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Of Bernoullied Euler and Eulered Lagrange and Other Gianted Giants - Part III

Posted on July 22, 2021

We can now finally unveil the Euler-Lagrange equations of motion with just a little bit more calculation. Recall that kinetic energy of a system of \(k\) particles is given by

\[K=\sum\limits_{i=1}^{k}\frac{1}{2}m_i\boldsymbol{v}_i^T\boldsymbol{v}_i\]

Now the following computations should be straightforward to understand

\[\frac{\partial K}{\partial q_j}=\sum\limits_{i=1}^{k}\frac{1}{2}m_i\left(\frac{\partial\boldsymbol{v}_i}{\partial q_j}\right)^T\boldsymbol{v}_i+\sum\limits_{i=1}^{k}\frac{1}{2}m_i\boldsymbol{v}_i^T\frac{\partial\boldsymbol{v}_i}{\partial q_j}\]

\[=\sum\limits_{i=1}^{k}m_i\boldsymbol{v}_i^T\frac{\partial\boldsymbol{v}_i}{\partial q_j}=\sum\limits_{i=1}^{k}m_i\boldsymbol{v}_i^T\frac{d}{d t}\left(\frac{\partial\boldsymbol{r}_i}{\partial q_j}\right)\]

\[\frac{\partial K}{\partial\dot{q_j}}=\sum\limits_{i=1}^{k}\frac{1}{2}m_i\left(\frac{\partial\boldsymbol{v}_i}{\partial \dot{q_j}}\right)^T\boldsymbol{v}_i+\sum\limits_{i=1}^{k}\frac{1}{2}m_i\boldsymbol{v}_i^T\frac{\partial\boldsymbol{v}_i}{\partial \dot{q_j}}\]

\[=\sum\limits_{i=1}^{k}m_i\boldsymbol{v}_i^T\frac{\partial\boldsymbol{v}_i}{\partial \dot{q_j}}=\sum\limits_{i=1}^{k}m_i\boldsymbol{v}_i^T\frac{\partial\boldsymbol{r}_i}{\partial q_j}\]

Now recall from the previous article that

\[\sum\limits_{i=1}^{k} m_{i} \ddot{\boldsymbol{r}}_{i}^{T} \frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}=\sum\limits_{i=1}^{k}\left\{\frac{d}{d t}\left[m_{i} \dot{\boldsymbol{r}}_{i}^{T} \frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}\right]-m_{i} \dot{\boldsymbol{r}}_{i}^{T} \frac{d}{d t}\left[\frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}\right]\right\}\]

Substituting \(\dot{\boldsymbol{r}}_{i}=\boldsymbol{v}_i\) above, we obtain

\[\sum\limits_{i=1}^{k} m_{i} \ddot{\boldsymbol{r}}_{i}^{T} \frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}=\sum\limits_{i=1}^{k}\left\{\frac{d}{d t}\left[m_{i} \boldsymbol{v}_i^{T} \frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}\right]-m_{i} \boldsymbol{v}_i^{T} \frac{d}{d t}\left[\frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}\right]\right\}\]

And using the tricks from the previous article, we obtain

\[\sum\limits_{i=1}^{k} m_{i} \ddot{\boldsymbol{r}}_{i}^{T} \frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}=\sum\limits_{i=1}^{k}\left\{\frac{d}{d t}\left[m_{i} \boldsymbol{v}_i^{T} \frac{\partial\boldsymbol{r}_i}{\partial\dot{q_j}}\right]-m_{i} \boldsymbol{v}_i^{T} \frac{\partial\boldsymbol{v}_i}{\partial q_j}\right\}=\sum\limits_{i=1}^{k}\frac{d}{d t}\left[m_{i} \boldsymbol{v}_i^{T} \frac{\partial\boldsymbol{r}_i}{\partial\dot{q_j}}\right]-\sum\limits_{i=1}^{k}m_{i} \boldsymbol{v}_i^{T} \frac{\partial\boldsymbol{v}_i}{\partial q_j}\]

