the 'Latifundia' was due to a series of complex factors not described here. Nevertheless, the result was that a tiny minority of very wealthy people came to own large swathes of land both in southern Italy and abroad and consequently became wealthier. As Crassus once stated: ''You can't be considered wealthy unless you can afford to raise an army''.  

In 133 BC, Tiberius was elected as a 'Tribune of the People' for that year. This ancient political position was designed to protect the ordinary folk (plebians) from the rapacious abuse of the noble class, patricians. At this time, tribunes numbered ten and carried sweeping political power for the year they were elected. Not only were tribunes sacrosanct, but they had the power to veto the proposals of the head magistrates of the Roman Senate (consuls). As tribune, Tiberius proposed a land reform bill (Lex Sempronia Agraria). This law proposed the break-up and redistribution of public land owned by wealthy landowners for the use of poor citizens. This bill was received poorly by the senatorial class as if carried, it would no doubt reduce their own wealth and power. The influential members of the Senate were uncompromising in their opposition. In response, Tiberius rallied vociferous support from Rome's urban poor. Both sides began attracting supporters, which inevitably led to a violent confrontation in the streets of Rome. During the fight, Tiberius was slain together with hundreds of his supporters. Thus ended Tiberius' gallant attempt at significant land reform. His body was thrown into the Tiber to sleep with the fishes. 

In hindsight, Tiberius's attempt at land redistribution was way too ambitious for the Rome of the time. The wealthy elite were not going to hand over land without a struggle. They were well aware that the law was likely to pass, and therefore, in time-honoured tradition, they used violence to squash the issue. But this was not to be the end. Despite the risk, Tiberius' younger brother, Gaius, decided to carry on his brother's noble work. Like his brother before him, he was elected as 'Tribune of the People' (123 BC). Gaius was a skilled orator and agitator, and during his tenure, he proposed similar reforms but expanded their scope. Not only did he propose extensive land reform, but he also wanted to supply subsidised grain to the poor and grant Roman citizenship to Rome's allies. He also wanted to introduce a political counterpoise to the senatorial elite; he sought to introduce measures to bolster the power of the extensive equestrian class. The backlash from the patricians was, again, inevitable. In 121 BC, Gaius was seeking reelection as tribune, and during the count, Gaius and his supporters were killed by an angry mob of senators. Although some sources avow that he anticipated his murder and committed suicide by falling on his sword. There were no more Gracchi brothers to carry on the legal legacy, and, therefore, the ambitious reforms were not enacted. As an aside, it is to be noted that both brothers were killed after their period as tribunes. The arrogant patricians were not stupid enough to kill the brothers during their tenure. The law was very strict about the killing of an actively standing tribune.    

The legacies of Tiberius and Gaius Gracchus are complex and multifaceted. On one hand, they are celebrated as champions of the common people, whose efforts laid the groundwork for later reforms in Roman society. Their advocacy for land reform and social justice resonated with generations of reformers and revolutionaries throughout history. On the other hand, their methods were controversial, and their actions ultimately led to political instability and violence.

The brother's saga was a culmination of centuries of conflict between the 'haves' and the 'have nots' of Roman society. This time, however, the stakes had never been higher. The surge of wealth due to conquest and the extensive destitution it indirectly caused did not bode well for societal cohesion. As usual, the wealthy senators were happy with things as they were and were not open to change. The Gracchi tried, by legal routes, to alleviate the suffering of the poor. The patricians were well aware that if the laws were passed, they would suffer financially. And so they funded and initiated violence to prevent the bills from being passed. Whilst this strategy was highly effective in the short term, it could never provide the basis for a solid long-term political system. The following 90 years would see profound changes in the political landscape of Rome. Rome would enter into the era of the 'Strong Men'. Men of military and political merit would dominate Rome in successive waves, beginning with Marius and culminating in Caesar's perpetual dictatorship in February 44 BC. One month later, he was dead. Civil war would follow as men fought to rule. This ended in 27 BC as the victor, Octavian, later Augustus, took hold of the rein/reign of political power. Emperors would rule Rome until the end. Although the Romans would never admit it, the hated rule of kingship had been reinstated.