# Does energy conservation imply time invariance?

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).

• 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. – ACuriousMind Jun 5 at 15:04
• @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? – gardenhead Jun 6 at 4:48
• 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 – ACuriousMind Jun 6 at 9:00

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