Energy conservation and time uniformity

Question

Reading Peter Atkin's book Four laws that drive the universe, I cannot see the link between time uniformity and conservation of energy the way it is tackled in this excerpt:

" Noether’s theorem, proposed by the German mathematician Emmy Noether (1882–1935), which states that to every conservation law there corresponds a symmetry. Thus, conservation laws are based on various aspects of the shape of the universe we inhabit. In the particular case of the conservation of energy, the symmetry is that of the shape of time. Energy is conserved because time is uniform: time flows steadily, it does not bunch up and run faster then spread out and run slowly. Time is a uniformly structured coordinate. If time were to bunch up and spread out, energy would not be conserved."

Answer 1

That's just an unfortunate way of framing Noether's theorem. There is no meaning to the assertion that time flows steadily, time always flows at the rate of $\frac{1~\mathrm{sec}}{1~\mathrm{sec}}$ out of necessity and I can't imagine what the alternative could possibly be. Furthermore, even to the extent to which the assertion that time flows steadily could be made meaningful$^\dagger$, it has nothing to do with Noether's theorem which is about the invariance of the laws of physics with time, and not about some intrinsic property of time itself.

OK, so what does the invariance of the laws of physics mean? It means that if you do an experiment at some time $t_0$ and do the exact same experiment (with the same set-up and same initial conditions, etc.) at some other time $t_0+\Delta t$ then the outcome of the experiment would turn out to be the same. In other words, the equations of motion that describe the underlying laws of physics do not have time as an explicit parameter in them. If the laws of physics for a given system have this property then the given system (or, its physics) is called invariant under translations in time, or, it's called to exhibit time-translational symmetry. Noether's theorem deals with this symmetry. In particular, it says that there exists a conserved quantity (that can be identified with what we know as energy) whenever the laws of physics governing a system have time-translational symmetry.

It is hard to explain why the symmetries of a system are linked to conserved quantities without getting into technical details. However, it can be understood intuitively by someone who is already a little bit familiar with the subject. See, for example, the two highest-voted answers on this question: Can Noether's theorem be understood intuitively?.

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$^\dagger$The way in which this assertion can be made somewhat meaningful is by understanding that time is defined in such a way that such processes that we expect to be periodic are indeed periodic. For example, in some sense, you can say that the time measured by a solar clock is less uniform compared to the time measured by an atomic clock. See my answer here. However, this has to do with how time is defined or what we deem to be a good clock. It has nothing to do with whether the laws of physics governing a given system are invariant under translations in time.

Answer 2

I cannot see the link between time uniformity and conservation of energy the way it is tackled in this excerpt...

It is very difficult to understand what the author means in detail based solely on the posted excerpt. The author is trying to explain something in everyday language that really only can be explained in equations.

Conservation of energy can be seen to hold in a variety of systems ranging from classical mechanical systems, to relativistic systems, to quantum mechanical systems. The requirement for conservation of energy turns out to be that there is no explicit time-dependence in the Hamiltonian (e.g., no time dependent constraints, no time dependent potential, etc.).

To see how this comes about, consider a simple example of a single particle (with mass $m$, position $x$ and velocity $v=dx/dt$) in one dimension in a constant conservative potential (U). The Hamiltonian for this systems is: $$ H = \frac{m}{2}v^2 + U(x) $$

The time derivative is: $$ \frac{dH}{dt} = \frac{m}{2}2v\frac{dv}{dt} + \frac{dU}{dx}v $$ $$ =mva-Fv, $$ since $\frac{dU}{dx} = -F$, for a conservative potential.

But, we know that $F = ma$, from Newton's second law, so the RHS is actually zero: $$ \frac{dH}{dt} = mva - mav = mva - mva = 0 $$

This property of the Hamiltonian turns out to hold much more generally[1] such that: $$ \frac{dH}{dt} = \frac{\partial H}{\partial t}\;, $$ which means that if there is no explicit time dependence in the Hamiltonian H (i.e., $\frac{\partial H}{\partial t}=0$) then $\frac{dH}{dt}=0$. This latter equation is what we mean when we say that energy is conserved.

To restate again in words: If there is no explicit time dependence in the Hamiltonian then energy is conserved. This is really "all" that can be understood from the quoted flowery excerpt.

[1]: For the Classical case, see, for example, Whittaker's "A Treatise of The Analytical Dynamics of Particles and Rigid Bodies" (2nd Edition) at page 62.

Answer 3

Energy is conserved because time is uniform: time flows steadily

What I can understand from this is that some periodic movements keep the same ratio of their periods. And exactly for that reason they were chosen historically to measure what we call time.

The ratios between days (earth rotations), weeks (lunar phases), months (moon phase to phase approximately) and year (repetition of solar position with respect to the stars, are fairly stable.

Ratios in the oscillation of springs or pendulums are also stable when changing their parameters and comparing the periods.

On the other hand, all that periodic movements can be modelled by equations where a quantity $E = E_p + E_k$ is constant.

So, there is a relation between our capacity to measure time as an objective stuff and energy conservation.