The above expression can be simplified in terms of the kinetic energy as follows

\[\sum\limits_{i=1}^{k} m_{i} \ddot{\boldsymbol{r}}_{i}^{T} \frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}=\frac{d}{d t}\left[\frac{\partial K}{\partial\dot{q_j}}\right]-\frac{\partial K}{\partial q_j}\]

Making use of the above expression, we can obtain the following

\[\sum\limits_{i=1}^{k} \dot{p}_{i}^{T} \delta \boldsymbol{r}_{i}=\sum\limits_{j=1}^{n} \left\{\frac{d}{d t} \frac{\partial K}{\partial \dot{q}_{j}}-\frac{\partial K}{\partial q_{j}}\right\} \delta q_{j}\]

To obtain the above, we simply need to recall from the previous article that

\[\sum\limits_{i=1}^{k} \dot{p}_{i}^{T} \delta \boldsymbol{r}_{i}=\sum\limits_{j=1}^{n} \left(\sum\limits_{i=1}^{k} m_{i} \ddot{\boldsymbol{r}}_{i}^{T} \frac{\partial \boldsymbol{r}_{i}}{\partial q_{j}}\right) \delta q_{j}\]

Now since

\[\sum\limits_{i=1}^{k} \boldsymbol{f}_{i}^{T} \delta \boldsymbol{r}_{i}-\sum\limits_{i=1}^{k} \dot{\boldsymbol{p}}_{i}^{T} \delta \boldsymbol{r}_{i}=0\quad \mathrm{and} \quad \sum\limits_{i=1}^{k} \boldsymbol{f}_{i}^{T} \delta \boldsymbol{r}_{i}=\sum\limits_{j=1}^{n} \psi_{j} \delta q_{j}\]

then it is obvious that

\[\sum\limits_{j=1}^{n}\left\{\frac{d}{d t} \frac{\partial K}{\partial \dot{q}_{j}}-\frac{\partial K}{\partial q_{j}}-\psi_{j}\right\} \delta q_{j}=0\]

From the independence of the generalized displacements, it follows that

\[\frac{d}{d t} \frac{\partial K}{\partial \dot{q}_{j}}-\frac{\partial K}{\partial q_{j}}=\psi_{j}\quad\mathrm{for}\quad j=1,\cdots,n\]

The generalized forces \(\psi_j\) acting on each particle may arise from the potential energy fields (electrostatic, gravitational or magnetic) or they may simply represent an externally applied force denoted by \(\tau_j\). Let the net potential energy field acting on particle be denoted by \(P(q)\). From classical mechanics, we have

\[\psi_j=-\frac{\partial P}{\partial q_j}+\tau_j\]

I have frequently thought about the classical fields (whether gravitational or electric) as denoting both a Force and an Energy field. Both the Energy and Force fields are then dual manifestations of a single object called Field. This is in a similar vein to the cumulative distribution function and probability density function being two manifestations of the same probability distribution for some particular random variable. The Energy part aims to quantify the motion and vibrations undergone by a particle in the past while the Force part aims to quantify the motion and vibrations to be experienced by that particle in the future. Thus, Field basically aims to track the trajectory of a particle through spacetime, both its past and future; and why worry about the present because that is already there! At any moment, Energy sums up the effects caused by the Field on the particle in the past and Force is the information that decides what effects the Field will cause to the particle in future. However, there is no unique way to interpret these concepts of science.

