The Principle of Least Action at the ACPA

The following is a presentation given at the recent meeting of the American Catholic Philosophical Association. It is a part of an ongoing project on the principle of least action, and this version will be incorporated in some manner in a longer paper, hopefully by the end of this year. Comments are most welcome.

 

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The Action and Power of the Universe (ACPA Version)

Ubi eras quando ponebam fundamenta terrae? Indica mihi, si habes intelligentiam.

– Job 38:4

Juvat integros accedere fontes.

– Lucretius, d.R.N. I:927

I. Introduction¹

The principle of least action (the PLA) was lauded by its discoverer as follows:

There is a principle truly universal, from which are derived the laws which control the movement of elastic and inelastic bodies, light, and all corporeal substances; it is that in all the changes which occur in the universe . . . that which is called the quantity ‘action’ is always the least possible amount.²

This principle has been the focal point of debates about the foundations of modern physics ever since this enunciation by Pierre Louis de Maupertuis in the 18th century and the subsequent formalization of the PLA by Euler, Lagrange, and Hamilton. This paper defends an Aristotelian-Thomistic interpretation of the PLA.³

Some physicists propose the PLA as the centerpiece for reforming the introductory physics curriculum.⁴ The PLA is typically forgone in such introductions due to its technical difficulty. It doesn’t deal in the more readily imaginable forces-and-particles Newtonian model, but with expressions that need higher-level undergraduate or graduate courses in mathematics so as to be fully utilized when practicing physics. This air of being the “inner sanctum” of the physics curriculum leads one to suspect that Newton is the milk, while Lagrange and Hamilton are the meat. As one physicist observes: “[Newtonian] forces are crucial to the understanding of simple problems, yes, but just as children use counting numbers while mathematicians have graduated to the use of real numbers, so we must graduate from forces to kinetic energy and to the energy ‘structure’ functions.”⁵ Such comments make it sound as if initiated are those who have exchanged a Newtonian analysis for a least action analysis.

Aristotle, we may recall, begins natural philosophy with the admonition to start with what is better known to us and to end with the principles, causes, and elements that are more knowable in themselves.⁶ Is the PLA a principle in the natural order or only in a mathematical order? Our thesis lies in between: the principle of least action, at the level of classical mechanics, captures natural motion in its mathematical formulation. If our interpretation of the least action principle permits us to defend natural motion, especially natural motion due to gravity at universal scales, this is a promising advance for articulating a contemporary Aristotelian-Thomistic philosophy of cosmology.

II. Analogies and Precursors

Our first approach to the PLA is an analogy given by Richard Feynman (Figure A).⁷

Imagine a lifeguard (L) who sees a swimmer (S) in trouble some distance away from shore. What is the most efficient path for the lifeguard to take to reach the struggling swimmer? It is not the straight line, that is, “the path with less sand,” LAS. Nor is it the path of least water, LBS. The quickest path is somewhere in between. This path of least time, LCS, is between the first two because the lifeguard traverses the given media—running on sand, swimming through water—at different rates. He economizes on transit time using the middle route. This note of “economy” is our first hint at the PLA.

Feynman bases his analogy on Pierre de Fermat’s Principle of Least Time: “Out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.”⁸ Again, notice economy and optimization here. From Fermat’s Principle we derive Snell’s Law (Figure B).

Snell’s Law states the relationship between the sines of the angles of a refracted beam of light and the velocity of light in each medium. Where exactly does the point of refraction occur? Fermat’s Principle permits us to find it.

As we vary the distance x, we change the point of refraction and thus we change the total travel time, T. (In our lifeguard analogy, this is analogous to changing the lifeguard’s point of entry into the water.) In the derivation, we invoke Fermat’s Principle when we set the first derivative of the expression for the total time of travel equal to zero, since this corresponds to a minimum value—that is, the most economical amount of the total time as a function of x.

From Feynman and Fermat we can draw a general lesson to prepare for the PLA. Certain processes can exhibit economizing or optimizing features. These processes arise in situations, whether belonging to art or to nature, where one seeks to “optimize” a desired result in view of some fixed, given quantity and another, variable quantity. What shape with a fixed perimeter length has the greatest area?⁹ Does light reflect off a mirror by taking the shortest distance?¹⁰ Do bees use hexagonal honeycomb cells so as to hold the maximum amount of honey for the least amount of wax?¹¹ And so forth. Such optimizations are at least heuristic clues to the inner workings of nature. For more insight, we turn to the PLA itself.

