Proof of conservation of energy?

How is it proved to be always true? It's a fundamental principle in Physics, that is based on all of our currents observations of multiple systems in the universe, is it always true to all systems? Because we haven't tested or observed them all. Would it be possible to discover/create a system that could lead to a different result?

How are we 100% sure that energy is always conserved? Finally, why did we conclude it's always conversed? What if a system keeps doing work over and over and over with time?

Answer

How is it proved to be always true? It's a fundamental principle in Physics, that is based on all of our currents observations of multiple systems in the universe, is it always true to all systems? Because we haven't tested or observed them all. Could it possible that we discover/create a system that could lead to a different result?

A physical theory, and also the postulates on which it is based can only be validated, that means that every experiment done shows that the theory holds and in this case energy conservation holds. A physical theory can be proven only to be false even by one datum, and then the theory changes. Example: classical mechanics fails at relativistic energies, when mass turns into energy, and classical energy is not conserved. A relativistic mechanics was developed that still has conservation of a more generally defined energy.

How are 100% sure that energy is always conserved? Finally, why did we conclude it's always conversed?

We cannot be sure, as I said above. If we find even one case where the newer energy definition fails than the postulate fails and new propositions will be studied. It has not failed up to now in our laboratory and observational experiments.

In any case, the framework is important, classical conservation of energy still holds for non relativistic energies, for example.

In general relativity conservation of energy-momentum is expressed with the aid of a stress-energy-momentum pseudotensor. The theory of general relativity leaves open the question of whether there is a conservation of energy for the entire universe.

For cosmological matters see another entry in this forum on the law of conservation of energy.