Does time invariance conclude conservation of energy?

Question

I find it hard to understand that time-translation invariance necessarily implies conservation of energy. As I understand it, Noether's theorem says that there is an energy conservation because the laws of nature do not depend on time.

The following statement is logically true: "Because there is an energy conservation and the laws of nature do not change over time, it doesn't matter which point in time we choose, the energy content of the universe is always the same".

The statement: "Because the laws of nature do not change over time, it doesn't matter which point in time we choose, the energy content of the universe is always the same" is not necessarily true in my view.

From a philosophical perspective I can imagine a possible world where (a) time exists and (b) the laws of nature do not change over time, but there is no conservation of energy. Imagine a world where a black hole is constantly creating energy or a fundamental force that is not conservative. I do not see that energy conservation is a priori given, something that Noether's theorem implies.

Is it that Noether's theorem in relation to energy conservation is circular reasoning in some way? The theorem is based on the assumption that the laws of nature are energy conserving - when these laws don't change over time - there is an universal energy conservation. But hypothetically speaking - if the laws of nature were not energy conserving and they wouldn't change over time - there would be no universal energy conservation.

Answer 1

The derivation of conservation of energy from Noether's theorem is somewhat pointless. The problem is that in Lagrangian mechanics (in which Noether's theorem applies), the definition of the Lagrangian involves assuming the existence potential energy. This means, that by definition, we assume there are no dissipative forces nor there are "energy-gaining" loops.

As a counter-example, consider the following hypothetical universe, in which $\vec F = \vec C \times \vec x$, where $\vec C = \textrm{const}\;.$ Of course, a particle could gain some kinetic energy by just doing some loops with respect to origin of coordinates (if we would further restrict it's motion to a circle, the particle would come to the very same point after some time, just with a larger kinetic energy). However note that potential energy can not be defined, thus Lagrangian can't be too.

However, it seems very likely, that the fundamental law's of physics can be defined in terms of Lagrangian.

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Edit:

Actually, non-conservative forces could be simulated by varying the potential, however that would obviously conflict with the assumption that the law's of nature do not vary in time (because then the Lagrangian function does vary).