Riemann For Anti-Dummies Part 63: Dynamics not Mechanics
by Bruce Director
Despite the prevailing popular opinion to the contrary, human beings are not mechanical systems. So, if you wish to begin to understand the science of physical economy, you must know the science of dynamics, as distinct from, and superior to, the Aristotelean sophistry called mechanics.
The distinction between dynamics and mechanics is not semantic. It is fundamental. Dynamics concerns causes. Mechanics concerns effects. Dynamics treats processes as a whole. Mechanics treats the interaction among some of its parts. They are as different as truth from rhetoric, ideas from words or, love from sex. Though this distinction arises directly through an investigation in the domain of physical science, its implications, as Leibniz himself emphasized, are universal. For without an understanding of the dynamics of situation, it is impossible to know anything about politics, history, science or art.
All Matter is Animated
Though he had developed its essential concepts much earlier, the science of dynamics was founded, under that name, by Leibniz, in his 1691 {Dynamics: On Power and the Laws of Corporeal Nature} as “{ the new science of power and action, which one might call {dynamics}.”
As the name implies, Leibniz’s science of dynamics {dynamis} has its origins in the Pythagorean-Platonic science of Sphaerics, as that science was developed into its modern form by Cusa and Kepler, and it lays the foundations for the development of the higher form of anti- Euclidean geometry associated with Gauss, Abel, Jacobi, Dirichlet and Riemann. Leibniz further identified this new science of dynamics as superior to mechanics: “Through more profound meditation…I learned a truth higher than all mechanics, namely, that everything in nature can indeed be explained mechanically, but that the principles of mechanics themselves depend on metaphysical and, in a sense, moral principles, that is, on the contemplation of the most perfectly effectual , efficient and final cause, namely, God and cannot in any way be deduced from the blind composition of motions.” ({On the Nature of Body and the Laws of Motion}.)
As he enunciated it, the immediate impetus for the introduction of this new science was to correct an error that had been introduced into physics by Descartes and his followers. The Cartesians had insisted that the motions and interactions of physical bodies were completely determined, mechanically, by the visible characteristics of those bodies, that is, their mass and their speeds. Thus, from a Cartesian point of view, a 2000 kilogram body moving at one kilometer per hour would have the same “quantity of motion” (mass times velocity) as a one kilogram body moving at 2000 kilometers per hour. But, as would be obvious to anyone standing in the paths of these two objects, the smaller object moving at the faster speed would be capable of delivering much more power than the heavier object moving at the slower speed. By several different demonstrations, Leibniz determined that this greater power could be measured by the mass times the square of the velocity. Thus, though the two objects in our example would carry the same “quantity of motion”, that is, 1 x 2000 or 2000 x 1 = 2000, their power would be far different. The first object would carry a power proportional to 2000 times 1^2 = 2000, while the second would carry a power proportional to 1 times 2000^2 = 4 million.
Leibniz demonstrated this same principle inversely for the case of a pendulum. As can be demonstrated by physical measurement, the speed of a falling body is proportional to the square root of the vertical distance dropped, or conversely, the vertical distance is proportional to the square of the speed. For a simple pendulum, that means that if the bob were raised to a vertical height of four feet, it would attain a speed of two units at its lowest point and it will possess sufficient power to ascend back to a vertical height of four feet as it continues its motion. On the other hand, if that same body were raised to a vertical height of only one foot it would attain a speed of only one unit at its lowest point and possess sufficient power only to ascend to a vertical height of one foot. Thus, a body moving twice the speed has the power to produce four times the effect, or, more generally, the power is proportional to the square of the speed. Leibniz also emphasized that power (potential) is a characteristic of the physical process as a whole. “Furthermore, I have discovered that this {law of nature} holds instead, namely, that the {whole effect has the same power as its full cause}….” That is, that the power of the cause is completely reflected in the effect. Thus, if something is moving at a certain velocity the cause of that motion must have had sufficient power to have produced that effect. For example, in the case of a pendulum, if the maximum velocity is two units, the pendulum’s bob must have been raised to a height of four units.