III. The Origin Story

Our version of the PLA’s origin story tracks the growth of classical mechanics from Newton to Lagrange; we will leave out Einstein (relativity) and Feynman (quantum theory). This story traverses a path of both increasing abstractive generality and the power to model entire systems in motion using functions that describe the energy of the system. By “system” we mean a group of mobile bodies conceived of in isolation from their surroundings, whatever their surroundings may or may not be, and where “isolation” implies a lack of pertinent causal interactions with those surroundings. This story-line departs from the forces-and-particles method of a Newtonian analysis to consider the work, energy, and general coordinates of whole systems in motion all at once, without building up a system from individual particles.

Historically, this was accomplished in two steps. First, theorists conceived of the motion of a system from moment to moment using the principles of statics (that is, static equilibrium defined as net zero “virtual work”). Second, from this “moment to moment” dynamics, theorists generalized to the motion of a whole system from start to finish. Coopersmith summarizes:

The Principle of Virtual Work yields the condition for static equilibrium: it applies at one instant and then for all time (in other words, time doesn’t come into it). D’Alembert’s Principle, being a special case of the Principle of Virtual Work, also applies at just one instant, but as we’re now in the realm of dynamics the conditions do change with time and so d’Alembert’s Principle must be reapplied at the very next instant, and then again at the next instant, and so on and so on. However, what we would like is a method that frees us from the need to explicitly re-apply d’Alembert’s Principle, and, instead, enables us to mathematically track the motions continuously and over the whole time-interval of the problem.¹²

In other words, the Principle of Virtual Work tells us the conditions for the equilibrium or lack of motion of a system. This “virtual work” is imagined, conceptually possible work (force through a distance). One counterfactually imagines work being done on the system so as to deduce the required forces needed for its equilibrium. D’Alembert’s Principle then extends this by iterating these conditions of static equilibrium. The iteration of static equilibrium from moment to moment is a mathematical simulacra of the dynamism of motion. The method for calculating this dynamism all at once, for a whole motion, is what Lagrange, Euler, and Hamilton jointly perfected, using the principle of least action (which, in a certain variation, is also called “Hamilton’s Principle”).

Instead of imagined changes in the path of a lifeguard, or imagined changes in the path of a light ray, or the virtual work done to a system in equilibrium, the mathematics of the PLA allows us to imagine alternative paths of motion generated by key functions defining the kinetic and potential energy of the mobile system in question. More formally, the “action” that is made “least” in the PLA is a scalar quantity, the integral denoted by S (see the formula above). The amount corresponds to the “path of least time” for the lifeguard or the light ray. Action measures the whole motion, start to finish, and, since it is an integral, it is also the summation of what is true at each point of space and time along the way. Somehow, the PLA attends to both the whole motion and all of its infinitesimal parts. The units used to measure action have the dimensions energy-time, or units of Joule-seconds (J•s); or, equivalently, momentum-meters (p•m). This definition of action by energy-through-time or momentum-through-space is significant because it adds qualitative powers (energy and momentum) to the general consideration of motion only in terms of quantities (space and time).¹³

One obtains the action from the definite integration between a starting time and an ending time. These correspond to the points L and S in Feynman’s analogy, or Q and P in Figure B for the light ray. The integrand L is called the “Lagrangian.” It is the difference between the kinetic and potential energy of a physical system in motion: L = K – P. (In a moment we will discuss why kinetic and potential energy define the Lagrangian in classical mechanics. The Lagrangians of quantum or relativistic physics are beyond our scope here, but I think an analogous account could be given.) For the purposes of the integration, the Lagrangian is expressed using two functions at once: a function of position designated by q, and a function of velocity designated by q̇.

Now, “the calculus of variations” developed by Euler and Lagrange permits one to compare this scalar quantity of the action to all other conceptually possible paths of motion by varying the integrated functions of q and q̇. This variation in the action is symbolized by the little ‘δ’ out in front of the integral sign and corresponds to paths that are alternatives to in Feynman’s analogy or alternative paths for the light ray. In contrast to the conceptually possible alternative values of , the true path has an action (energy-through-time) with a “stationary” or “minimum” point, just as the first-derivative of the total-time path of the light ray has a zero value. Figure C illustrates some of these unrealized, “alternative” paths that do not minimize action. (The true path is right through the middle of these alternatives.) Nature’s behavior is only captured if we ignore these mathematical alternative possibilities and use the path where the action is least or stationary or minimized.¹⁴

