I have studied high school physics but only have a pop-sci kind of understanding of ideas like Noether's theorem so go easy.
As far as I know, Noether's theorem simply states that any kind symmetry of a physical system is accompanied by a particular conserved quantity.
The normal example of this is that if we consider a system to be time symmetric (the laws that govern the system are the same at all points in time), then energy conservation emerges. There are, however, supposed to be examples of systems where energy is not conserved, implying that the laws that govern the system isn't symmetric.
My problem is with this idea that laws are not symmetric over time. What does this really mean? Does this mean that the laws literally change over time? Is it not a very basic assumption for the study of physics that the laws that govern the universe never change? If we found systems where laws changed, how could we ever study the laws?
I believe that my confusion here is down to a misunderstanding of what it means for a system to be time symmetric but maybe it's something else. Thanks for any answers.
Best Answer
It's a little more involved than you say because the object that possesses the symmetry is the action for the system. The equations that describe the motion, i.e. the laws of physics for this system are then derived from this action using the Euler-Lagrange equation.
But we can describe a very simple example. Suppose you're in your spaceship and you plan a flyby of some large object e.g. the Sun. You start far from the Sun with some velocity $v$, and the Sun's gravity accelerates you inwards with the usual Newtonian equation for gravity:
$$ F = \frac{GMm}{r^2} $$
You swing in towards the Sun, accelerating as you go, then as you move away from the Sun again its gravity slows you back down and you depart with the same velocity $v$ that you started with. Your kinetic energy is unchanged.
However suppose it turns out that Newton's constant $G$ is actually time dependent and is decreasing with time:
$$ F(t) = \frac{G(t) Mm}{r^2} $$
That means $G$, and therefore the force, is lower for your outward journey than your inward journey. You are decelerated less on your outward journey and you leave with more energy than you started. Energy has not been conserved, and it's the time dependence of $G$ that is responsible.