Isaac Newton proposed a model of mechanics as a "game" in which objects
affect each others' motion. He proposed that these interactions be
described in terms of "forces". This Newtonian force concept is
quantitatively defined by what we now call Newton's three laws of motion
(N1, N2, N3).
In watching this game of interactions which objects play, our human
penchant for analysis/synthesis asks: "Is what I see describable as a
'zero sum' game? Can we define a mathematical quantity whose value never
changes in time, even though this quantity is a function of the
continually changing states of all of the participating objects?"
We have invented three such mathematical entities which are thusly
"conserved" for a wide scope of ordinary experience: 1) a vector (linear
momentum); 2) a pseudo-vector (angular momentum) and 3) a scalar
(energy). These rest upon quite different theoretical foundations.
The "conservation status" of 1) (linear momentum) follows rigorously from
N1, N2 and N3;
The conservation status of 2) (angular momentum) requires N1, N2, and N3
"in the strong form" eg. central forces;
The conservation status of 3) (energy) requires the specific ad hoc
assumption that IT IS CONSERVED - this assumption is not (for the general
case) reducible to any more primitive property of object interactions
(the semantics of this last statement may spawn debate).
Requirements and definitions must be adjusted as the scope of experience
is enlarged. For example, electromagnetic interactions require that all
three conservation statements include "fields", endowed with appropriate
properties, as participating objects in the zero sum games (the "strong
form" requirement for angular momentum conservation can then be relaxed).
"Clearly there is no room for disagreement about simple mathematics. But
there may be disagreement about the physical significance of it." - J.S.
Bell, Speakable and Unspeakable in Quantum Mechanics.