(The reader should be aware that Leibniz’s concept of conservation of power is not the same as the reductionist’s notion of “conservation of energy”. The latter is a sophistry based on a fraudulent assumption of the existence of a linear, scalar quantity called energy that is associated with bodies. Leibniz’s concept of power is a characteristic of the principle that is causing the motion of the bodies. As we will indicate below with respect to Riemann’s treatment of Dirichlet’s Principle, power itself is non-linear, as demonstrated, for example, by the power of human creativity on to increase the physical power of man over nature.) This error of the Cartesians, Leibniz insisted, was not merely a technical one limited to the domain of physics, but it indicated a profound epistemological defect. The Cartesians had adopted the false dogma of Aristotle, that the universe was separated into two distinct departments, the department of “physics”, which concerned visible objects and dead material substances, and the department of “metaphysics”, which concerned immaterial things such as ideas and principles. The former were, for both the Aristoteleans and Cartesians, “in the world”, while the latter were “outside the world”.
But, through his simple demonstration, Leibniz showed that the visible characteristics of material bodies were insufficient to explain the real effect that these bodies had “in the world”. This effect was not determined by the material substance alone, but by the {force (kraft) which animated it.} Leibniz called this force, appropriately, {vis viva} or {living force}. Contrary to Aristotle and the Cartesians , Leibniz insisted that this {living force} was more real than the visible, material effects of motion:
“This consideration, the distinction between force and quantity of motion, is rather important, not only in physics and mechanics, in order to find the true laws of nature and rules of motion and even to correct the several errors of practice which have slipped into the writings of some able mathematicians, but also in metaphysics, in order to understand the principles better. For if we consider only what motion contains precisely and formally, that is, change of place, motion is not something entirely real, and when several bodies change position among themselves, it is not possible to determine, merely from a consideration of these changes, to which body we should attribute motion or rest, as I could show geometrically, if I wished to stop and to do this now.
“Now, this force is something different from size, shape, and motion, and one can therefore judge that not everything conceived in body consists solely in extension and in its modifications, as our moderns have persuaded themselves. Thus we are once again obliged to reestablish some beings or forms they have banished. And it becomes more and more apparent that, although all the particular phenomena of nature can be explained mathematically or mechanically by those who understand them, nevertheless the general principles of corporeal nature and of mechanics itself are more metaphysical than geometrical, and belong to some indivisible forms or natures as the causes of appearances, rather than to corporeal mass or extension.” ({Discourse On Metaphysics, sec. 18.})
Thus, to understand a physical process it is necessary to know its dynamics.
The Dynamical Phase-Space
Because the dynamic of a physical process reflects principles that are not directly susceptible to sense perception, it cannot be expressed by a description of the visible appearances of the action. Therefore, Gauss, Dirichlet and, most importantly, Riemann, developed the means to express the dynamic by the determining characteristics of a Riemannian manifold, or what is sometimes referred to in modern science as a “phase space.”
The paradigmatic example of such a Riemannian phase-space is Kepler’s expression of the planetary orbits as the visible pathways determined by the harmonic ordering of the solar system as a whole. These harmonic relationships are not “heard with the sensual ear”, but the sensed motions of the planets, nevertheless, faithfully follow their polyphony. (Newton’s fraud was to replace this celestial dynamics of Kepler with his celestial mechanics, the which merely describes the observed effects. The failure to recognize this distinction is symptomatic of the endemic illiteracy prevalent in the teaching of science today.)
While Kepler established this fundamental basis for determining the interaction of physical bodies as the effect of the dynamics of physical powers, he specified the need to develop a more general approach, specifically the development of the calculus and elliptic functions. The former was developed by Leibniz and the latter was elaborated by Gauss, Abel, Jacobi, Dirichlet and Riemann.
To get a firmer grasp of the implications of this general approach, let us look again at the case of the pendulum from the standpoint of both Leibniz and Gauss.
In a simple pendulum, a material body hangs from a fixed point by a rigid string. and is set in motion by being raised to some height and then released. The bob then descends along an arc, reaching a maximum speed at its lowest point, and ascending along a continuation of that arc to its maximum height, at which point it reaches its minimum speed, and then descends back along the arc just traversed, repeating its previous motion.
Something dramatic emerges when we examine this problem more closely from the standpoint of Leibniz’s principle of power. (For ease of calculation in what follows, we set the length of the string and the mass of the bob equal to one).
It is physically evident that the speed of the pendulum is always changing, from its minimum, at its maximum height, to its maximum, at its minimum height. Thus, like a planet in an elliptical orbit, the time elapsed is non-uniform with respect to the arc traversed. Further, as Leibniz emphasized, the speed of the pendulum along the arc is an effect of the power (potential) of gravity to produce that speed.
