Of Bernoullied Euler and Eulered Lagrange and Other Gianted Giants

This is an important question that deserves to be revisited again and again. Clearly, today the S.T.E.M. community is more concerned with simulation modeling, algorithm engineering and data analysis as opposed to theory building. Fast and powerful computers have shifted intellectual pursuit from theory building to these other more fruitful endeavours. This is not a mistake in itself and may arguably be considered more useful to society (because simulations are far more relatable, algorithms are the technology of future, and we are producing so much data with respect to the 3 V's). However, lest we forget the measure of the theoretical feats achieved in the past, repeating often what a genius is and what a genius does is required from time to time.

There are many books of excellent quality¹ ² ³ which celebrate such geniuses. It is not completely clear how genius is produced. There's that debate between raw intelligence and experiential intelligence. And then many also wonder about its root cause. Is it mostly training? Is it mostly society? Is it mostly genes? Is it some X factor we inadvertently gloss over? Intuition will nudge us into thinking genes may be playing a vital role because after all whatever we physically become from a single cell is encoded in those genes but I am aware that geneticists advise extreme caution on making such simplistic arguments. Still if it turns out that genetic phenomena play a significant role in determining such traits,⁴ then the fact that both the words, genes and genius, sound similar would be such a whimsical coincidence.

Every academician pays tribute to these geniuses sooner or later. The preferred method is almost always to elaborate upon Newton's immortal words, If I have seen further it is by standing on the shoulders of Giants.

But why don't we randomly explore a gem from the times past such as the Euler-Lagrange equation which is one of the most remarkable achievement (maybe the most) of classical mathematics and physics. However, the equation gets repeated so often in so many syllabi and curricula of schools and colleges that most students hardly realize the tremendous excitement behind its development. It is like having visited an exquisitely architected and sculpted historical site but only realizing its significance in terms of human achievement after several years have passed and then your wise old grandfather tells you interesting stories behind the construction of the site. Then you want to hit yourself with utter regret for why did not anyone make you wiser about it at the right time because now it's too difficult for you to create time to visit that site again or even if you could get time, you would now lack the child-like raw intelligence which the prodigious monument relies upon to magically inspire new achievements out of you.

How much of the excitement associated with the Euler-Lagrange equation can be reimagined in a light blog article! But we can still attempt to briefly glimpse this ancient temple of wonder. I shall elaborate the mathematical details behind the equation so that the reader can zoom through the text without much hassle because the goal is not to help the reader master the mathematical aptitude required for this topic. The goal is to motivate thoughts about the mathematical circles that excelled in building far-reaching scientific theories and to realize that genius is not an individual but a collective. A genius spans over several lifetimes of several individuals.

Boring Modeling

We show how to recast the classical equations of motion into Euler-Lagrange equations of motion. This is more in line with the approach taken by Lagrange whose primary focus was classical mechanics. However, Euler-Lagrange equations were originally derived because of the goal to solve trajectory optimization problems but more on that later.

We begin with a rigid body that we want to analyze. In practice, no object is perfectly rigid. However, most objects of value to classical engineering can be approximated to be a rigid body.

A rigid body is assumed to be made up of particles. In physics, a particle can refer to fermions, atoms, molecules or even a sizeable box-shaped container. Here we vaguely use it to mean the smallest unit of resolution of an object beyond which finer analysis will yield no benefit. This is analogous to finite element analysis where finer discretization leads to improved analysis but there is a trade-off against computational efficiency. Now such a smallest unit can really be very small compared to the object of interest. For example, a robot arm will consist of countably infinite such particles. Thus, we have reduced the object of our interest to a rigid system of particles.

Each particle can apply force on any other particle whether that particle is part of the considered rigid body or external to the rigid body. The forces that can be exerted on any particle belonging to this rigid system are classified into

Choosing an arbitrary reference frame with fixed coordinate axes allows us to assign a position vector to each particle. This could be a 3-dimensional Euclidean coordinates which everyone is usually familiar with. We can then define the position vectors in terms of those coordinates.

Suppose that there are \(k\) particles in the rigid body, then the particles can be denoted by \(\boldsymbol{r}_1,\boldsymbol{r}_2,\cdots,\boldsymbol{r}_k\). Clearly, we can now define the evolution of the system with respect to the dynamics involving these \(k\) position vectors. However, we have already established that \(k\) can generally be a very large number and we are constrained by computational tractability not to solve large-sized problems.

Fortunately, one can always study the evolution of these \(k\) position vectors in terms of \(n\) generalized coordinated vectors. This leads to a new coordinate system, called the generalized coordinate system, comprising \(n\) coordinate axes, where all of our analysis shall now happen. The position vectors representing the evolution of the system are essentially the \(n\) axes of this coordinate system. This transformation is useful because we can usually obtain \(n\leq k\).

