In his paper on Emmy Noether’s theorem, Miles Mathis argues that the theorem is nothing more than a tautology without any conceptual advantage over Newtonian mechanics. While it can be argued that all theorems are tautologies, Miles shows a lack of understanding of Lagrangian mechanics that render his arguments invalid.
Miles begins with a rather philosophical introduction that deserves its own critique. However, I feel that it makes more sense to concentrate on his statements that can be compared to an actual physical theory. Miles writes
In the simplest situation (no potential or acceleration) action is just KΔt, where K is the kinetic energy.
In the case of classical mechanics, the action is a functional defined to be an integral over time, given a path of the system, of the so called “Lagrangian”
While it is true that the integral gives along the path of free particle that starts at and ends at , this tells us very little about the stationary points of the functional. The actual value of the action along a path is usually of significantly less importance than obtaining a solution for the resulting Euler-Lagrange equations.
In this stripped down equation it is quite clear that time symmetry implies a conservation of energy. If Δt is a constant then K must be “conserved” since it is the only variable you have left.
How ‘time symmetry’ implies that is a constant is unclear to me. Given the correct definition of the action, it simply depends on our choice of and . The stationarity of the action does not imply that it must be a constant for all possible paths, rendering his argument for the constancy of invalid.
From his arguments about the action of a free particle, Miles infers that they are correct for a particle in a potential as well:
This is equally true if we have a potential energy as well, since the energies are summed before being integrated.
Miles mistakes the Lagrangian of a system for its Hamiltonian (see (1) as well). If the total energy is a constant of the motion, an integral over it for a given path indeed yields a similar expression to the one calculated above. However in this Lagrangian the potential and kinetic energies are not summed but substracted. The difference between potential and kinetic energy is certainly not a constant of the motion for a general system.
Miles misunderstands the formalism and thus misses its advantages, one of which is formulated in Noether’s theorem.
(1): Note that even though it is true for the system considered, not all Lagrangians have the form where is a kinetic and is a potential energy.