## Classical Mechanics, Philosophy of

## Classical Mechanics, Philosophy of

# CLASSICAL MECHANICS, PHILOSOPHY OF

Classical physics is the research tradition beginning with Isaac Newton's *Mathematical Principles of Natural Philosophy* (often called simply the *Principia* ) of 1687, which was overtaken by relativity theory and quantum mechanics in the early twentieth century and is still undergoing lively development in such areas as chaos and catastrophe theory. The "Newtonian" physics canonized in textbooks includes many elements added long after Newton, such as vector notation, the analytical mechanics that Joseph-Louis Lagrange and William Hamilton developed in the late eighteenth and early nineteenth centuries, and the laws of energy conservation and of electromagnetic phenomena formulated in the mid-nineteenth century. Indeed, Leonhard Euler in 1749 was the first to express Newton's second law as the familiar relation between a body's instantaneous acceleration and the momentary force that the body experiences; Newton's own version of the law set the body's change in momentum during a finite period of time equal to the impulse on the body (the force times the period's length, for constant force over the period). However, the subject of this entry is the anachronistic classical mechanics found in textbooks.

Though classical mechanics is false, as relativity and quantum mechanics reveal, there are many reasons for philosophers to continue investigating its proper interpretation (i.e., what the world would be like if classical mechanics were true). Many of the difficulties encountered in trying to interpret modern physics also arise in connection with classical physics, but in a simpler context. Moreover, many of the venerable metaphysical and epistemological ideas vigorously developed by modern philosophers such as George Berkeley, David Hume, and Immanuel Kant are best understood in connection with the classical physics that originally prompted them. Furthermore, one should not wait to deploy one's interpretive faculties only after physics has secured the final theory of everything; if one did, progress in both physics and philosophy would suffer. Finally, although classical physics lacks some of the provocative features exhibited by relativity and quantum mechanics, it has long served philosophers as the exemplar of what genuine scientific understanding would be. By studying it, one can learn about the concepts, logic, and limits of science. This entry touches on only a few of the metaphysical and epistemological questions that classical physics provokes.

## Basic Ontology: Mass and Matter

Philosophers have worked to identify the ontologically fundamental objects, properties, and relations that classical mechanics posits. Among the candidates proposed have been distance, time interval, velocity, force, matter, mass, electric charge, and inertial reference frame. All raise difficult questions.

Mass is the single parameter relating a body's motion to the force on the body. Remarkably, that relation in classical mechanics is the same for macroscopic bodies as for their constituents; classical mechanics "scales up." Newton defined mass as measuring the amount of matter composing a body, though he did not define *matter* itself. In contrast, the nineteenth-century Scots physicist James Clerk Maxwell (1952), who formulated the laws of electromagnetism, defined mass in terms of momentum and energy, which he believed more fundamental. An alternative approach later pursued by Ernst Mach (1960), the Moravian physicist and philosopher, characterizes mass operationally: The masses of two bodies are related as the inverse ratio of their mutually induced accelerations when isolated from other bodies. If mass is not defined operationally, but instead is an intrinsic property responsible for resistance to force, then according to many philosophers, one knows the effects for which mass is responsible, but one cannot know what mass is in itself. Similar considerations apply to electric charge.

Classical mechanics is sometimes interpreted as deeming a macroscopic body to be a swarm of point bodies in a void. In an alternative interpretation classical mechanics takes bulk matter to be continuous space-filling stuff, or instead to be composed of many bodies of a small but finite dimension made of continuous media, having no internal structure, and separated by empty space. Newton's laws and Charles-Augustin de Coulomb's electrostatic force law are often codified in terms of pointlike bodies, whereas the basic equations of hydrodynamics and the theory of elastic solids are typically expressed in terms of continua. Mass points present obvious difficulties; when two collide, they must be inside each other and the gravitational force between them becomes infinite. Continuous media avoid the latter problem, since at a point, there is no finite quantity of mass; there is only mass density, defined as the limit of mass per volume as the volume becomes arbitrarily small. But collisions still present a problem: When two bodies collide, do they occupy a common point? Or is there simply no finite volume between them? (If a point separates them, then how are they in contact?)