We also note that \(\partial P/\partial \dot{q}_j=0\) because potential energy does not depend on the velocity of the particle. Also define Lagrangian of the system as \(\mathcal{L}=K-P\). Then it is not difficult to obtain the equations given below which are referred to as the Euler-Lagrange Equations of Motion.

\[\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial \dot{q}_{j}}-\frac{\partial \mathcal{L}}{\partial q_{j}}=\tau_{j}\quad\mathrm{for}\quad j=1,\cdots,n\]

As long as constraint forces do no work and only holonomic constraints are involved, we can apply this equation. Otherwise when nonholonomic constraints are involved or the frictional forces must be taken into account, the only straightforward route to take is that of falling back to the Newtonian approach which itself is quite enthralling in its own right.¹

The historical account behind the discovery of Euler-Lagrange equations is much richer than can be indicated here. The above theory is an equal mix of mechanics and calculus. If you want to dig up the interesting stories from the Physics point-of-view, the best route is to find more mechanics material to read in connection with the Brachistochrone problem. This is a pivotal problem in Euler-Lagrange theory which due to scope-beyondness issues I will skip although it is unfair to discuss the mathematics of Euler-Lagrange equations without discussing the mathematics behind the Brachistochrone problem. And for the calculus point-of-view, need you a better starting point than Dunham!²

Of S.E.C.T. in S.T.E.M.

Shouldn't STEM be regarded as the most beautiful abbreviation but well, beauty lies in the eyes of the beholder³ as they say. Talking about Genius generally comes to be associated with STEM fields though other fields may be equally worth exploring.

To motivate what follows, we need to briefly touch upon certain historical accounts⁴ surrounding the Brachistochrone problem which was proposed by Johann Bernoulli to the \(``\)most brilliant mathematicians in the world.\("\) While solving the isoperimetric problem (these two problems along with the least resistance aerodynamical problem must definitely form the Holy Trinity in the context of computing trajectories that optimized functionals, the so-called optimal curves), Euler had discovered a geometric approach to these equations much earlier. Lagrange discovered an analytic approach to these equations to which Euler said: \(``\)Your solution to the isoperimetric problem leaves nothing to be desired.\("\) Inspired by Jakob Bernoulli's solution to the Brachistochrone problem, Euler was able to use Lagrange's ideas to rigorously develop the theory behind calculus of variations. At the moment when he was about to publish his detailed report, he resisted the desire to publish his own ideas about this equation (a rare moment in academia, indeed!). He must probably have felt that Lagrange's ideas were fresh and more insightful, and could be applied to many complicated problems in mechanics in a rather elegant way, so he wrote to Lagrange: \(``\)the importance of the matter has led me to outline, with the aid of your light, an analytical solution to which I will give no publicity until you yourself have published the whole of your research, so that I do not take away any part of the glory that is due to you.\("\)

In fact, thanks largely to Lagrange's efforts (who undoubtedly received a huge assistance from Euler but still), Analytical mechanics rose and prospered while being distinctly different from Newtonian mechanics. Though both types of mechanics combine to form classical mechanics, we know that analytical mechanics receives much deeper appreciation from the scientific community today (at least when considering idealized problems which are free from real world complications and where both approaches are equally applicable). Moreover, knowing Euler's hunger to publish,⁵ one has to appreciate how difficult it must have been for him to resist the desire but at the same time his commitment was stronger towards growth of human knowledge and he believed that Lagrange had developed the theory better. Another profound Brachistochrone episode⁶ occurred when Newton famously said "I do not love to be dunned and teased by foreigners about mathematical things." How can we forget to talk about Newton here!

In fact, the most vital feature of Newton is excellently identified by Richard Westfall⁷: "In his age of celebrity, Newton was asked how he had discovered the law of universal gravitation. By thinking on it continually, was the reply. No better characterization of the man can be given, not only in its delineation of a life whose central adventure lay in the world of thought rather than action, but also in its description of his mode of work."

Tikhomirov⁸ says: "Newton solved the aerodynamical problem completely. Why was Newton's curve relegated to the role of the unfortunate Cinderella for 300 years? Why was Newton's idea not fully understood for so long? Newton's problem belongs to the category of problems of optimal control (which was not rigorously understood back then). Within the framework of this theory Newton's problem has a natural and standard solution. On the other hand, this problem has no natural and standard solution within the framework of the calculus of variations (which was the only sophisticated tool available to attack such problems in those days). Thus this problem put Newton 300 years ahead of his time! Newton's problem is certainly a remarkable mathematical event. For 250 years it seemed likely that it had no physical basis and its solution was absurd. But the \(``\)mistake\("\) of a genius turned out to have been an insight. In a word, hasty judgments are sometimes just that. It can happen that the thought of a genius, which we regard as a mistake carries within it the imprint of truth — a truth clear to him but hidden as yet from us."

