A recent paper by Lemos titled,1 “Breakdown of the connection between symmetries and conservation laws for semiholonomic systems,” may unintentionally lead the reader to suppose that Noether's theorem suffers a “breakdown” in the example of a mass that slides without friction inside a cylinder that rolls without slipping on a horizontal plane.

In classical mechanics, Noether's theorem2 is a restatement of an insight of Lagrange that if the Lagrangian L of a system is invariant under coordinate q (that is, independent of q), then the CANONICAL (or generalized) momentum pq=L/q̇ is a constant of the motion (i.e., a conserved quantity). Unfortunately, this theorem is often misinterpreted/oversimplified to mean that if the Lagrangian of a system of total mass M is independent of the spatial coordinate x, then the total LINEAR momentum Mẋ is conserved. However, the linear momentum is conserved only if px=L/ẋ=Mẋ.

In the example of Lemos, the Lagrangian is independent of the horizontal coordinate x of the center of the cylinder, but px=L/ẋMẋ. Although px is a constant of the motion in this example, Lemos's paper suggests that this is a breakdown of Noether's theorem because px does not equal Mẋ. This inference is a disservice to Lagrange and to Noether, as a conserved momentum related to an invariance/symmetry does exist in Lemos's example, exactly in accordance with Noether's theorem.

Lemos cited Noether's theorem in the Introduction to his paper, and immediately afterwards wrote,1,3 “The conservation of linear momentum, angular momentum, and energy for many-particle systems is associated with invariance of the action under translations, rotations, and time displacements, respectively.” In that context, readers might assume that the quotation is Noether's theorem, although it is rather only an important special case of it. Lemos then went on to show that this statement does not hold for his example of the rolling cylinder, because, as is shown in his textbook,4 this statement holds only for “proper” holonomic mechanical systems (as well as for ones with no constraints), but not for semiholonomic systems or for nonholonomic systems.5 

According to the narrow but popular view of the association/connection between conservation laws and invariance stated in the above quote from Lemos's paper, its title, “Breakdown of the connection between symmetries and conservation laws for semiholonomic systems,” is valid. But, the wording of Lemos's paper may lead readers to think that Noether's theorem suffers a breakdown for semiholonomic systems. This letter affirms that the full power and validity of Noether's theorem, for any and all systems describable by a Lagrangian, is unaffected by Lemos's result.

1.
N. A.
Lemos
,
Am. J. Phys.
90
,
221
224
(
2022
).
2.
See, for example, Sec. 13.7 of Ref. 1 of Lemos's paper,
H.
Goldstein
,
C. P.
Poole
, Jr.
, and
J. L.
Safko
,
Classical Mechanics
, 3rd ed. (
Addison Wesley
,
San Francisco
,
2001
). Noether wrote about “invariance” rather than “symmetry” (mainly in the context of general relativity), although the term symmetry is now popularly associated with her theorem.
3.
For systems with constraints, the term “action” (used, but not defined in Lemos's paper) can be construed to mean (in the context of Lemos's paper) the “extended” Lagrangian L defined in his Eq. (25), which includes the constraints via a Lagrange multiplier.
4.
The only previous demonstration of this fact may be that on p. 69 of Lemos's book,
Analytical Mechanics
(
Cambridge U. P.
,
Cambridge
,
2018
).
5.
The terms “holonomic” and “semiholonomic” are discussed on p. 264 of Ref. 9 of Lemos's paper,
J. G.
Papastavridis
,
Analytical Mechanics
(
World Scientific
,
Singapore
,
2014
). Holonomic systems have constraints of the form fi({qj},t). Semiholonomic systems have velocity-dependent constraints of the form gi({qj,q̇j},t) that can be integrated to the holonomic form, but which include constants that depend on the initial conditions. Hence, semiholonomic systems are a subset of holonomic ones. A distinct subset of holonomic systems is sometimes called proper, for which constants in the constraints of these holonomic subsystems are independent of the initial conditions. The term “holonomous” = integral (őλoς) laws (νoμóς) was introduced by H. Hertz in Sec. 123, p. 80 of The Principles of Mechanics (Macmillan, New York, 1899); see also p. 91 of the original German edition (Barth, Leipzig, 1894).