Category Archives: Lagrange and variational mechanics

Lagrange, variational mechanics, and holism in the sciences


Maxwell and the Treatise on Electricity and Magnetism:


Maxwell Between Two World-Views

Many of us may know what it means to feel “at sea”: without beacons to steer by, without terra firma on which to set our feet. A dialectical passage between two world-views is like that, and James Clerk Maxwell’s life-story might be read as the log-book of just such an expedition: a lifelong search for a clear and coherent view of the physical world. Maxwell’s voyage would almost precisely fill his lifetime, but it would in the end be rewarded by his recognition of one single principle, the principle of least action, which would be key to a virtually complete inversion of the Newtonian world order from which he was escaping.



In a sense, Maxwell was born into a dialectically-divided family. His father was Scottish, and Maxwell spent his early, formative years at the family home of Glenlair in rural southwestern Scotland. His mother on the other hand was English, and though she died while Maxwell was very young, her family was to have a strong influence on his career. While the English spirit would lead him eventually to Cambridge and the epicenter of an aristocratic, Newtonian concept of both science and society, the Scottish channel would lead him to a democratic view of society, and with it an appreciation of experiment and the evidence of the senses, together with a profound mistrust of the mathematical abstractions Newtonian theory.

These two themes met in abrupt confrontation when he was dispatched to Edinburgh to enter a new academy, designed to prepare students for entrance to English universities. In an encounter which must been a rude awakening, he was beaten up by his new fellow-students for his rural attire and his country ways. He stood his ground and soon became a leading student, but the encounter must have thrown light on an issue which would abide throughout his life.


Edinburgh University

Maxwell was clearly ready for entrance to Cambridge, for which his interest in science and his skill in mathematics surely qualified him; but he delayed for a year at Edinburgh University, and then, against the advice of family and friends, persisted in continuing there for a second year. At Edinburgh, he was encountered with excitement a truly liberal education; he loved, as he affirmed later, his professor of natural philosophy, and he became confirmed in his skepticism by the metaphysics of Kant as taught by Sir William Hamilton. Maxwell was not so much following Kant as agreeing with him: he left Edinburgh with a lifelong disbelief in the inert particles and forces upon them, on which Newtonian science rested.

After these two years he went on to Cambridge, where his skill in mathematics earned him entrance to Trinity College–the college of Newton–and high standing in the rigors of the tripos examinations. But he had brought his Edinburgh education with him, utterly abandoning Newton’s world, as we shall see when we turn to the first of his scientific papers.


Three Papers on Electricity and Magnetism

At this point, we find Maxwell, having obtained a fellowship at Cambridge, fully embarked on his voyage on the open seas. He is fascinated especially by the phenomena of electricity and magnetism, but he has no interest in joining the scientific community of his time elaborating Newtonian “laws of force” acting on electric “charges” or magnetic “poles”. He has met Michael Faraday, the self-proclaimed unmathematical philosopher, who has been dong brilliant experiments at the Royal Institution in London. Maxwell has, I think we can say, begun a lifelong devotion to this unassuming character, who represents the very opposite of the Cambridge/Newtonian concept of science–and almost defiantly takes up Faraday’s cause as his own. These are open seas: how to proceed?


Paper 1: An Analogy

Maxwell turns, not to theory, but to analogy. He shares common ground with Faraday regarding an interest in visual thinking–Faraday has presented his insights into the magnetic field by way of patterns formed by iron filings. Maxwell perceives these as the very lines of flow of a fluid. Here, then, is a gift from Faraday, a visual scientific language Maxwell can use! So is conceived the first of three papers on electromagnetism: On Faraday’s Lines of Force. This instrument of analogy, and with it, the goal of writing for the common man (for the democratic intellect, as one student of Scottish thought has put it) is to become one of the sure signs of the new world-view, towards which Maxwell has already begun steering.


Paper 2: A Physical Theory Maxwell has a great propensity for wit, though his friends remark on the difficulty of catching his intent. It may be a shield for a person who is living between worlds: not fully a member of the world friends suppose him to share with them. Maxwell is now in the position of having arrived at a whole view of the interrelations among the electric and magnetic phenomena: yet having no structure of theory in which to compose such a vision. I have proposed that his recourse is that of Aristophanes, in Peace, or the Birds–to stage his vision in the mode of comedy. Maxwell has been playing with the subject of mechanism (he and Karl Marx happen to have taken a course from the same teacher in London, though perhaps not at the same time!). He cannot mean that he proposes that such a mechanism actually exists, but he invents a great machine, for which he writes all the appropriate equations, which would do all the things the electromagnetic field actually does. Maxwell calls it A Physical Theory of the Electromagnetic Field, but he is not proposing that these vortices and idler-wheels actually exist. Like Aristophanes’ world of peace, it is an object for the mind, a project of pure thought. Again, this is a major step toward the goal Maxwell is seeking: in the new world, we will not take mechanisms seriously.