The generalized coordinates are very easy to identify in practice as one basically has to track down all the moving parts of the rigid body mechanism. Each moving part denotes one or more degrees of freedom. The total degrees of freedom form the \(n\) generalized coordinates.

Now if a rigid body mechanism is kept completely unconstrained, then it would have \(3k\) degrees of freedom, and then \(n=3k>k\) which is a contradiction. This contradiction is usually avoided in the case of rigid body mechanisms because the motion is constrained. Each such constraint associated with a rigid body can be holonomic or nonholonomic. But we shall restrict ourselves to only holonomic constraints for the purpose of deriving Euler-Lagrange equations. The latter case is ignored as Euler-Lagrange equations are not applicable and one has to fall back to the tedious route of using Newton's Laws from scratch. So suppose the system is subjected to \(l\) holonomic constraints. Each holonomic constraint is of the form

Note that each holonomic constraint must be expressible in closed-form as a function of only the position coordinates. If we were to pick particles 1 and 2 belonging to the system, then the rigidity constraint demands that the distance between these two particles remain constant (fixed value of \(L\)) at all times.

\[g_i\equiv(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\boldsymbol{r}_1-\boldsymbol{r}_2)-L^2=0\]

In principle, we would be required to write out each holonomic constraint and then reduce the system of the particles' position vectors to a system of generalized coordinates. But this is gets complicated so the intuitive method of exploiting the degrees-of-freedom is usually followed. Thus, each of the original position vector can, in theory, be expressed in terms of the generalized coordinates.

Since the evolution of a system cannot be studied without explicitly bringing time into the picture, so we have

This simple-looking equation coupled with certain tricks and definitions forms the basis for deriving the Euler-Lagrange equation for our rigid body.

Principle of Virtual Work

We must first introduce the concept of virtual displacement. Virtual displacement is an assumed infinitesimal change of system coordinates occurring while time is held constant. So first let us try to understand the infinitesimal change of system coordinates. The definition of total derivative comes to our rescue here.

\[d\boldsymbol{r}_i = \frac {\partial \boldsymbol {r}_i}{\partial t} d t + \sum_{j=1}^n \frac {\partial \boldsymbol {r}_i} {\partial q_j} d q_j\]

But suppose that time is held constant by setting \(dt=0\) which makes no physical sense because no actual displacement (not even infinitesimal amount) can take place without the passage of time, hence the term virtual is applied. Virtual displacement is denoted by the symbol \(\delta \boldsymbol{r}_i\) to differentiate from the usual definition of infinitesimal displacement. Think of it informally as freezing the world at a particular time instant to be able to solely analyze its topology at a leisurely pace so that \(\boldsymbol{r}_i\) is not dependent on time, meaning the function form of \(\boldsymbol{r}_i\) can be written as \(\boldsymbol{r}_i(q_1,\cdots,q_n)\).

\[\delta \boldsymbol{r}_i = \sum_{j=1}^n \frac {\partial \boldsymbol {r}_i} {\partial q_j} \delta q_j\]

We can take a look at a quick application of this concept. Suppose two particles in a rigid body have position vectors \(\boldsymbol{r}_1,\boldsymbol{r}_2\). Under virtual displacement, their position vectors are transformed to \(\boldsymbol{r}_1+\delta \boldsymbol{r}_1,\boldsymbol{r}_2+\delta \boldsymbol{r}_2\). But due to rigid body assumption, the distance between the two particles must not change. Therefore,

\[(\boldsymbol{r}_1+\delta \boldsymbol{r}_1-\boldsymbol{r}_2-\delta \boldsymbol{r}_2)^T(\boldsymbol{r}_1+\delta \boldsymbol{r}_1-\boldsymbol{r}_2-\delta \boldsymbol{r}_2)=(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\boldsymbol{r}_1-\boldsymbol{r}_2)\]

We can perform further simplification to arrive at an equation that will be used later.

\[((\boldsymbol{r}_1-\boldsymbol{r}_2)^T+(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)^T)((\boldsymbol{r}_1-\boldsymbol{r}_2)+(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2))=(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\boldsymbol{r}_1-\boldsymbol{r}_2)\]

\[(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\boldsymbol{r}_1-\boldsymbol{r}_2)+(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)^T(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)+(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)+(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)^T(\boldsymbol{r}_1-\boldsymbol{r}_2)=(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\boldsymbol{r}_1-\boldsymbol{r}_2)\]

\[(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)+(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)^T(\boldsymbol{r}_1-\boldsymbol{r}_2)=0\]

\[(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)=0\]

We are now ready to state the principle of virtual work. But let us first make two assumptions because a theory's accuracy rests on how well its assumptions are specified.

