What does Noether's theorem imply about conservation?

Noether's theorem is a fundamental principle in theoretical physics that connects symmetries and conservation laws. Specifically, it states that for every continuous symmetry of a physical system, there is a corresponding conservation law. In the context of time, if the laws of physics are invariant or unchanged over time—meaning they do not change with the passage of time—this leads to the conservation of energy.

When a system exhibits time translation symmetry, it implies that the behavior of the system remains the same regardless of when you observe it. Consequently, this invariance results in the conclusion that energy remains constant in that system; therefore, energy is conserved. This is why the assertion that energy is conserved if laws do not change with time accurately reflects the application of Noether's theorem in this context.

Understanding this principle helps clarify why conservation laws arise in physical systems and how they are deeply linked to the symmetries of those systems.