## Basic Ontology: Motion and Force

A body's velocity is its position's instantaneous rate of change, and its acceleration is its velocity's instantaneous rate of change. As ordinarily defined, a quantity's rate of change at an instant is its average rate of change during a finite interval around that instant, in the limit of an arbitrarily short interval. Hence, a body's velocity at time t is just a mathematical property of the body's trajectory in a neighborhood of t, which includes some of the body's trajectory after t. But its velocity at t is supposed to be an initial condition in the causal explanation of its subsequent trajectory. That would apparently require points in that subsequent trajectory to help causally explain themselves. This is puzzling. Furthermore, consider a body moving uniformly across the surface of a smooth horizontal table and then falling off the edge. At the final moment that the body is on the table (assuming that the table includes its edge), its trajectory's second derivative is undefined; taken from the left it is zero, but taken from the right it is equal to the gravitational acceleration. Presumably, though, a body has a well-defined acceleration at all times.

Force is characterized by William Thomson and Peter Guthrie Tait in their canonical mid-nineteenth-century physics text as "a direct object of sense" (1895–1896, p. 220). However, other natural philosophers regarded forces as redundant in classical mechanics once fields are admitted as local causes or remote charges and masses are acknowledged as acting at a distance. The late nineteenth-century German physicist Heinrich Hertz regarded forces as mere calculational devices between cause and effect, "simply sleeping partners, which keep out of the business altogether when actual facts have to be represented" (1956, p. 11).

## Fundamental Laws

Newton's three laws of motion, with his inverse-square law of gravity, are commonly regarded as the fundamental laws of classical physics. In the nineteenth century the laws of electromagnetism were added to them. The status of the conservation laws and the variational principles of classical physics remains more controversial, as will be seen.

Newton's second law is sometimes taken to be "The net force on a body, divided by its mass, equals its acceleration in any inertial frame of reference." But how is *inertial frame* defined? Mach suggested that inertial frames are frames where the universe's average matter is not accelerating. But this definition leads to predictions that depart from those made by Newtonian mechanics regarding, say, a body in otherwise empty space or in a universe where (according to Newton) all other matter is accelerating. Sometimes *inertial frame* is taken to be defined by Newton's first law: a frame is inertial exactly when a body feeling no forces remains at rest or in uniform rectilinear motion in that frame. But then Newton's first law is true by definition.

The chief alternative is to presuppose points of absolute space, as Newton did, and to define an *inertial frame* as a rigid Euclidean frame at rest or in uniform rectilinear motion with respect to those points. However, even disregarding objections to absolute space as either empirically inaccessible or in contravention of metaphysical scruples, Newton's approach contains surplus ontological structure. A particular frame need not be privileged as at rest. Rectilinear uniform motion need only be distinguished from other paths; the frames pursuing such trajectories are inertial. *Inertial frame* is thereby defined independent of Newton's first law, which is not tautologous but just a consequence of Newton's second law. (Newton's first law is never instantiated, since every body feels some component gravitational forces.)

In this "neo-Newtonian" space-time, there is no fact of the matter regarding a body's velocity. (But there is a fact regarding its acceleration and its velocity relative to another body.) There is also no fact regarding the distance between two nonsimultaneous events, unlike in Newton's absolute space and time. All inertial frames are equal in neo-Newtonian space-time.

However, absolute velocity figures in the classical laws of electromagnetism. Absolute space is then no longer superfluous. This fact opened one of Albert Einstein's paths to relativity theory.

## How Much Does Classical Physics Say?

Thomson and Tait interpret Newton's second law as requiring every acceleration to be caused by some force (1895–1896, p. 223). But simply as an equation, Newton's laws make no explicit mention of causes and effects; they merely relate a perpetually isolated system's past and future states to its current state. Accordingly, Bertrand Arthur William Russell concludes that the notion of a causal relation (insofar as it goes beyond a correlation demanded by the laws) has no place in physics, but is "a relic of a bygone age" (1929, p. 247). David Lewis (1983–1986) draws a different moral, arguing instead that since classical physics reveals causal relations, those relations must supervene on the laws and the actual course of events.