As noted above, the maximum speed is an effect of the total power (potential) of the process as a whole, whose measure is proportional to the square of the speed it produces. Consequently, in the case of the pendulum whose motion is due to the power of gravity, the maximum speed is “the whole effect”, and must reflect the “same power as its full cause”. Since the maximum speed is proportional to the vertical height, the potential of gravity to produce that speed is also proportional to that same height. That height can be measured as a function of the cosine of the angle that the pendulum’s string makes with the vertical. (See Figure 1.)
Thus, there is a connected relationship between the rate at which the pendulum’s bob moves along the arc and the rate at which that cosine increases or decreases. In this relationship can be found the difference between the mechanics and the dynamics of the pendulum. The mechanical approach takes the motion along the arc as primary, and the increase and decrease of the cosine as the effect. This is because, from the standpoint of visible geometry, the non-uniform cosine appears to be an effect of the uniform angles and arcs.
(It should be noted, that this mechanical approach is the one taken by Galileo who claimed to have measured the pendulum’s period simply by the angle. This was similar to his attempt to explain the catenary by the parabola. Such mechanical methods appear to produce approximate descriptions of the observations. But those approximations hold only if the action measured is a very small interval, such as for small swings of a pendulum or small parts of the catenary. When the total potential is considered the discrepancy between the curve-fitting approximation and true principle becomes evident. However, the true genius is always able to recognize this discrepancy as a matter of principle, such as Kepler’s famous 8 minutes of an arc in the case of Mars’s elliptical orbit, or Gauss’s famous 16 seconds of an arc in determining the length of the meridian from Goettingen to Altona. Yet history is littered with fools who ignored the advice of such geniuses and made their decisions based on such curve-fitting approximations. )
But, in the real world of physics, the motion of the pendulum determines the angle, and the pendulum’s motion is a function of the continuously changing relationship between the total potential and the effect it has produced up to any moment. Since the potential is measured by the square of the speed, and the speed is proportional to the square root of the vertical distance dropped (which is measured by the cosine), the continuously changing relationship between the potential and its effect can be expressed as a function of the cosine. Thus, from the standpoint of the dynamics of the pendulum, the rate of change of the cosine, which measures the changing relationship between the potential and its effect, must be understood as the measure of the cause of the change of the angle from moment to moment. That is, we must measure the change in angle as a function of the cosine. But, as we will indicate below, this inverse function that expresses the pendulum’s motion is a different inverse function than the simple circular inverse cosine.
To make this point a little more concrete, take the case of Huygens’ isochrone, which is the pendulum that traverses arcs in equal times. To determine the curve of this pendulum, it is necessary to begin with the cause (the potential), and determine the curve that reflects a uniform relationship between that potential and the produced effect. Therefore, we cannot start with a curve and measure the effect that a pendulum moving on that curve has on the rate of increase and decrease of the cosine. Rather, we must start with the physical requirement that the cosine must increase and decrease at a uniform rate, and determine the resulting curve as an effect. Huygens demonstrated that this would produce a cylcoidal path. (See Animated Figure 2.)
Thus, the visible curve that reflects a {uniform} relationship between the pendulum’s potential and the effect produced, is the {non-uniformly} curved cycloid.
But, in the case of a circular pendulum, whose visible shape is uniform, the speed along the arc is changing non-uniformly. That means that there is a non-uniform relationship between the potential and the produced effect from moment to moment.
Again, this continuously changing relationship between the total potential and the effect (speed) it has produced, is proportional to the changing rate of the vertical distance dropped as the pendulum swings along its circular arc, and can be expressed as a function of the rate of change of the cosine. Since the speed in the circular pendulum is non-uniform, its changing relationship to the potential is also non-uniform. Consequently, the rate of change of the cosine of a circular pendulum is non-uniform. (See Figure 3.)
This non-linear function is a different non-linear relationship than the non-linear relationship simply between the cosine and the angle in simple circle motion. (See Figure 4.)
This double incommensurability, which is similar to the double incommensurability found in the elliptical motion of a planetary orbit, is a characteristic which Gauss, Abel, Jacobi, Dirichlet and Riemann called “elliptical functions”. Elliptical functions are so named because they emerged out of the investigation of the Kepler problem, which also presents a double incommensurability the incommensurability between the arc and the sine and the arc and the angle.
