Joshua Eisenthal (University of Pittsburgh)
I argue that the notion of force in classical mechanics does not have a uniform meaning. In particular, I argue that the traditional formulation of classical mechanics appeals to at least two distinct notions of force which are not obviously compatible with one another. This equivocation at the heart of the theory challenges the standard construal of classical mechanics as a candidate fundamental theory. I claim that the apparent fundamentality of mechanics stems in part from the tacit unification of a diverse range of problem-solving strategies; strategies which may not sit comfortably within a single unified framework.
There are two distinct traditions in the history of mechanics: the vectorial tradition and the variational tradition. (Cf. Lanczos (1962) xvii and Sklar (2013) pp. 76-79.) The vectorial tradition is most recognisable in Newton’s canonical laws of motion. Paradigm problems in this tradition involve distance forces acting between point-masses, such as gravitational forces in a simple model of the solar system. In contrast, the variational tradition is most recognisable in the ‘analytic’ methods of Lagrange and Hamilton. The paradigm problems in this tradition involve applications of extremal principles such as the principle of least action. The ‘Newtonian’ notion of force operative in the vectorial tradition is the familiar notion of a kind of push or pull, typically represented by a three-dimensional vector in ordinary space. In contrast, the ‘Lagrangian’ notion of force is an abstract vectorial quantity which can have as many dimensions as a system’s degrees of freedom.
Many of the homely properties of Newtonian forces are not applicable to Lagrangian forces. For instance, although a Newtonian force can be regarded as acting from one body on another and causing the second body to accelerate, Lagrangian forces cannot typically be interpreted in this way: a Lagrangian force is an atemporal property of a system, comparable to the system’s total energy. Furthermore, although Newtonian forces often depend only on the relative distances between bodies, a Lagrangian force can often depend on a body’s velocity.
These conflicting demands on the notion of ‘force’ have important implications for standard interpretations of classical mechanics. According to the standard view, Newton’s laws are intended to be universal and exceptionless, applying equally well to molecules, chairs, and galaxies. However, I argue that the apparent fundamentality of classical mechanics depends on the tacit unification of a diverse range of problem-solving strategies. (This criticism of the standard interpretation of classical mechanics takes its cue from Wilson (2013).) The contrasting conceptions of force evident in the vectorial and variational traditions makes the differences between these strategies vivid, and calls into question the standard construal of classical mechanics as a candidate fundamental theory.
