Why does time-translational symmetry imply that energy (and not something else) is conserved?

Question

I'm trying to understand Noether's theorem from an intuitive perspective. I know that time-translational symmetry implies the conservation of energy. Is it possible to convince oneself that time-translational symmetry implies that energy is the only conserved quantity, and not linear momentum for instance, without going through the mathematical derivation?

Answer 1

1. The main point is that if an action has a given infinitesimal quasi-symmetry with a single infinitesimal parameter then there is a single non-ambiguous & unique on-shell continuity equation; not 2 or more as OP seems to suggest.

2. Importantly, a translation quasi-symmetry in one variable implies that the canonically conjugate variable is conserved. This is easier to see in the Hamiltonian formalism, cf. e.g. my Phys.SE answer here.

3. Concerning time-translation quasi-symmetry and energy conservation, see e.g. this related Phys.SE post.

Answer 2

Is it possible to convince oneself that time-translational symmetry implies that energy is the only conserved quantity and not linear momentum, for instance, without going through the mathematical derivation?

Short answer: it is not possible because it is not true.

To explain energy conservation in terms of time-translational symmetry, one needs some starting definition of what is a description of a dynamical system and how to derive what energy is. Noethers' theorem provides such starting point in a completely general way (not only for time-translational symmetry) with the Lagrangian of the system. Notice that as soon as we have the Lagrangian, we have, at the same time, the equations of motion and the expression for the conserved quantity in the case of continuous symmetries. The main point I would like to stress is that the kind of dynamics is a critical ingredient in translating symmetries into conservation laws. In these terms, the idea that one could derive the conservation of an observable quantity by intuition without a sound mathematical derivation looks like an impossible task.

However, without the need for Dr. Noether's theorem in its full fledge, we can go quite far in providing an elementary answer to your questions.

Energy conservation and momentum conservation are two independent conservation laws. For example, the simple harmonic oscillator (a mass $m$ attached to a spring of elastic constant $k$) conserves the energy $$ E(t)=\frac12 m v^2 (t) + \frac12 k x^2 (t) $$ but not the momentum $p(t) = m v(t)$. On the other hand, in an inelastic collision, energy is not conserved, while momentum is. These two examples should justify the independence between the two conservation laws.

Moreover, we may have conservative systems (energy is conserved) where there may or may not be additional conserved quantities. For example, a $2D$ harmonic oscillator conserves energy and angular momentum. Energy conservation can be related in an elementary way to the time symmetry of the energy. Conservation of angular momentum depends on the central character of the interaction law (rotational symmetry of the potential energy). These are independent symmetries. Introducing two different elastic constants along the $x$ and $y$ directions is a trivial exercise. The resulting energy is still time-symmetric (constant of motion), while the rotational symmetry is broken, and angular momentum is not conserved anymore.

Answer 3

You can intuitively think about energy (energy density to be precise) as a flux of "something" in time. Take a resting mass in a point $p_0$, then you can describe the mass density as a flux through time of such a quantity. It doesn't move in space but only goes forward in time, so its a "time-stream", called flux, of stuff (the mass). At every instant it leaves the "past-$p_0$" and enters the "future-$p_0$". If your laws are time symmetric, this means that the flux of stuff is time symmetric: if at $t_1$ you got $X$ flux of substance entering from the past $p_0$ and $Y$ of a substance leaving the same point from the future, the same will be valid for the same point $p_0$ at $t_2$ due to time symmetry. Since it will be the same for every $t_1$ and $t_2$ of your choice, it means that at a point $p_0$ at every time you have the same amount of $X-Y$ substance, this is conservation of energy.

The same argument applies to every type of energy, I used mass because you can get a better intuition behind it than using EM energy density or chemical potential binding energy and so on, but the argument is the same.

Answer 4

This answer is for the context of classical mechanics.

Among the conserved quantities of classical mechanics there is one for which the association is quite vivid: angular momentum.

As we know, Kepler anticipated the concept of angular momentum conservation in the form of his second law of celestial mechanics. As a planet is orbiting: in equal intervals of time equal areas are swept out.

In the Principia Newton derived the area law from first principles. Moreover, Newton demonstrated a more general area law. When the force that is moving the object around is a central force then the circumnavigating object will in equal intervals of time sweep out equal areas.

Years ago I created a diagram for Newton's demonstration, it is available in the wikipedia article about angular momentum, in the sub-section Law of Areas.

(Newton's demonstration makes use of the property that the same reasoning applies for any orientation; the rotational symmetry is key.)

The Coulomb force and gravity are examples of a force with rotational symmetry. The magnitude is a function of radial distance only, there is no dependence on orientation. When a force has that property: the resulting motion has the property that angular momentum is conserved.

In general:
What counts as a conserved quantity:
A quantifiable property of an isolated physical system that does not change as the system evolves over time.

That is: symmetry under time translation is implicit in every conservation principle.

In the case of angular momentum:
The explicitly associated symmetry is a spatial symmetry: independence of orientation.

With momentum the symmetry and the conservation are in different dimensions. The symmetry is spatial, the conservation is over time.

In the case of Energy:

Yeah, that is not a clean cut situation; there is no opportunity to associate energy conservation with some spatial symmetry.

It seems to me that in the case of Energy conservation we should simply acknowledge that there is no associated spatial symmetry.

That would make Energy conservation a simpler case; its associated symmetry is in the same dimension as its conservation: the time dimension.

Answer 5

Here is a simple way to think of this.

Time translation symmetry means that in an experiment on a dynamical system in which we are using (among other things) a stopwatch to timestamp our observations, we do not have to re-zero the stopwatch hands for each measurement: we will get the same result in our subsequent calculations if we everywhere replace the time variable t with any arbitrary (t+x). This is the same thing as saying if we do the experiment tomorrow (that is, at t+24 hours) we'll get the same result. Algebra then reveals to us that a derived quantity in the description of the dynamical system with the units of energy is invariant when t is swapped out for (t+24 hours). In this way time translation symmetry implies energy conservation.