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This is similar to this question: Is the converse of Noether's first theorem true: Every conservation law has a symmetry?. However, the answer given there is very technical and general. I am only interested in the specific case of energy conservation (mostly because dark energy seems to break energy conservation / time invariance).

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    $\begingroup$ How are you defining "energy" other than as "the charge of time translation"? If we define energy as the charge of time translation, then it is a tautology that its conservation implies time invariance. Also, "dark energy" suggests you are thinking about cosmology, which as a general relativistic theory is far from ordinary applications of Noether's theorem in classical Lagrangian mechanics. $\endgroup$ – ACuriousMind Jun 5 at 15:04
  • $\begingroup$ @ACuriousMind I was defining energy in as the sum of the kinetic and potential energies of the system. To be more precise, the system consists of $N$ particles each with kinetic energy $\frac12 m_i v_i^2$ and pairwise potential energies $V(q_i, q_j), i \neq j$. Is this an unsatisfactory definition of energy? $\endgroup$ – gardenhead Jun 6 at 4:48
  • $\begingroup$ Well, that definition doesn't work for field theories (EM fields can store energy, too!) and certainly excludes the dark energy term, so I'm not sure how it is relevant to your question $\endgroup$ – ACuriousMind Jun 6 at 9:00
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It is actually the other way around. Time translation symmetry refers to Energy conservation. We define the Hamiltonian as $$\normalsize {H} = \Large{\Sigma_i}\normalsize{p_i\overset{.}{q_i} - L} $$ This says that the Hamiltonian in other words the energy is conserved when the Lagrangian has no explicit time dependance. i.e. $$\frac{dH}{dt}=\frac{\partial L}{\partial t}$$

This means as long as the laws of motion are time translation invariant, the energy of the system in consideration is conserved.

The converse is true as well. As you can see the equations say that

$$\frac{dH}{dt}=\frac{\partial L}{\partial t}$$

Which means that if energy is conserved it means that the Lagrangian has no explicit time dependance. Now even though it might seem in certain systems that the energy is not conserved, we must remember that the system is not necessarily isolated, so when we see that the energy is not conserved it just means that the flow of energy is from the surroundings

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    $\begingroup$ I think the OP is aware of Noether's theorem and is asking if the converse is also true (for energy/time specifically). $\endgroup$ – Rococo Jun 5 at 17:08

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