I'm afraid my paper1 has been misunderstood by Professor K. T. McDonald. He puts words in my mouth that I never said. The phrase “breakdown of Noether's theorem” is never to be found or suggested in my paper. Nor do I ascribe this supposed breakdown of Noether's theorem to the fact that px does not equal $Mẋ$ in my example. In the Introduction, I say that “Noether's theorem establishes the most general correspondence between invariance under continuous transformations and constants of the motion.” Next, I mention that conservation of linear momentum, angular momentum, and energy are associated with “invariance under translations, rotations, and time displacements, respectively.” These results are, obviously, particular cases of Noether's theorem because translations, rotations, and time displacements are not the most general continuous transformations. In no way whatsoever have I suggested that these particular results are Noether's theorem. McDonald has misread my paper.

Let $L=T−V$ be a standard Lagrangian, where V does not depend on velocities. When all constraints are holonomic, invariance of both the Lagrangian and the constraints under translations implies conservation of the total linear momentum, which turns out to coincide with the corresponding total canonical momentum.2 Velocity-dependent integrable constraints, which are said to be semiholonomic, seem completely equivalent to holonomic constraints because, in their integrated form, they restrict coordinates alone. So, it came to me as a surprise that semiholonomic constraints are not equivalent to holonomic constraints as regards the connection between symmetries and conservation laws. My example shows that the Lagrangian is invariant under translations but the corresponding conserved canonical momentum is not the total linear momentum, which is not conserved. In short, contrary to expectations, the conserved quantity associated with translation invariance is not the total linear momentum.

According to McDonald, I misinterpret Noether's theorem by stating or intimating that it provides a connection between symmetries and conservation laws only for standard Lagrangians and holonomic systems. But this is nowhere to be found in my paper. The connection between translation invariance and conservation of the total linear momentum for a holonomic system with a standard Lagrangian is a particular case of Noether's theorem because, for transformations that leave time unchanged, the invariance of the Lagrangian is equivalent to the invariance of the action. Never do I hint that the case of standard Lagrangian and holonomic constraints is the only instance of a connection between symmetries and conservation laws.

Finally, McDonald incorrectly states that “semiholonomic systems are a subset of holonomic ones.” If this were true, one would face the contradiction of a result that holds for all members of a set A but does not hold for members of a subset of A. It's just the other way around, holonomic systems are a subset of semiholonomic systems.

I find McDonald's criticism unwarranted because it rests on a misinterpretation of my paper.

1.
N. A.
Lemos
, “
Breakdown of the connection between symmetries and conservation laws for semiholonomic systems
,”
Am. J. Phys.
90
,
221
224
(
2022
).
2.
N. A.
Lemos
,
Analytical Mechanics
(
Cambridge U. P
.,
Cambridge
,
2018
), Chap. 2.