There are many similar questions about how richly or austerely classical physics describes the world. For instance, the law of energy conservation might be interpreted as specifying that the universe's total quantity of energy is fixed. But it might instead be taken as saying more: That for any volume over any temporal interval, the change in energy within must equal the energy that has flowed across its boundary. Alternatively, the conservation laws (of mass, energy, linear momentum, and angular momentum) might not be understood as laws of classical physics at all. They do not follow immediately from Newton's laws of motion and gravity.

## Time-Reversal Invariance

Newton's laws are time-reversal invariant. Roughly speaking, if a sequence of events is permitted by the laws, then the laws also permit those events to occur in reverse order. The laws recognize no difference between past and future just as they fail to discriminate among spatial directions. However, certain macroscopic processes are never observed to occur in reverse. For example, when two bodies of unequal temperature touch, heat flows from the warmer to the cooler body. Although there are configurations of the bodies' molecules that would lead by Newton's laws to the warmer body's becoming still warmer, many more configurations would produce the result one sees. So irreversibility can be reconciled with Newton's laws if, roughly speaking, all the possible microrealizations of a system's macrostate are equally likely.

But this equiprobability is not required by Newton's laws. Its origin remains puzzling. Furthermore, even if a closed system far from equilibrium (e.g., with an unequal distribution of heat) were much more likely to head toward equilibrium (i.e., to increase its entropy) than away from equilibrium, entropy's increase in the space-time region one observes would remain unexplained. It would still be mysterious why one's space-time region is so far from equilibrium in the first place.

## Mechanism and Determinism

Classical mechanics suggests that the universe is like a majestic clockwork, the laws fully determining the universe's past and future states given its present state, and a body changing its motion only because another body touches it. But gravity and electromagnetism apparently operate by action at a distance. Newton famously offered no hypotheses (*hypotheses non fingo* ) regarding the means by which gravity operates. Accordingly, some natural philosophers ceased to seek local causes for all effects. In contrast, Michael Faraday and Maxwell regarded fields of force as existing on a par with bodies. The field at a given location would cause a body there to feel a force. The field picture avoids positing action at a distance but departs significantly from the picture of material particles in the void: Fields occupy all locations, even where there is no ordinary matter.

In 1814 Pierre Simon de Laplace invoked his famous "demon" to explain the determinism of the clockwork universe:

Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. (1951, p. 4)

Twentieth-century research revealed that Laplace may have overstated the determinism of a universe governed by classical physics (although there is no obvious way in which the indeterminism of classical physics supports the freedom of the will). When two point bodies collide, their mutual gravitational interaction becomes infinite, yet the laws of energy and momentum conservation nevertheless allow an analytic solution to the classical equations of motion to be extended uniquely through the collision singularity. However, this extension is generally impossible when three bodies collide. Furthermore, Newton's laws enable a closed system of point bodies to undergo an infinite number of triple near collisions in a finite time (as the sequence of encounter times converges to some particular moment). By the slingshot effect resulting from these close approaches, certain bodies attain infinite acceleration in finite time and so afterward are absent from any finite region of the universe. They are literally nowhere to be found. Since Newton's laws are time-reversal invariant, they permit this sequence of events to proceed in reverse, so that "space invaders" suddenly appear in the system from nowhere. Determinism is thereby violated without a collision occurring. Of course, the invaders' unanticipated appearance violates mass, energy, and momentum conservation, illustrating that these principles fail to follow from Newton's laws alone.

## Analytical Mechanics

In 1661 Pierre de Fermat derived the law of refraction from the postulate that in traveling from one location to another, a ray of light takes the path that minimizes the travel time. To some (such as the eighteenth-century French mathematician Pierre-Louis Moreau de Maupertuis), Fermat's principle suggested that nature produces effects with the greatest economy, efficiency, or ease—demonstrating God's wisdom. However, this metaphysical moral was undermined somewhat by the discovery that light may also take the path of greatest travel time. For example, consider a point light source at the center of an ellipsoidal mirror. The points around the mirror's margin that can reflect light back to the center are exactly the two points along the mirror's minor axis (i.e., where the edge is closest to the center) and the two points along the mirror's major axis (i.e., where the edge is farthest from the center).

