When a potential explicitly depends on time, energy is not conserved. However, if we take into account what is causing this potential (for example, a machine moving some object(s)), would the total energy of the combined system be conserved? I'd like to know if there's any proof one way or the other in the most general case.
Asked
Active
Viewed 178 times
1
-
Related: https://physics.stackexchange.com/q/19216/2451 and links therein. – Qmechanic Oct 28 '23 at 11:29
2 Answers
6
Energy conservation can be derived via Noether's theorem. If a physical system's Lagrangian does not explicitly depend on time, then the system's energy is conserved. If it does, as you pointed out, energy is not conserved, but this non-conservation is local. By including whatever is causing the time dependence, one can construct a Lagrangian for this "bigger system" that does not have explicit time dependence and energy conservation will be "restored".
Alex
- 1,183
-
But how do we know that the Lagrangian of the larger system no longer depends on time? Is it because the four fundamental forces dont depend on time? – Denn Oct 29 '23 at 00:01
-1
Depends.
Technically, external energy is supplied as free energy of a thermodynamical system. Paradigma: Storage of mechanical energy in a pressure sphere with ideal gas. Ideal gas can store energy by increase of temperature only. On compression of an thermical isolated sphere the mechanical input energy $p dV = R dT$ energy is completely stored in the kinetic energy of the gas by inrease of temperature.
But if the storage sphere is thermically connected to an external atmosphere of constant temperature, the internal energy is constant and the energy is stored in the atmosphere of the environment. Neverthless, the compressed sphere can be shipped elswhere and release the same amount of energy (at equal temperature).
It was a long historical path until Mayer conjectured conservation of total energy even in this cases, theoretically always enlarging the system until its completely isolated energetically from the rest of the universe. This is a concept of physics now, since Emmy Noether showed, that constants of motion are in a 1-1 correspondence with symmetry conserving change of variables. In the case of energy ths symmetry is movement in time, or in other words, if physics is the same at any time for a system, total energy, including all its different forms of appearance. is conserved.