Before we try to deepen our understanding of the PLA, let’s note two things. First, the PLA survives in realms where Newtonian conceptions don’t. Sir Arthur Eddington observes that

the law of gravitation, the laws of mechanics, and the laws of electromagnetic fields have all been summed up in a principle of least action. . . . Action is one of the two terms in pre-relativity physics which survive unmodified in a description of the absolute world. The only other survival is entropy.¹⁵

Eddington’s words also suggest that the PLA is somehow “in things,” not only in our conception of them. If so, the PLA is predicable of a naturally ordered whole or system of mobiles. “In such systems, action is minimized,” one might say, or perhaps, “Some physical systems are Lagrangian.”¹⁶ Yet the heavy dose of mathematics required to express the PLA gives us pause. Perhaps the PLA is predicable only of a certain family of equations. Consider again our derivation of Snell’s Law. The step of “setting the first derivative equal to zero” is clearly something said of the mathematics used to derive the law and not a description of the inner workings of light itself. Second, the PLA permits arguments that are not available to Newtonian particles-and-forces mechanics. Using the PLA and assuming the homogeneity and isotropy of space or the homogeneity of time, one can derive the three central conservation laws: the conservation of linear and angular momentum and the conservation of energy.¹⁷ One can even derive Newton’s Second Law of Motion!¹⁸ Of course, such derivations follow a mathematical order: the Second Law can also be used to derive the PLA! Indeed, the natural conservation of physical quantities is what yields the symmetries that we capture mathematically, and not vice-versa.¹⁹ So the real question is not about priority in a mathematical argument but priority in physical causes.²⁰ Indeed, both of these points emphasize that the sound interpretation of the PLA must respect how nature as a cause is prior to our formulations and conceptions of nature.

IV. Lagrangians and Energy

We must now answer two questions. First, why is the classical (i.e., non-relativistic and non-quantum) Lagrangian defined by kinetic and potential energy? At each instant of a motion conceived in the mode of mathematical physics, there are measurable kinetic and potential energies belonging to the system. So, if one integrates the mathematical expressions for the entire motion from start to finish, it stands to reason that one would obtain two sorts of components in the entire integral, one for kinetic and one for potential energy. In actual fact this is how these defining elements of the classical Lagrangian come about. One extends this application of virtual work in statics by re-describing it dynamically—this is what D’Alembert’s Principle accomplishes. Consequently, the integration sets the equations of static equilibrium into motion, obtaining two parts to the resulting integral: one part corresponds to the kinetic energy and the other to the potential energy. It stands to reason that in order to provide some natural philosophical basis for the PLA, we must understand what this “integration” represents.

Our second question is as follows. Given below are the mathematical expressions for kinetic energy and the potential energy due to gravity.

The measure of this potential energy is equal to the product of the force of gravity (g) on a body of mass m at a height h near the earth’s surface. If one uses the PLA to examine systems involving gravitational potential energy (e.g., a falling body, a projectile, a pendulum), the Lagrangian would involve kinetic energy and an expression derived from gravitational potential energy. Note that kinetic energy always and everywhere has the same formula, while the formula for potential energy depends upon the character of the system in question. Consequently, different types of physical systems or arrangements will possess different Lagrangians, and the PLA can only be used if one is able to define that system’s Lagrangian. To repeat: the Lagrangian of a system is tied to the character of the system in question, and this character’s distinguishing feature is its potential energy. It stands to reason that in order to provide some sort of natural philosophical grounding for the PLA, we must understand not only kinetic but also and especially potential energy. This is our second question.

In order to accomplish the first task, we must keep in mind Aristotle’s treatment of the physical continuum from Physics, Book VI.²¹ The general lesson of this treatise is that appealing to a body’s materiality and divisibility alone will not explain motion.²² Looking at motion from the infinitesimal part, or motion’s occurrent point, or the instantaneous now of time is not enough. The full being and intelligibility of motion derive from the other three causes. Now, reflect on the very idea of a mathematical integral—our example is the action integral S. As an integral, is a continuous sum of the values of two functions (q and q̇) taken at each momentary “now” between the beginning and ending times. So, in each mathematical “now” there is a measured value of position and velocity (q and q̇) and therefore a value for kinetic and potential energy. (We discuss why and yield these energies in answer to the second question.) Notice that the mathematical formulation signifies these momentary values of energy in abstraction from or indifference to actual motion, for mathematics abstracts from motion. This is in harmony with the origin of the PLA as a generalization from and extension of the mechanics of static equilibrium. The formalism is empty of the reality which motion possesses.

