Breakdown of a misinterpretation of Noether's theorem Free

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 $p q = ∂ L / ∂ q ̇$ 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 $M x ̇$ is conserved. However, the linear momentum is conserved only if $p x = ∂ L / ∂ x ̇ = M x ̇$⁠.

In the example of Lemos, the Lagrangian is independent of the horizontal coordinate x of the center of the cylinder, but $p x = ∂ L / ∂ x ̇ ≠ M x ̇$⁠. Although p_x is a constant of the motion in this example, Lemos's paper suggests that this is a breakdown of Noether's theorem because p_x does not equal $M x ̇$⁠. 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.