Maxwell’s jeu d’esprit is so successful that he can calculate from it the speed at which vibrations would be transmitted: it is very close to the speed of light! He is the point now of announcing the electromagnetic theory of light. But his discovery hangs in the air (or floats on the waves) it is no more than a beautiful play of thought.


Paper 3:Dynamical Theory

At last, the gods smile on Maxwell’s endeavor. He meets dynamical theory–and a new world begins to take shape. The mode of this encounter is deeply ironic, and correspondingly confusing. One of the most obdurate and imperious of Newtonian advocates is Lord Kelvin, once more modestly Maxwell’s colleague, William Thomson. He, with Maxwell’s close scientific friend P. G. Tait, have undertaken to write an ambitious, one might say proud, Treatise on Natural Philosophy. It’s intended to lay, once for all, the secure foundations of Newtonian science. An edifice of all physical science is to be built on this solid foundation, of which they’ve published only Volume 1.

At sea in uncharted waters, very strange things can happen! Kelvin and Tait, building their arguments on solid Newtonian foundations, expound a new approach to physical problems in terms of energy, rather than force: it is termed dynamical theory (they are importing it to England from the Continent, where it has been developed.) Though Kelvin resolutely insists that it is really still Newtonian, and nothing new, Maxwell sees light at the end of his tunnel (or a beacon on a new continent!) If equations can be written in terms of energy rather than force, nothing further needs to be said about forces acting upon those underlying particles, which he has always been convinced, do not exist!


The Treatise

The new dynamical equations are named after Pierre Lagrange, who wrote them, and Maxwell now uses them to characterize the electric and magnetic fields as regions on energy and momentum. Lagrange makes no explicit reference to ponderable mass, but that no longer matters–the equations carry all the energy that reaches Earth from the Sun. Maxwell publishes his Dynamical Theory of the Electromagnetic Field, and confidently announces his electromagnetic theory of light, based on the new equations.

Maxwell’s problem is not yet solved. Either the equations stand, as Kelvin maintains, on Newtonian theory – in which case we have only avoided the issue by not referring to some underlying particles, hardly more than a subterfuge, certainly not worthy of Maxwell, or they flow from some higher principle which Maxwell has not yet named. This is perhaps the darkest night of his voyage: he has glimpsed the new shore, but it has slipped away in the obscurity of this night.


The Principle of Least Action

Blessedly, Lagrange’s dynamical equations of motion can be derived from another source: indeed, this new source is their natural home, for this new origin is itself expressed in dynamical terms, i.e., in terms of the potential and kinetic energies of the system as a whole. Causality of the whole natural world is at stake here, so this “derivation” of Lagrange’s equations is no mere mathematical question! For Newton, causality flows from below to the whole: the “reason” things happen is mechanical, the whole moves as a consequence of the motions of its parts. So it was with Maxwell’s joking physical theory; he knows very well there are no such underlying parts. The new derivation of Lagrange’s equations flows from above–and with it, causality likewise flows downward, from some inclusive whole.

That inclusive whole–from which all the motions of he natural world flow–is the Principle of Least Action. The motions of the natural world arise ultimately from potential energies, such as the calories in a loaf of bread, or the BTUs in a gallon of gasoline. The conventional symbol for potential energy is V. Motions arise as potential energy is converted to kinetic energy, whose symbol is T. The difference (T–V) is called the Lagrangian, and the action (A) associated with any motion is nothing more complicated than the product of the Lagrangian and the time (t) the motion takes:

 A = (T – V) x t

 With that modest introduction, we can now state the principle on which it seems, nature runs. For any system:

The motion will be such that the action is least.

It can get complicated when systems are complex, or when relativity or quantum principles are involved, but it works, too, for systems as simple as a falling stone. Since each system is characterized first of all as a whole, it is inherently organic, and applies especially well to ecologies, which nature appears to see primarily as wholes, and organic.