As Abel, and later Riemann, elaborated, such elliptical functions are the simplest case of an extended class of transcendental functions now known as Abelian functions. These functions are distinguished by the number of principles combined to produce a single effect. In the general, an elliptic function is a function in which two different principles are connected. This is reflected in the circular pendulum by the appearance of the doubly incommensurable inverse function. Thus, the path of a circular pendulum, in a dynamic sense, is not a circular arc, but an elliptical pathway in the dynamic “phase-space” of a continuously changing potential, whereas, the non-uniform visible path of the cycloidal pendulum is a uniform pathway in the dynamic phase-space of a potential.
How Things Hang Together
In the above examples, the pendulum’s action was shown to be a function of the characteristic of the potential and its visible path was shown to be an effect of the dynamic path in the phase-space of that potential.
A similar relationship can be expressed through the example of the catenary, whose shape is a pathway with respect to the potential for gravity to produce an effect on a hanging chain. Like the pendulum, the visible shape of the catenary can be characterized as the effect of the dynamic path in the phase-space of the gravitational potential. (For the latest pedagogy on the catenary, the reader is referred to the Boston LYM’s pedagogical models. Pictures of those models are posted here.)
Now turn our attention directly to the characteristics of the phase space, so that we can begin to form a more general concept of such phase-spaces and draw from it the epistemological implications. This phase-space is not the infinitely extended, flat Euclidean space of Descartes and Newton’s fantasy. Rather, it is a bounded manifold with a definite characteristic curvature.
The boundaries of the manifold are expressed by the minimum and maximum relationship of the action with respect to the potential. In the case of the pendulum, those are the points of maximum height-minimum speed, and minimum height-maximum speed. In the case of the catenary, that minimum-maximum relationship is expressed by the relationship between the lowest point and the hanging points. The characteristic dynamic curvature of the manifold is elliptical for the pendulum (considering the circle as a special case of an ellipse for the case of the cycloid), and exponential for the case of the catenary.
The dynamic path of the pendulum or catenary is determined by the distribution of its potential for action between these boundaries. The physical least-action characteristics determine, in general, how the potential is distributed. The boundary conditions determine the specific pathway of action. Since in these cases the manifold is what Riemann would call simply- extended, the potential is distributed along a curve.
A change in these boundary conditions, produces a corresponding change in the specific distribution of the potential throughout the manifold. For example, a change in the position of the hanging points of the catenary produces a corresponding change in the positions of the links in the chain. A change in the rate of increase or decrease of the cosine of the angle of the pendulum produces a change in the shape of the arc of its swing, and vice versa. But, the general characteristics are unchanged.
The relationship between the boundary conditions and the distribution of the potential is associated with what Riemann called the connectivity of the manifold, as will be further indicated below. For example, though the links of the chain may be physically attached, they are connected to one another, not pair-wise, but through their direct relationship to the manifold as a whole. As we will also indicate below, Riemann showed that this characteristic of “connectivity”, in its general sense, is a fundamental, defining, characteristic of physical manifolds more fundamental than the particular conditions at the boundary.
To set the stage for Riemann’s approach, we must first look at Gauss’s extension of Leibniz’s dynamics.
Gauss extended Leibniz’s dynamics from the investigation of the distribution of a potential along a curve, as in the case of the pendulum or catenary, to the distribution of a potential over a surface or a volume. Such multiply-extended distributions are expressed by the notion of a {potential field}, such as in the case of geomagnetism or geodesy. When such a potential field takes the form of a surface or a volume, the boundaries of that field become, respectively, curves and surfaces, and the characteristic curvature of these fields is expressed, respectively, by the curves and surfaces of minimum and maximum potential within that boundary.
But even more crucial is Gauss’s recognition that the defining characteristics of these potential fields are determined by boundary conditions that appear, from the standpoint of the potential, to be outside the field itself. For example, the characteristics of the distribution of the magnetic potential outside a magnet, are determined by the distribution along the surface of the magnet itself. But the surface of the magnet is formally “outside” the potential field, appearing in Gauss’s potential function as a mathematical discontinuity. However, as Cusa emphasized in {On Learned Ignorance}, what appears to be mathematically infinite should not be considered outside the universe, but rather should be considered as the indicator of the existence of a higher principle, and the need to form a less imperfect conception of the universe as a whole.
