Rationale for Conservation of Energy

[junaid314159]

I know that the Law of Conservation of Mechanical Energy can be derived using Newton's Laws and Kinematics. I believe that at very small scales, where Newton's Laws no longer apply, that Conservation of Mechanical Energy is still true (or is it Conservation of Energy in general, not sure).

1) Is there a way to derive the Law of Conservation of Mechanical Energy in these settings using different laws that are valid there?
2) Is there a way to derive the Law of Conservation of Mechanical Energy in traditional Newtonian settings without appealing to Newton's Laws?

I just feel that intuitively there must be some underlying set of physical laws that lead to Newton's Laws and Conservation of Energy (through them) that lead to Conservation of Energy even in the absence of their validity. On a side note, can it be said that some of Newton's Laws are a result of the Law of Conservation of Energy?

In general, is physics interested in questions like which laws lead to other laws, that is which is more fundamental, or are they seen as interdependent, interwoven truths without the need to look into which cause or lead to the other.

Thanks,
Junaid

 

[Dale]

At very small scales you typically use Lagrangian mechanics. So, given a Lagrangian which describes your small system, if that Lagrangian is invariant under time translations then by Noether's theorem there is a conserved quantity which is the energy.

 

[junaid314159]

Thanks. I will read up on that.