Fermat's principle was generalized by Euler, Lagrange, and Hamilton into the variational principles of analytical mechanics. Given the system's initial configuration (the initial positions and velocities of its particles) and final configuration, there are various paths (through configuration space) by which the system may get from one to the other. These paths may differ, for instance, in the time it takes the system to arrive at its final configuration and in the configurations through which the system passes along the way. Roughly speaking, the Euler-Lagrange "principle of least action" states that the time integral of the system's total kinetic energy is "stationary" along the actual path as compared to all sufficiently close possible paths. That is, roughly speaking, the sum of the kinetic energies at all the points along the path actually taken is a minimum, maximum, or saddle point as compared to the sums for similar paths that are not taken. (So "the principle of least action" does not demand that the action be "least.")

Similarly, Hamilton's principle states roughly that of all the possible paths by which the system may proceed from one specified configuration to another in a specified time, the actual path as compared to other possible, slightly different paths makes stationary the time integral of the system's Lagrangian (i.e., the difference between the system's total kinetic and potential energies). A possible path may violate energy conservation and other laws; Hamilton's principle picks out the path demanded by the laws. So to apply Hamilton's principle, scientists must contemplate counterlegals: what would have been the case, had the system violated natural laws in certain ways. But a possible path must respect the constraints on the system, which may include a body's having to remain rigid or in contact with a certain surface.

These constraints may be plugged into the variational principles without the forces that constrain the system having to be specified. This gives variational principles a practical advantage over Newton's laws, since the forces of constraint may be unknown, and emphasizes the style of explanation that variational principles supply. Newton's laws are differential equations; they determine the instantaneous rates of change of the system's properties from the system's conditions at that moment, such as the forces on it. The system's trajectory over a finite time interval is then built up, point by point, and the forces are efficient causes of the system's acceleration. In contrast, variational principles make no mention of forces; instead, they invoke the system's energy. The explanations they supply specify no efficient causes. Variational principles involve integral equations; they determine the system's trajectory as a whole, rather than point by point.

## Teleology

Explanations that use variational principles sound teleological; the system appears to aim at making a certain integral stationary. But then the system's final configuration apparently helps to explain the path that the system takes to that destination; later events help to explain earlier ones. That is puzzling. How does a light ray "know," at the start of its journey, which path will take less time? How can the light adjust the earlier part of its route to minimize its later path through optically dense regions (where it cannot travel as fast) unless it knows about those distant regions before it sets off?

Some natural philosophers (such as Max Planck) suggested that variational principles are more basic laws than Newtonian differential equations, especially considering that unlike Newton's laws, variational equations of the same form apply to any set of variables sufficient to specify the system's configuration. Other natural philosophers (notably Gottfried Wilhelm Leibniz) embraced both mechanical and teleological explanations as equally fundamental. Leibniz declared that there are

two kingdoms even in corporeal nature, which interpenetrate without confusing or interfering with each other—the realm of power, according to which everything can be explained mechanically by efficient causes when we have sufficiently penetrated into its interior, and the realm of wisdom, according to which everything can be explained architectonically, so to speak, or by final causes when we understand its ways sufficiently. (1969, pp. 478–479)

Other natural philosophers (such as Mach) denied final causes but also denied efficient causes as well (allowing only the relations specified by natural laws). The most common view, however, has been to reject teleological explanations as a relic of anthropomorphic characterizations of nature and to regard variational principles as logical consequences of more fundamental, mechanical laws. The variational principles follow from the Newtonian differential equations roughly because the entire path can minimize the integral only if each infinitesimal part does (since otherwise, by replacing that part with another, one would create a new path with a smaller integral), and the minimum for each infinitesimal part reflects the gradient of the potential there, which is the force. The variational principle thus arises as a byproduct of the relation between the force and an infinitesimal section of the path.

** See also ** Berkeley, George; Chaos Theory; Determinism, A Historical Survey; Faraday, Michael; Hamilton, William; Hertz, Heinrich Rudolf; Hume, David; Kant, Immanuel; Laplace, Pierre Simon de; Laws, Scientific; Leibniz, Gottfried Wilhelm; Lewis, David; Logic, History of: Precursors of Modern Logic; Mach, Ernst; Maxwell, James Clerk; Newton, Isaac; Philosophy of Physics; Planck, Max; Quantum Mechanics; Relativity Theory; Russell, Bertrand Arthur William.

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*Marc Lange (2005)*