Consequently, this discontinuity at the boundary of the magnet and the potential field of the magnetic effect does not indicate the existence of two separate worlds– a world of fields and a world of matter, as Aristotle, Descartes, Newton, Faraday and Maxwell falsely insisted. But, just as ideas do not exist except in connection with human personalities, but have an effect outside the human body, a magnetic effect do not exist except in connection with magnets and the material bodies on which it acts. Thus, the mathematical discontinuity of Gauss’s potential function at the surface of the magnet does not mean that the physical magnet doesn’t exist, or that the potential field doesn’t exist. Rather it signifies that to understand the phenomenon of magnetism it is necessary to form a concept of how the distribution of the potential throughout the manifold is being determined by the conditions at the boundary.
Riemann recognized that this relationship between the characteristics of the distribution of that potential and the conditions along the boundary, is the most important fundamental characteristic of a physical manifold, and that the { power of a physical manifold is a function, not of the specific conditions at the boundary, but by the number of boundaries (principles) acting on that manifold.}
Let us illustrate Riemann’s concept through a series of animations. Animated figure 5 illustrates the distribution of a potential over a circular disk.
The blue curves signify curves of minimum potential and the red signify curves, of maximum potential. In this case the boundary remains circular, but the position of the intersection point changes, which in turn changes the positions, i.e. density, of the intersections of the red curves with the boundary changes. In animated figure 6, the shape of the boundary changes, which produces a corresponding change in the distribution of the potential within the surface.
In animated figure 7, the intersection point of the red curves is replaced with two focal points, which is associated with more complex re- distribution of the potential within the field.
Though all three of these examples appear to be very different, from Riemann’s standpoint, they have a common characteristic. There are no discontinuities, so the potential is defined for every point in the field. Riemann called such manifolds, “simply-connected”. Riemann noted that though, as in our examples, the distribution of the potential in such simply- connected manifolds can vary quite widely, a function can always be found that transforms one simply-connected manifold into another. (This discovery has come to be called the “Riemann Mapping Theorem”.)
But all this changes when one or more discontinuities are introduced into the manifold. Such manifolds Riemann called, “multiply-connected”. In such manifolds, each discontinuity introduces a new set of boundary conditions. Thus, the distribution of the potential in such a manifold is determined by the connected effect that these separate boundary conditions impose on the manifold. This is illustrated in animated figures 8a and 8b for the case of a function with two boundaries–the so-called elliptic functions.
This is the type of function that appeared, for example, in our investigation of the circular pendulum. There the elliptical characteristic did not appear in the visible domain, but as the double incommensurability when the angle is expressed as a function of time.
The animation illustrates the elliptic function mapped onto a sphere so that the entire function, including the discontinuities, can be visibly represented all at once. In this case, the distribution of the potential within the manifold is changing with respect to the independent, but connected, changes of the boundaries defined by both discontinuities.
Compare this with the previous examples where all the changes took place with respect to only one boundary. Riemann emphasized that it was only the introduction of new discontinuities that could fundamentally change the connected relationship among all the action in a potential field.
(For example, though Riemann’s mapping theorem holds for all simply-connected manifolds, no such relationship exists among multiply-connected manifolds.)
From the standpoint of Leibniz, Gauss, Dirichlet and Riemann, a potential field expresses a dynamic under which all potential for action is connected through the characteristics of the boundaries and the curvature of the distribution of the potential. A change in the boundary conditions changes the distribution of the potential, but not the way that distribution is connected. However, with the addition of a new boundary, through the effect of an additional principle acting on the manifold, the connectivity of the manifold is changed, thus, changing fundamentally all action in the manifold. Further, a potential of greater power is generated if the connectivity of that potential is changed by the introduction of a greater number of principles acting on and in the manifold.
Typical of such a change in connectivity of a manifold is the effect of a revolutionary new discovery on the noosphere. Such a discovery, which appears as a discontinuity with respect to the prevailing view, has its effect on the noosphere by changing the way individual minds interact, with each other, and with the noosphere itself.
This points to a new type of dynamic, a dynamic of change from a manifold of lower to a manifold of higher potential. Such a dynamic is typified by Vernadsky’s concept of the noosphere as a dynamic system that is always moving to states of higher potential. Or, in the case of the development of the physical economy, which must be understood as a multiply-connected manifold whose boundaries are all multiply-connected manifolds.
This raises, in a different, but connected form, what Cusa had raised centuries earlier. Cusa considered that all action is the effect of a potential for that action to exist. “Nothing happens in the universe that is not possible”. Therefore, to understand any physical action, it is necessary to know what made it possible (its dynamics). But, the most important question, what Cusa called, “The Summit of Vision”, is what makes possible that possibility is possible. Or, what is the dynamic of dynamics?