Maxwell learned of this from the writings of William Rowan Hamilton of Dublin; he jokes of his “two Hamilton’s, saying their metaphisics are valuable in proportion to their physics. He means, I think, that the Kantian metaphysic espoused by Sir William Hamilton of Edinburgh was geared to the Newtonian world-view. The “new” Hamilton of Dublin is geared to a new, very different world-view in which the whole is primary as such, and not an assemblage of parts, and causality flows organically from whole to part. Wholes of course do not have to be big, the quantized hydrogen atom, a protein molecule, or the living cell, are instances.

We spoke earlier of Maxwell’s devotion to Faraday. Now we must ask, has he brought Faraday with him to this new land of Least Action? The answer, I can say confidently, is Yes.

How do we characterize a “system”? In the old, Newtonian way in which the parts were causal, it was important to describe a system in terms of those parts which constituted it and caused it to move. But now, parts are no longer causal. Our concern will be, instead, to characterize the state of a whole connected system. Interestingly, there is no one right way to do that! Any set of measurements sufficient to characterize the state of the system will serve. They don’t have to be readings of meters; Faraday’s diagrams of lines of force will serve very well to characterize a magnetic field. His intuitive interpretations of the behavior of his galvanometers serve him better than columns of numbers. Further, Maxwell’s analogy to fluid flow may serve very well to comprehend the structure of the magnetic field. Indeed, the Principle of Least Action in effect restores life to nature, which tends to move, as Faraday observed of his magnets. We have indeed arrived at a whole new world, yet one which Faraday, and Maxwell in his devotion to Faraday, already had in view.

Why has the modern world so resisted recognition of this principle, leaving it to rather esoteric studies within mathematical physics rather than teaching and embracing it generally as a far better way of understanding and caring for the natural world? Any thoughts on this will be very much appreciated.

What Do we Mean by the Term “Elementary”?

What do we mean when we use the term ,”elementary”, in relation to a science? Does it refer to an easy introduction, as contrasted with an “advanced” treatment of the same subject? Or does it mean a solid account of the very foundations of the science? Or, for that matter, are these the same thing?

Maxwell had a tendency toward writing “elementary” texts: he wrote one on heat, and another on mechanics, both for use in classes for workingmen – a project to which he was deeply committed. Finally, at the time of his death he was at work on his “Elementary Treatise on Electricity and Magnetism, intended to serve as the Cambridge text to support a new degree in experimental natural philosophy at Cambridge University.

My sense is that Maxwell endowed each of these with earnest attention – that he regarded the “elements” not as evident, but as a topic to be approached with great care. Our decision as to what is elementary in a science has a great deal to do with our sense of the form the finished product will take – so that the most difficult issues may focus on the most elementary beginnings.

For example, Maxwell wrote his workingmen’s text in mechanics, Matter and Motion, only after he had hit on the fundamental idea, new to him, of Lagrnagian mechanics and generalized corrdinates. This would not be a mechanics in Newtonian form, in which the elements would be assumed to be hard bodies acting upon one another according to laws; rather, elements of this sort would be the least known components of the system, represented by generalized coordinates.

In this view, what we observe initially is a whole system of some sort; it is this whole which is fundamental, and truly elementary. The parts which compose it, we may never know. Our science can be complete and secure even if that question remains unresolved, or unresolvable.

This is the point of view I believe Maxwell had come to, underlying his approach to the new program at Cambridge as well. If so, must it not represent a truly revolutionary inversion of our very concept of scientific knowledge?

It fitted the primacy he – following the path of Farday – was giving to the concept of the electromagnetic field. In this view, he field would not be a secondary phenomenon, a composite or consequence of simpler “elements”, but itself both simple and whole.

If the elementary is what is primary, then in the case of the field it is the whole which is the element, from which we deduce what we can, concerning lesser components. Faraday had felt strongly that in the case of electricity, there was no “charge” lying on the surface of a charged body, but what we call a “charge” was a field, which filled the room.

Isn’t it the case that when we ask for the “explanation” of a physical system, we are asking for an account in terms of its elements? If so, then the field is itself explanatory, and we would not seek explanation in terms of the actions of some lesser parts. What will be the consequences if we extend this view to physical explanation – or explanation beyond the realm of physics — more generally?

“The Dialectical Laboratory”: A lecture on behalf of holism in the sciences


My lecture, the “Dialectical Laboratory ” (see the “lectures” section of this website) , was given as a sort of parting statement to the St. John’s College community in Santa Fe.  But though directed to the college, and expressed by way of references to certain of the “great books” of that tradition, its message is of far broader import.  The “dialectical” issue – meaning, a watershed of western thought – is between a science based on mechanical actions between disparate parts, and a holistic science in which wholeness is respected, and whole systems are regarded as fundamental, not as mere aggregations of parts.  

