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Noether's theorem enables us to say the time translational invariant of any Lagrangian is the energy.
Heat has dimensions of energy. My question is can heat too be thought as a type of Noether invariant?
P.S: Please mention the role of inexact differentials of the first law in the derivation which enables one to make (or counter) the claim?
$$ dU + \delta w= \delta q $$
It is not obvious that the could be any way to establish a link between Noethere invariant and heat. The fact that energy and heat have the same physical dimension is a too weak argument. Let me remind that even torque has the same physical dimensions as energy, but is not a conserved quantity.
A more specific objection is that heat, as defined in physics, is not a property of a physical system (like for example the energy). It is a measurement of the change of energy of a system as a consequence of a process (and it depends on the process).
If by heat you we referring to thermal energy of a system, it would be possible to argue that some connection with Noether's theorem is present, but that's not at the level of description of Thermodynamics. Indeed, from a Statistical Mechanics perspective, the internal energy of system is nothing but the average total energy (in the center of mass reference frame) of the internal degrees of freedom. Therefore, as soon as the fundamental lagrangian controlling their dynamics is time invariant, conservation of the total energy will result.
The problem of such an approach in thermodynamics is that at a macroscopic level it is not possible to achieve a complete control on the microscopic states, making impossible to measure the energy. Therefore one has to use an indirect way of controlling internal energy through the net work and heat exchanged between system and outer world.