What is that reality?²³ The definition of motion, the actuality of what exists in potency as such, implies that motion possesses a twofold order:

An imperfect act fulfills the definition of motion both insofar as it is compared to a further act as a potency and insofar as it is compared to something imperfect as an act. Thus, motion is neither a potency existing in potency, nor is it an act existing in act, but it is an act existing in potency, (i) such that “act” designates the order of [the mobile] thing to a prior potency, and (ii) such that “existing in potency” designates its order to a further act.²⁴

Consider a local motion. The potency of a local motion as such orders the mobile to the act of being in the terminus ad quem. The act of the local motion as such orders the mobile to a prior potency. Neither the act nor the potency signified in the definition orders the mobile to that place where the mobile is occurrently (i.e., at that very “here” and “now”). Otherwise, the mobile would not be in motion but would be at rest. Therefore, the mobile in motion exists in the infinitesimal “here” or “now” in an odd way. The twofold order provides a formal wholeness to the motion that unifies each moment of the motion to the motion’s final, yet-to-be-realized act. This twofold order obtains for the duration of the motion and integrates it ontologically. This ontological integration signified by motion’s definition is paralleled by the mathematical integration of the action integral , but they signify different unities. However, these different signified unities can be meaningfully held in parallel precisely because a body actually in motion possesses values of kinetic and potential energy at each moment, even though we do not actually stop the body’s motion to measure values. Since the Lagrangian mathematically permits us to conceive all these moments in a single grasp, we propose that the action integral mathematically models the whole of a unified, continuous motion defined in Aristotelian terms. That is, the mathematical physicist uses the integral to mentally unify the mathematical and the physical orders and “do physics.”²⁵

Now, we must provide support our key assertion that a body actually in motion possesses values of kinetic and potential energy at each moment. This is different than the assertion that there are kinetic and potential energy values for each point of time between and , for this is a mathematical formulation. We are asking after its natural philosophical interpretation. So, we turn to our second task: What are kinetic and potential energy? Here we build on arguments of Tom McLaughlin.²⁶ He maintains that “kinetic energy is an instance of the Thomistic notion of the act or activity of motion,” while “gravitational potential energy is an instance of Thomistic potentiality, specifically passive potentiality.”²⁷ We examine each in turn.

The case of kinetic energy is a bit more straightforward. Even physicists describe kinetic energy as “the energy-of-motion,” and this comports with the historical genesis of the concept.²⁸ However, what justifies the view that it tracks the actuality of Aristotelian local motion? We argue dialectically, following McLaughlin, that kinetic energy is defined in physics by the measure of how much work that a body in actual motion can accomplish.²⁹ This argument can be amplified using the scholastic notion of quantitas virtutis or “quantity of a power,” as emphasized in a little-known work by Fr. Charles Bonaventure Crowley, OP. So, this “virtual quantity” names something that is analogously called a quantity, it does not name a potential quantity or the virtual presence of a quantity. This “quantity of a virtue” or virtual quantity is a quantification of habits, powers, or dispositions—i.e., qualities—which can involve some type of measurement of the operations and effects of those qualities.³⁰ Now, kinetic energy manifests itself in the very occurrence of work. “Work” in this technical sense (force-through-a-distance, which is measured in Joules), is itself a virtual quantity, the measure of the ontological ability of some body to cause physical effects. Since kinetic energy is the measure of this reality insofar as a body is in motion, it makes sense that kinetic energy as a mathematical quantity represents an instance of the activity of motion.³¹ Furthermore, mathematical derivations of the expression for kinetic energy () appeal to the conservation of energy in the interaction of two moving bodies.³² This involves an appeal to the principle of relativity: “To obtain an absolute or invariant [i.e., conserved] quantity we must always consider one quantity relative to another.”³³

However, a solution can sometimes lead to a new problem: what form shall we give to the kinetic energy of just one isolated particle? The resolution is that we are then compelled to define the kinetic energy as having the [form ]. This leads to consistency in the theoretical modeling.³⁴

Of course, neither would Aristotelian local motion have its full reality were a solitary body to be “set in motion” in a void-universe.³⁵ So, given that the universe is not composed of only one solitary body, the actuality of the motion of individual bodies is consistently modeled by the kinetic energy formula.