Each of these two very different scientific approaches has its rigorous theory, and either can be used to solve engineering problems.  But conceptually they are worlds apart, and I am convinced it’s crucial that we follow the way of holism, and learn, before it’s too late, to appreciate and work with systems – from the least living organism to the global environment – which are more than the sum of mechanical parts.  Science is moving in this direction, but there is now no time to lose! 

Comments on these remarks, as well as on the lecture itself, will be welcome in reponse to this posting. 




In Praise of Generalized Coordinates

I’ve been expressing my enthusiasm for a holistic approach to the understanding of nature — in relation to my favorite topic, the electromagnetic field, this takes the form of the Lagrangian equations for the field as a single, connected system characterized by its energy, not by forces.  It was crucial to Maxwell’s development of the equations of the field in his “Treatise on Electricity and Magnetism” that they be formulated as instances of such a connected system — i.e., in Lagrangian terms, and NOT on the basis of Newton’s laws of motion.  (The difference — very fundamental to our understanding of nature — is developed in “The Dialectical Laboratory”, in my “Lectures” menu.) Now, the question arises: “If we start in this way, from the ‘top down’, how do we ever arrive at the elements?”   The answer is, “We DON’T!” We move logically “downward” by finding the dependence of the energy of the whole system upon ANY set of measurements we want to make — provided only that it’s a complete (i.e. sufficient to determine the state of the system), with each measure “independent” of the others. We find such a set of measurements by doing experiments — and when we get them, they are called “generalized coordinates”.  The important thing is that there may be many ways we can define them, each set as good as the others: and in the whole process we never get any”real,underlying elements” — we don’t need them!  Reality is founded at the top, not the bottom, of the chain of explanation.   This is Maxwell’s new view of physical reality, founded upon the field.  It is the opposite of the notion of “mechanical explanation”, and it is the direction which our approach to nature desperately needs to take as we approach the challenges which lie before us today.  In terms of the philosophy of science, Maxwell it seems was far ahead of his time.  I propose to call this the “Maxwellian Revolution”. 

Indigenous Views of Nature and the Deep Roots of Western Science

When I wrote yesterday about the “deep roots” of Western science, I intended to point to a possible relation this opens up between the domain of “science” and Indigenous views of the natural world.  If we follow that line of development which leads from Aristotle through Leibniz to the holistic mathematical physics based on the Principle of Least Action, we find ourselves in a position much closer to that of Native American thinkers than we might have expected.Modern science in its mechanical mode cuts off “science” from any sense of wholeness or, especially, of purpose. It wants to reduce all quality to quantity, all motion to the operation of laws which bind matter apart from any sense of goal or meaning, and sees “nature” exclusively as an object from which we stand apart as mere observers. None of these limitations apply to the physics in the holistic mode.  Least Action applies to whole systems, and sees systems moving directionally toward the optimization of a quantity which applies to the system as a whole.  Although this goal may be no more than the optimization of a mathematical quantity, it opens the way to thinking of systems such as organisms or ecologies as moving as wholes toward ends — a line of thought of which the modern world is in desperate need.One more link in this line of thought: the modern computer is bridging the gap ;between “quantitative” and “qualitative” thinking.  What goes in as number typically comes out on the computer screen as a graphical image readily grasped by the intuitive mind and conducive to interpretation in terms of purposes and goals. We can see how systems are moving, and where they “are going”.   Nothing stands in the way of reading these in terms of purposes, and that is what we do on a daily basis — think for example of evidences of the consequences of global warming emerging from complex computer modeling.  Thinking in this way in terms of whole systems,  understanding their motions in terms of a mathematics of optimization, and bridging the gap between quality and quantity — all this is yielding an approach to science at once new and old — in a continuous thread leading from Aristotle into the age of the modern computer.  If we follow that path and think of modern science in terms like these, then it seems to me the gap between a holistic science and Indigenous relations to the natural world is not as deep as it had seemed.  Set aside mechanistic thinking, embrace the sense of nature as a whole of which we ourselves are part, admit goal as a category amenable to science — and then the old gap between Indigenous, or simply hunan views of the world, and those of “western science”, begins to dissolve.   Thus the Cosmic Serpent project, designed to consider this relationship, begins to look much more promising than it otherwise might have.