Symmetries and the explanation of conservation laws in the light of the inverse problem in Lagrangian mechanics

Abstract

Many have thought that symmetries of a Lagrangian explain the standard laws of energy, momentum, and angular momentum conservation in a rather straightforward way. In this paper, I argue that the explanation of conservation laws via symmetries of Lagrangians involves complications that have not been adequately noted in the philosophical literature and some of the physics literature on the subject. In fact, such complications show that the principles that are commonly appealed to to drive explanations of conservation laws are not generally correct without caveats. I hope here to give a clearer picture of the relationship between symmetries and conservation laws in Lagrangian mechanics via an examination of the bearing that results in the inverse problem in the calculus of variations have on this topic.

Introduction

There is frequent talk in philosophical circles and among some physicists of explaining conservation laws via symmetries. Within such a general explanatory program, it is typically thought that particular conservation laws are explained by particular symmetries. For example, it is typically thought that time translation invariance explains conservation of energy. Likewise, other conservation laws are believed to be explained rather cleanly by particular symmetries: conservation of momentum is explained by spatial translation invariance and conservation of angular momentum by rotational invariance.

Though many subscribe to this general view, various philosophers have different notions of what is required for explanation (both generally and apparently in this specific case) and they also seem to have different intended explanada in mind. For example, Marc Lange discusses the strong program of showing why conservation of energy had to hold for a particular system as a matter of physical necessity. Part of that program involves showing not only that the particular forces at work in the system are such as to conserve energy but that any other forces that there could have been would have conserved energy. Within this program, it is thought that force functions (or associated functions like Lagrangians which encode them) could not have taken certain forms if that would violate a “meta-law” symmetry principle.¹ For example, though Lange does not tell us exactly what these “meta-laws” come to in Lagrangian mechanics, one might think that they require that a Lagrangian (for a closed system) not explicitly depend upon time (except possibly in a gauge term). Presumably, there will be analogous requirements imposed on certain open systems. For example, since the part of the Lagrangian representing the internal interactions of an open system cannot contain the time (via the principle for closed systems), any time dependence that enters into the Lagrangian for an open system must represent the time-varying nature of the external world. Thus, if an open system is interacting solely with a time-unvarying external system, its Lagrangian ought not contain the time. From that requirement, one needs a deductive path to a conservation law. And, in this general program (not just with Lange's strong explanandum), it is typically supposed that Noether's “first” theorem provides that path.² Lange claims, “It is widely known that within a Lagrangian framework, each of the classical spacetime symmetry principles logically entails one of the familiar conservation laws” (Lange, 2007, p. 457). Thus, for example, the absence of explicit dependence upon time in a Lagrangian will (it is thought) guarantee that the energy is conserved. So, a symmetry principle that places some requirement on Lagrangians for a system along with Noether's first theorem (which supposes that the dynamics are given by the Euler–Lagrange equations) will (it is thought) imply some conservation law for that system. As Lange puts it,

“When a conservation law is explained by a symmetry principle, the symmetry principle functions as the “covering-law” and the fundamental dynamical law [e.g. Newton's Second Law or Hamilton's Principle] functions as the “initial condition.” That is, the dynamical law is governed by the symmetry principle; the symmetry principle would still have held even if the dynamical law had not. The symmetry principle $[\dots ]$ possesses a variety of necessity. It explains the conservation law by making it likewise necessary that the conservation law hold given the dynamical law. That is $[\dots ]$, the conservation law holds in every possible world where the dynamical law holds” (Lange, 2007, p. 478).

One might call this “Explanation of Conservation of Energy in the Strong Sense”: What one wants to do is show that conservation of energy had to hold given a particular symmetry principle (which has a sort of necessity) and a particular dynamical principle (like the Euler–Lagrange equations).

Of course, some who have discussed the explanation of conservation laws will find this reliance upon “physical necessity” to involve a notion about which they are rather skeptical. If this is what it takes to explain conservation of energy, then no explanation is likely to be had. Among those who is likely to feel this way is Bas van Fraassen who appeals to symmetry (in part) because it is a legitimate scientific notion free from the dubious metaphysics that is frequently invoked to ground physical necessities. Even so, van Fraassen speaks of “symmetries and the conservation laws they engender” (van Fraassen, 1993, p. 433) and he claims, “In the twentieth century we have learned to say that every symmetry yields a conservation law.” (van Fraassen, 1989, p. 258). So, a weaker explanatory program involves showing that time translation invariance engenders conservation of energy. Apparently, the evidence that a specific symmetry generally engenders a certain conservation law is (again) Noether's first theorem which van Fraassen glosses as: “[F]or every symmetry a conservation law” (van Fraassen, 1989, p. 287, italics in original). Though he makes the claim with respect to a Hamiltonian formulation of mechanics rather than a Lagrangian formulation, van Fraassen reads part of the content of Noether's theorem as follows: “[I]f H [the Hamiltonian for a system] is invariant under time translation, then H is conserved in time, and a fortiori the total energy is conserved.” (van Fraassen, 1989, p. 286). Many textbooks in mechanics state the essential content of Noether's theorem in a similar manner when it comes to conservation of energy³:

“A recurring theme throughout has been that symmetry properties of the Lagrangian (or Hamiltonian) imply the existence of conserved quantities$\dots$ [I]nvariance of the Lagrangian under time displacement implies conservation of energy. The formal description of the connection between invariance or symmetry properties and conserved quantities is contained in Noether's theorem”⁴ (Goldstein, 1980, p. 588).

Presumably, van Fraassen is not wanting to explain the necessity of conservation of energy (which he doubts). So, this is a weaker project than Lange's, but it presupposes the same general connection between a specific symmetry of Lagrangians (such as time translation invariance) and a specific conservation law (such as conservation of energy).

What seems to be the common thread in discussions of this explanatory program—even against a backdrop of differing requirements for explanation⁵—is, for example, that time translation invariance plays a rather straightforward and unique role in explaining conservation of energy via Noether's theorem. In this paper, I want to evaluate these variants of this general explanatory project within classical mechanics—in fact mostly within the classical mechanics of simple finite-dimensional systems since many provide illustrative examples from this arena (Lange, 2007, van Fraassen, 1989). To my mind, no one has given a sufficiently detailed description of how they expect time translation invariance to explain the conservation of energy of a system that appears both to accomplish its explanatory aims and to involve principles that are true as stated. In fact, I find that such explanations often rely on principles that are false without further specification (including descriptions of the content of Noether's theorem that issue from unmotivated restrictions of the scope of Lagrangian mechanics) or on unspecified (or underspecified) “symmetry principles” and “meta-laws”. Chief among the difficulties is that it seems still not to be widely appreciated—though it has been known since (Santilli, 1983)—that time translation invariance of a Lagrangian is neither necessary nor sufficient for conservation of energy. In the rest of this paper, I will spell out the research program—the inverse problem in the calculus of variations—that has led to the discovery that some of the principles appealed to within this program are false and will note other complexities that arise for this explanatory program. I do not think that there is no grain of truth to this general program.⁶ I think that in some sense the complexities noted in this paper increase our understanding of, say, conservation of energy via its relation to symmetries. And, if all that is required for explanation is increased understanding, then I do think that there is explanation in the vicinity. But, these complexities do suggest that the simple pattern of explanation often expected requires caveats that are not generally noted.