What about potential energy? In particular, we are interested in gravitational potential energy (GPE). We again expand upon McLaughlin:

The chief indication that potential energy is not an actuality but is an Aristotelian potentiality is that potential energy requires that we consider a body’s position both with respect to where the body is at some time and with respect to where it can be but is not. . . . Since a body with potential energy must include a reference to being located in a position that is attainable but unattained, potential energy must include such a position potentially. And once such a position has been attained—once the apple has fallen to the ground—it no longer has potential energy with respect to that position.³⁶

That is, an object in the potential field of a source of gravity possesses a certain virtual quantity of potential energy due to its position, and for this reason its measurement is expressed in terms of its height. Potential energy is the energy of possible interactions, and “the [gravitational] field tells what would happen if a test [body] were brought [to a certain height],” and when released “the [body] interacts gravitationally.”³⁷ Consequently, we can say that GPE, measured as a virtual quantity, is a natural disposition of a body for gravitational interaction due to two joint causes: its mass and its position with respect to other bodies in a gravitational field. As positional, GPE is tied to the Lagrangian’s function of q. Kinetic energy, measured as a virtual quantity, belongs to a body precisely insofar as it is in motion; therefore, it is tied to the Lagrangian function q̇.

If this is the case, then we have supported our assertion that a body actually in motion possesses kinetic and potential energy at each moment. These energies, formulated as measured virtual quantities, mathematically represent the “act” and “potency” signified in the definition of motion. The Lagrangian, defined by kinetic and potential energy, is now ready for an Aristotelian interpretation.³⁸

Before providing one, however, we should aid our natural philosophical imaginations by considering a thought experiment proposed by Nicole Oresme in the 14th-century. This thought experiment posits “that the earth is pierced clear through and that we can see through a great hole farther and farther right up to the other end where the antipodes would be if the whole earth were inhabited.”³⁹

Imagine dropping a stone down this hole. The stone would not “stick fast” at once in the center of the earth, but would first oscillate back and forth before coming to rest. Oresme proposes an analogy to a pendulum to support this intuition about “impetus” or “momentum.”

We can understand this more easily by taking note of something perceptible to the senses. If a heavy object b is hung on a long string and pushed forward, it begins to move backward and then forward, making several swings, until it finally rests absolutely perpendicular and as near the center as possible.⁴⁰

Oresme’s thought experiment forces us to consider a distinction among the body’s inner principles of motion and how those principles relate to surrounding positions, i.e., what the environment contributes to the natural motion of a body. Similar thought experiments are still presented to students of Newtonian mechanics (Figure D). Indeed, one can prove that in this idealized case of the “Antipodal Pendulum,” the formula for the period of the stone matches the formula for the period of an everyday simple pendulum.

Now, both the simple pendulum and analogously, Oresme’s Antipodal Pendulum, have a Lagrangian:

The simple pendulum thus has a definable quantity of action. Let us contemplate the pendulum and its action in the abstract space called phase space. Figure E plots the position and velocity of a pendulum in phase space, mapping its position (θ) against its velocity (θ-dot).

The “motion” of the pendulum through phase space tracks along the arrows, in a clockwise fashion, inside the cat’s-eye shape.⁴¹ So, as the pendulum swings back and forth (or as θ goes between left and right) its velocity increases and decreases; the pendulum’s velocity is zero when θ is greatest, and its velocity is greatest when θ is zero (when the pendulum is swinging through the vertical). We can also project this two-dimensional phase space into a third dimension (see Figure F).

In this diagram, the same position (θ) and (θ-dot) are plotted on the lower plane. The vertical axis, E, is the system’s total energy. The gray paraboloidal surface represents possible energies of the pendulum depending on its initial conditions S1, S2, and S3 (from lower to higher energy states); staying at these levels represents the conservation of energy. The circumference at each level is the definite integral of both and with respect to time, which is, of course, the invariant action, S, belonging to the system.

Imagine if we were to reintroduce the friction of the air or the heat generated by the pendulum’s motion. These are non-conservative forces and so energy would be lost. The clockwise “path” of the pendulum through the three dimensional phase space in Figure F would then “spiral” down and around the paraboloidal surface until it “settled” at the bottom, at rest, just as Oresme’s figmented stone eventually rests at the center of the Earth.⁴² We could consider other examples (for instance, the libration or “Lagrangian points” in Figure G which seem to indicate certain natural places). However, we are now ready to propose our interpretation of the PLA.

