I have read different accounts of time invariance leading to the conservation of energy, but have not encountered the specific logical explanation for it. Can someone provide it?
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Time independance of a Hamiltonian describing a system (or a Lagrangian) does not always mean that the energy is conserved. For example, if you describe a system in a non-inertial frame of reference (e.g.: a rotating frame), then, even if there is time invariance (i.e.: the Hamiltonian of the system is independant of time), there will be no conservation of energy.
In classical mechanics, in the Hamiltonian formalism, the Hamiltonian of a system (the Legendre transformation of the Lagrangian in canonical moments) is the time evolution generator. In general, if the Hamiltonian $H$ (same for the Lagrangian) does not explicitly depends of the time (i.e.: $\frac{\partial H}{\partial t} = 0$) then $H$ is conserved (i.e.: $\frac{d H}{d t} = 0$). But the Hamiltonian is not always equal to the energy of a system!
In order to make sure $H$ is equal to the system's total energy, the following conditions must be met:
- The frame of reference's coordinates expressed in terms of the generalized coordinates (the position coordinates of the Hamiltonian) must not depend of time and,
- The potential must not depend of the generalized coordinates total time derivatives.
When those conditions are met, then $H=E$ and if $\frac{\partial H}{\partial t} = 0$ then $\frac{d H}{d t} = \frac{d E}{d t} = 0$ so the energy is conserved. Of course, in most cases, we have that $H=E$ but it is not in all cases. In conclusion: time invariance does not always mean energy conservation.
All this formalism is explicitely defined mathematically and the proofs of these theorems are quite cumbersome to write and they come with a few definitions. For further details, I would refer to a classical mechanics book like the Goldstein's Classical Mechanics book which is a standard reference for this matter.
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