Classical Mechanics:
Energy Conservation
"The change of kinetic energy ΔK of a particle when moving from point a to point b via a path s under the influence of a force F(s) is defined by the work done by this force and hence by the path integral:
(1) ΔK(a,b) =
a∫
b[bold]ds F(s)[/bold]
( where the bold letters shall denote vector quantities.)
Newton's second law
F=ma results in the usual form for the kinetic energy:
ΔK(a,b) = 1/2.m.v
b^2 - 1/2.m.v
a^2
One now defines a
potential energy change ΔV(a,b) which is the negative of this change in kinetic energy, i.e.
(2) ΔV(a,b) = - ΔK(a,b)
Which means that one can define a
total energy ε for which the corresponding change
(3) Δε(a,b) = ΔK(a,b) + ΔV(a,b) = 0
"
"
By virtue of having introduced the
potential energy, energy conservation holds therefore by definition.
Of course this only makes sense if we have a
closed mechanical system (if no overall work is done on the system as a whole. In this case it is said that we have a
'conservative' force field under which influence the particles move).
The crucial point is that through
Eq.(1) the energy conservation law
(Eq.(3)) inevitably involves the presence of a corresponding
conservative force field which is responsible for changing the state of motion of the particle.
This circumstance is not immediately evident anymore in the mathematical Lagrange or Hamilton formalisms of classical mechanics, but it is nevertheless there as both the Lagrangian and the Hamiltonian function contain the potential energy V which in turn can not be defined without a
conservative force field.
It is therefore completely unjustified to assume that a relationship like Eq.(3) (or any formalism equivalent to it) could be a general principle in physics that would hold outside classical Newtonion physics
e.g. for light (in fact, for light it contradicts Maxwell's equations as the curl of a conservative force field must be zero, however, according to Faraday's law curl(E)=-dB/dt).
Noether's Theorem or other similar theoretical arguments that purport to derive a general law of energy conservation in physics are therefore flawed as they are implicitly based on assumptions that limit their validity to classical mechanics. In fact, by the very definitions of Eqs.(2) and (3), the notion of energy itself does not make any sense for anything but classical mechanics."