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According to Noether's Theorem, every physical symmetry leads to a conservation law. For example, time-translation symmetry (the laws of physics don't change over time) implies conservation of energy, and space-translation symmetry (the laws of physics are the same everywhere) implies conservation of momentum.

But, Noether's Theorem is both complex and abstract. I have no idea why that should be the case. It's easy to imagine a universe that has time-symmetry but not energy conservation. However, according to the theorem, that should be a contradiction.

So, is there a simple explanation for why that's the case? I'm not looking for a proof of Noether's Theorem; I'm looking for an intuitive explanation for why we should expect the theorem to be true.

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Here is an easier way to think about this.

Imagine that the laws of physics could be time dependent, and you arrange for the law of gravity to be turned off in the shank of an otherwise dead Thursday afternoon.

In anticipation of this useful event, you have a machine set up which uses falling weights to perform useful work, and a pile of weights ready to place inside a container mounted high above the machine, from which they are fed downwards into it.

Then comes Thursday, and with gravity shut off it is easy for you to lift all those weights up and into the container. Later that afternoon, when gravity comes back on again, you turn the machine on and it produces useful work, which cost you nothing because gravity was off while you loaded the machine with weights.

In this way, turning gravity on and off allows you to violate energy conservation.

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  • $\begingroup$ This hints that energy conservation implies time symmetry, but says nothing about the converse, which is what the question was about $\endgroup$ – BlueRaja - Danny Pflughoeft Apr 12 at 12:21
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It's easy to imagine a universe that has time-symmetry but not energy conservation.

It's certainly not easy for me. To me, time-translation symmetry means that if I start a system in some arbitrary initial state $x_0$ at some initial time $t_0$, then it will evolve the same way regardless of what $t_0$ is. But since evolution is generated by the Hamilton equations, if the system evolves differently then $H$ must be a different function of the phase space variables at different times (i.e. $H$ is explicitly time-dependent). Essentially the same idea holds in Lagrangian mechanics, if you'd like to translate the issue into that formalism.

So no time-translation symmetry $\implies$ $H$ is an explicit function of time, which means that the numerical value of $H$ is not preserved by time evolution. On the other hand, if $H$ is not an explicit function of time then the same argument shows that you will have time-translation symmetry.

In the reverse direction, the presence of time-translation symmetry implies that $\frac{\partial H}{\partial q}$ and $\frac{\partial H}{\partial p}$ both lack explicit time-dependence. It is possible that $H$ contains some time dependence like $H(x,p,t) = H_0(x,p) + H_1(t)$, but $H_1$ would be completely irrelevant to the dynamics of the system and $H_0=H-H_1$ would assume the role of the conserved quantity.

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You are correct that time invariance of laws does not require energy conservation.

It is a common misunderstanding of the content of Noether's theorem. It is not about "laws of physics being independent of time" implying "physics concept of energy is conserved". Those are simplified colloquial statements that are easy to state and remember, but they are inaccurate. There are two problems with such lazy formulations.

First, I can have time-invariant law of physics that every particle experiences both Hooke elastic force $-kx_k$ and friction force $-\gamma m \dot{x}_k$. Such a law of physics would mean the physical concept of energy is not conserved (due to friction force). You can see this hypothetical physics law is time-invariant, yet energy isn't conserved. So we can't formulate the Noether theorem in such a way - it does not work.

Second, Noether's theorem does not say "physics concept of energy is conserved", it says "there is a corresponding conserved quantity based on the action expression and the transformation that leaves $S$ invariant, its interpretation is on you, it may be energy, or not".

Let us formulate the Noether theorem for the special case of time coordinate shift invariance. We need an expression for the action integral, for example the expression

$$ S(t_0) = \int_{t_0}^{t_0 + \Delta t} L~ dt' $$

where $L$ is some function of coordinates and their derivatives. Noether's theorem states that if $S$ has the "symmetry" that time translations do not changes it (read: $S$ is invariant with respect to change in the time coordinate $t_0$):

$$ \frac{\partial S}{\partial t_0}(t_0) = 0 ~~~\forall t_0, $$ then the quantity

$$ Q = \sum_n\frac{\partial L}{\partial \dot{q}_k} \dot{q}_k - L $$ does not change as the system evolves in time - it is a "constant of motion".

However, neither Noether's theorem nor other general rule say this quantity has to be energy (in the physics sense)!

There are examples where the quantity $Q$ is energy, such as the Lagrangian of an harmonic oscillator $$ \frac{1}{2}m\dot{x}^2 -\frac{1}{2}kx^2 $$ and other examples, where it is not energy, like the Lagrangian of damped harmonic oscillator (Havas' Lagrangian [1]):

$$ L = \frac{2m\dot{x} +kx}{x\sqrt{4mK-k^2}}\tan^{-1}\left(\frac{2m\dot{x} + kx}{x\sqrt{4mK -k^2}} \right) - $$ $$ -\frac{1}{2}\ln (m\dot{x}^2 + kx\dot{x} + Kx^2) $$

So, to round this up, Noether's theorem says that existence of symmetry of action implies existence of the corresponding conserved quantity, period, full stop. Interpretation of these quantities is sometimes simple (total energy), sometimes not (Havas' Hamiltonian, which is conserved in time, and thus can't be total energy).

[1] Havas P., The Range of Application of the Lagrange Formalism - I,Nuovo Cim. 5(Suppl.), 363 (1957)

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Noether's theorem is completely general and makes it easy to give general results with a minimum effort. However, its generality and its relations with symmetries are sometimes misunderstood.

A typical misunderstanding is that it is necessary to explore the relations between symmetries and conservation laws. This is not the case, as shown in the answer by Chiral Anomaly to a related question.

In a similar vein, it is possible to relate a continuous time symmetry of mechanical laws to the conservation of energy in every formalism for mechanics. Assuming that Newton's equations of motion are more familiar, using them for exploring the connection between time translational symmetry and energy conservation can shed some light on a more general intuition about that.

Like momentum or angular momentum, the essential starting point is to correctly identify the elements of the theory that embodies the time invariance. In the Newtonian description, this is the force. A time-invariant force, in general, is a function of the positions and velocities. It cannot depend explicitly on time. If forces are conservative, this fact implies that the potential energy does not depend explicitly on time. Therefore by differentiating with respect to time the sum of kinetic and potential energy is trivial to check that the validity of Newton's equations of motion is equivalent to the conservation of energy.

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