Newton's greatest achievement may be the rigorous initiation of human thought process vis-à-vis the abstract concept of Force which he made concrete by allowing us to measure it. But soon Euler, Lagrange and others unleashed their Genius to replace Force with Energy as the standard way of thinking, and then finally Force and Energy got relegated to antiquity by Field when Maxwell, Faraday and others unleashed their Genius.

Genius is always identified only in hindsight and it is always very hard to deconstruct and reconstruct it. But nevertheless a Giant is almost always Gianted by another Giant. The human story has it that cerebral traditions have existed in various forms throughout the human history. Isn't the previously mentioned intellectual circle set up in the thoughtosphere among the Bernoullis, Newton, Euler, Lagrange, and others typifying some sort of a cerebral communal tradition? This circle's activities are summarizable only as fascinating and even fantastic. I choose to call such an event a Singular Event in Cerebral Traditions (S.E.C.T.). One could also whimsically refer to \(``\)Cerebral Traditions\("\) as Cereberations but that would destroy the proposed abbreviation which sounds better.

There is another terrific and more recent academic tradition of this kind but it may probably pale in comparison to the former. I am referring to the mathematicians at RAND Corporation who had triggered a similar wave of excitement. I call the former group "mechanics optimizers" and the latter "programming optimizers". The latter, in my opinion, get dwarfed by the former's intellectual heft (though this position is debatable and contradictable) but the quality of academic tradition and significance of achievements are similar for both. Vasek Chvatal and William Cook say⁹ that "The RAND Corporation in the early 1950s contained what may have been the most remarkable group of mathematicians working on optimization ever assembled: Arrow, Bellman, Dantzig, Flood, Ford, Fulkerson, Gale, Johnson, Nash, Orchard-Hays, Robinson, Shapley, Simon, Wagner, and other household names. Groups like this need their challenges."

Indeed, they do!

Brachistochrone problem (among others) was the former group's challenge around which they rallied. Travelling salesman problem (among others) was the latter's.¹⁰ Without a rallying point, they would not have been able to reinforce each other's thought processes; they would have remained in isolation and struggled with their academic pursuit in their individual capacities, probably failing to accomplish anything of much note and fading into oblivion like any layperson. An academic \(``\)sect\("\) has to form around such a problem.

Now there is a chicken and egg problem. Did the challenge problem bring the Giants together or did they come together and thereafter create a problem to exercise their brains? Given their capabilities, they must surely have intuited beforehand that the rallying problem, if solved satisfactorily, would lead to a paradigm shift and disrupt the status quo in their surrounding academic traditions in a good way. Anyway SECTs happen very rarely and when they occur, they come with an expiry date.

A third striking example that has been well recorded in historical texts seems to be the school of Greek geometers but discussing their legacy goes well beyond the present scope. Would it be fair to say that this SECT reached its E-L summit with Euclid's Elements? But of course, of course! And when Euclid's parallel postulate was picked upon by Gauss, Lobachevsky, and others for further nitpicking, it ushered in the era of non-Euclidean geometry. This derived SECT, in turn, reached its E-L summit via the progress made by Levi-Civita and Riemann. I could also mention the quantum circle but that seems to be just too obvious.