The second condition is called D'Alembert principle. When is it reasonable to make this assumption? It is easy to see that this assumption holds whenever the constraint forces between a pair of particles act along the vector connecting the position coordinates of the two particles. This is true in the case of rigid bodies. Consider two such particles and denote the force exerted on particle 1 by particle 2 as follows where \(a\) is a constant scaling factor

Then the work done by the constraint forces corresponding to a set of virtual displacements can be easily shown to be zero as follows

\[(\boldsymbol{f}^{(c)}_1)^T\delta\boldsymbol{r}_1+(\boldsymbol{f}^{(c)}_2)^T\delta\boldsymbol{r}_2=a(\boldsymbol{r}_1-\boldsymbol{r}_2)^T(\delta\boldsymbol{r}_1-\delta\boldsymbol{r}_2)=0\]

Now going back to the first assumption, equilibrium implies that the net force acting on each particle is zero. Let \(\boldsymbol{F}_i\) denote the total force on particle \(i\) which is equal to zero. Then the total work performed due to any possible set of virtual displacements is given by

The total work is clearly zero because each individual term is zero. The total force exerted on each particle, \(\boldsymbol{F}_i\), consists of the externally applied force \(\boldsymbol{f}_i\) and the constraint force \(\boldsymbol{f}_i^{(c)}\). So \(\boldsymbol{F}_i=\boldsymbol{f}_i+\boldsymbol{f}_i^{(c)}\). By D'Alembert principle, \(\sum\limits_{i=i}^k\left(\boldsymbol{f}_i^{(c)}\right)^T\delta \boldsymbol{r}_i=0\) summing over each particle \(i\). Therefore,

\[\sum\limits_{i=i}^k\boldsymbol{F}_i^T\delta \boldsymbol{r}_i=\sum\limits_{i=i}^k\left(\boldsymbol{f}_i^{(c)}\right)^T\delta \boldsymbol{r}_i+\sum\limits_{i=i}^k\boldsymbol{f}_i^T\delta \boldsymbol{r}_i=0\]

This equation is referred to as the principle of virtual work which allows us to analyze the dynamics of a system without having to evaluate the constraint forces.

Rest shall be continued in a later article. Let us end by quoting Koestler's words from his famous Sleepwalkers book. Historians as sharp as Koestler end up serving society by sharing their remarkable insights, and erudite historical insights may sometimes help society even better than scientists, engineers or entrepreneurs.

\(``\)...there are the twin threads of Science and Religion, starting with the undistinguishable unity of the mystic and the savant in the Pythagorean Brotherhood, falling apart and reuniting again, now tied up in knots, now running on parallel courses, and ending in the polite and deadly "divided house of faith and reason" of our day, where, on both sides, symbols have hardened into dogmas, and the common source of inspiration is lost from view.\("\)

\(``\)The jerky and basically irrational progress of knowledge is probably related to the fact that evolution had endowed homo sapiens with an organ which he was unable to put to proper use...The history of discovery is, from this point of view, one of random penetrations into the uncharted Arabias in the convolutions of the human brain.\("\)

\(``\)The muddle of inspiration and delusion, of visionary insight and dogmatic blindness, of millennial obsessions and disciplined double-think, which this narrative has tried to retrace, may serve as a cautionary tale against the hubris of science or rather of the philosophical outlook based on it.\("\)

\(``\)...the example of Plato's obsession with perfect spheres, of Aristotle's arrow propelled by the surrounding air, the forty-eight epicycles of Canon Koppernigk and his moral cowardice, Tycho's mania of grandeur, Kepler's sun-spokes, Galileo's confidence tricks, and Descartes' pituitary soul, may have some sobering effect on the worshippers of the new Baal, lording it over the moral vacuum with his electronic brain.\("\)

References

Arthur Koestler, The Sleepwalkers: A history of man's changing vision of the universe. Penguin UK, 2017.(back)
Ioan James, and James Ioan, Remarkable mathematicians: from Euler to von Neumann. Cambridge University Press, 2002. All of his other books in the same series are also equally excellent.(back)
Martin Aigner, Gunter M. Ziegler, Karl H. Hofmann, and Paul Erdos, Proofs from the Book. Springer, 2010.(back)
Many already consider Asperger's syndrome as one such trait. An interesting perspective is presented by Ioan James in the book Asperger's syndrome and high achievement: Some very remarkable people.(back)

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

Boring Modeling

Principle of Virtual Work

References