 

V. The Aristotelian-Thomistic Interpretation of the Principle of Least Action

Recall that position in a gravitational field enters into the definition of gravitational potential energy, and therefore into the definition of that system’s Lagrangian. Now, in its mathematical formalism, the PLA encodes a motion “by a continuous process of ‘instantaneous’ conversion of potential energy into kinetic energy.”⁴³ By “encode” we mean “to signify through the symbolic mathematics of functional expressions.” In other words, the PLA’s formula directly signifies certain logical and mathematical relationships that obtain among the kinetic and potential energy values and their functions in the Lagrangian. The PLA as a mathematical formulation acts as an extrinsic formal cause.⁴⁴ Physically, however, this formula can be used to indirectly signify what belongs to a mobile system in motion, for a body in motion actually possesses both kinetic and potential energy as measured virtual quantities. These values encoded by the formula at each instant can be sewn together in our thinking by seeing that these instants are aspects of a single whole, namely, the motion that we know thanks to an Aristotelian analysis. Now, our example case is gravity; we therefore argue as follows:

(1) The principle of least action is a time-integrated Lagrangian with a minimum or stationary value.
(2) Such a Lagrangian encodes the essential principles of a natural motion, measured as virtual quantities.
— Kinetic energy, measured as a virtual quantity, represents the actuality of motion.
— Potential energy, measured as a virtual quantity, represents the potentiality of the motion (e.g., due to the relationships of position in a gravitational field, or other potential fields where force is a function of distance).
— The definite integral (from beginning to end) represents the terminus a quo and terminus ad quem.
— The Lagrangian for the true path—the one with least or stationary action—is natural, for nature takes this path.
(C) Therefore, the principle of least action mathematically encodes a natural motion by integrating the Lagrangian over time.

As a corollary, this interpretation implies a conception of natural places.⁴⁵ Natural places, at least for the PLA in classical mechanics, are dynamically established positions of equilibrium in the configuration of gravitating systems, which such systems possess by natural necessity.⁴⁶ By “position” I mean situs, the relative arrangement of physical bodies. They are dynamically established insofar as the gravitational potential and the initial conditions of massive bodies are contingent, subject to interference, and self-influencing. Such a system cannot but have these positions (consider the libration points in Figure G), and they are natural properties insofar as they arise from the system as a whole by nature and not by art, violence, or chance.

Most importantly, they are positions of equilibrium in the configuration of the system. Here, “equilibrium” names a feature of the phase space and not the real, physical position of the system in question. It thus refers to the virtual quantities involved. As a name, “equilibrium” comes from the sense of “balancing” a weight or finding a point where a body rests due to nature, and so it is an apt analogical term. In the case at hand, bodies in orbit follow the contours of gravitational potential and thus exhibit a type of equilibrium in regard to their potential and kinetic energies. This configuration permits stable orbits based upon energy conditions and the least action principle. Using different Lagrangians, the PLA would encode other sorts of natural motions (e.g., particles in an electromagnetic field, or in relativistic or quantum dynamics). If this is true, and bearing in mind that least action figures prominently in astrophysical studies and the Einstein Field Equations that drive the standard model of Big Bang cosmology, then this interpretation of the PLA is a promising step towards a central element of the Aristotelian-Thomistic philosophy of modern cosmology.

– Appendix

A brief indication is possible that “natural motion” is captured by the PLA not only in classical mechanics but also in relativistic and quantum mechanics. The general notion one requires is that of a “worldline,” or the mathematical characterization of a motion through spacetime. Worldlines in classical, relativistic, and quantum mechanics are each closely related to economizing principles of action.⁴⁷ We reproduce the figure from Edwin Taylor’s seminal article on the use of the PLA in the introductory physics curriculum. This figure also outlines the argument for the PLA’s centrality as a unifying principle in physics:

Taylor himself utilizes the notion of the “principle of maximal aging” to provide a definition of natural motion in the context of relativity, namely, “Natural motion is the motion that maximizes the wristwatch time between any pair of events along its path.”⁴⁸ Thus, the existence of geodesics or straight-line or “free fall” worldlines are the current replacement for natural motion in relativistic physics.⁴⁹

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These presentations were produced as part of my postdoctoral research project.
FONDECYT Postdoctorado, Proj. Nº 3170446