And what about the early thinkers who laid down the first rules of language! Who were they who constructed the rules of Sanskrit language (like Pāṇini and others) and which SECT's genius (like Saptarishi and others) contributed to the compilation of the Rig Veda? How did early societies come up with language, or did a few Geniuses invent language that led to the formation of human societies? Language is probably the only medium for intellectual thinking. There may surely have been other languages before Sanskrit, but Sanskrit seems to stand out among others for some reason. Vedic Sanskrit must have played a very important role in our early intellectual traditions. And though I can only present this as an intuition right now, it seems to be powerful enough to pay attention towards. Imagine the cerebral traditions that must have been required to create a writing system as pivotal as Brahmi and a natural language as sophisticated as Sanskrit! Of the origins of Vedic Sanskrit, I have not found a more insightful description than Sri Aurobindo's ideas below¹¹

\(``\)...words, like plants, like animals, are in no sense artificial products, but growths — living growths of sound with certain seed-sounds as their basis [...] In their beginnings language-sounds were not used to express what we should call ideas; they were rather the vocal equivalents of certain general sensations and emotion-values. It was the nerves and not the intellect which created speech...\("\)

\(``\)...language expresses at first a remarkably small stock of ideas and these are the most general notions possible and generally the most concrete, such as light, motion, touch, substance, extension, force, speed, etc. Afterwards there is a gradual increase in variety of idea and precision of idea. The progression is from the general to the particular, from the vague to the precise...\("\)

\(``\)...one remarkable feature of language in its inception is the enormous number of different meanings of which a single word was capable and also the enormous number of words which could be used to represent a single idea. Afterwards this, tropical luxuriance came to be cut down. The intellect intervened with its growing need of precision, its growing sense of economy...\("\)

Even the most trenchant western academicians will realize the importance of Vedic Sanskrit. Doesn't NASA, building up on Rick Briggs' work, believe Sanskrit to be one of the most suitable natural languages to guide the development of an AI programming language? Even Donald Knuth who may be the most impactful computer scientist the world has ever known, took to learning Sanskrit while developing his formal language and parsing theory. Now if one accepts the importance of Vedic Sanskrit, then one should not fail to realize the importance of the Rig Veda, probably the first poetic composition to worship nature. Its exact date of origin can never be known because of its oral traditions. Acceding importance to Vedic Sanskrit and not according an equal (or more) importance to the Rig Veda is like being an ardent follower of Newtonian achievements but not having the slightest urge to browse over the unabridged version of Philosophiae Naturalis Principia Mathematica. Newton is Newton to us due to his Principia.

A superficial skimming of the Rig Veda may persuade someone not well trained in interpreting ancient poetic compositions, let alone the surrounding sociocultural nuances (like how was the society thinking in those days, or what was the zeitgeist of those times?), to label it as an inconsequential collection of rituals or just praising nature in the anticipation of building such rituals. But that is a blatant mistake and Sri Aurobindo brilliantly challenges such claims. His message is unmistakably scientific and intellectual. If only he used simpler English to communicate, his ideas would not have been that difficult to get access to. Consider, for example, that immortal line by William Wordsworth, \(``\)The Child is father of the Man.\("\) What did he mean by it? What is the deepest interpretation possible? He probably meant that those things that evoked the strongest emotions in him towards nature's beauty during his childhood will stay with him for his entire life. A more pragmatic interpretation would be that an adult is formed as a result of the habits and sensibilities that he develops during his childhood. So, the Child basically refers to childhood of the person and the Man refers to adulthood of the same person. Our entire adult life is spent paying homage to our childhood dreams, aspirations and experiences. Thus, a poem can have several interpretations. And the more you think about it, you come up with interpretations and incisive insights which even the poet may not have been aware of. And that may be the goal of the most sublime poems which is to allude to the subtlest and deepest truths — and a higher truth. Sri Aurobindo shows that something similar is true for the Rig Veda because there is a deeper interpretation possible for this epic poetry. However, it is difficult to recreate its exact hidden meanings because we live in very different times now and have got very different sensibilities today. That is the price we pay for forgetting all about a SECT, or even worse, not noticing a SECT as it sweeps by us.

Such SECTs seem to show up whenever a new language of sufficient intellectual calibre gets created. Although it is not absolutely accepted among experts, I regard mathematics too as a language, even in the strictest sense. Sure, its grammar is quite rigorous and abstract which makes it a rather peculiar language differing drastically from the natural languages. But think about it, mathematicians keep discovering theorems that are more and more abstract while inventing proofs for those theorems that keep getting more and more rigorous. It is their form of novel writing. Maybe more precisely, Mathematical Logic is a meta-language or a language template so that all the major sub-branches of mathematics such as probability and geometry instead are closer to our definition of language, in the same way that (Vedic) Sanskrit is seen to be a language template from which most Indian languages are derived. I also regard Music as a language expressing human emotions. Typically, a certain type of creative person turns to music to express himself because of the inadequacy of such inherent expressiveness in natural languages. Then the vocal cord of the creative or an instrument in the adept hands of the creative produces music. Repeated practice of that music creates in his mind some suitable lyrics. Applying the lyrics to the music produces a song. And finally, repeated singing by the creative may motivate him to devise a dance to accompany the song so that he could express himself better. This is most commonly the total lifespan of such a creative\(/\)artistic process, which in itself is a unique kind of intellectual\(/\)cereberal endeavor and definitely deserves much appreciation. I would, if I could, do a Noam Chomsky to justify both these intuitions but let them be hand-waved for now. However, more importantly, where and how do we find the other SECTs!

When a SECT is identified, the academic community impulsively feels the need to sustain the newly created traditions. In fact, if the goal of academic pursuit is to discover new knowledge which it is, then it feels almost ironical that the urge to sustain the newfound cerebral traditions is generally higher than the urge to search for new knowledge. But it is not actually so because new knowledge cannot simply be created by craving for it. New knowledge appears when the Homo sapiens species is ready for it but to be ready for it, we must cherish and preserve the spirit and practice revealed to us by the SECTs. It is due to this power of the Giants to install their traditions in society in such a strong way that sustaining those traditions in itself becomes the goal of academic pursuit. SECTs inspire many circles to repeat the same feat and such circles may, more frequently than expected, match or even exceed the level of achievement secured by their inspirational figures but will generally fail to generate the same level of amazement in society as the Pioneers. Just like pandemics sweep the planet once in a century or two, so do such SECTs with an unknown (albeit probably lower and more uncertain) frequency. These Genius-types appear to exist in a parallel, nonmaterial, Platonic-spiritual world, and just like during pandemics, a very few of them get infected on some rare occasion by a highly potent and virulent strain; then they infect the rest of the society but in a good way. In fact like Hilbert said, no one shall expel us from the paradise which these very few great men of the past have created for us. I am referring here to those very few who built the world of thoughts. This is not to say that those clever and enterprising doers who excelled in the world of actions thereby lifting us from physical misery and bringing us all into the paradise of material comfort are any smaller; only that this article is not about the doers but about the thinkers both of whom are remarkable coordinates albeit living in different yet related manifolds.

Can we now get back to the original question that was posed, viz. what is a Genius?

And the answer is...

A Genius is what a Genius does.

Okay, Okay! I know that is a non-answer.

But then asking for a definition of Genius is like asking for a definition of God. It's pretty much redundant to do so. You will fail to satisfy yourself with any kind of answer, let alone others. The vital characteristic of a Genius is that his thinking is directed towards Nature instead of people, towards science rather than money, that he professes cerebral thinking (PFC!) over thinking originating from the limbic system. Geniuses can be heroes or titans. Heroes expand a field. Titans expand the field and also understand it with so much clarity that they can explain its intricate concepts lucidly even to laypeople. For example, Faraday was a Hero and Weinberg a Titan.

But much about Genius as with any complicated or complex notion is hazy and fuzzy. Instead of talking about Genius, talking about SECT makes more sense. So how is a SECT created? Go figure!

The best way to understand Genius in a deep sense is by spending time reading the original manuscripts written by Geniuses otherwise it is strongly advisable to steer clear of them and move on with the immediate urgencies of daily lives.¹² Now in what spirit should we end this three-part discussion?

Newton had climactically announced in Principia: I feign no hypotheses.

And to paraphrase what Descartes famously said in First Meditation: I bring in for questioning all those things that may be called into doubt.

Well, let that linger on...

References