The laws of energy and momentum conservation are probably the most frequently quoted laws in physics, but also the most frequently abused ones. This is because the notions of energy and momentum are only strictly defined in classical mechanics but not if one considers non-mechanical physical entities like light for instance. Nevertheless, most physicist seem take it for granted that in the latter case the same physical principles apply.
Let's have a closer look at the definition of energy and momentum and the corresponding conservation laws in classical mechanics:
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∫bds F(s)
where the bold letters shall denote vector quantities.
(Note: it is in fact straightforward to show from the above definition that Newton's second law F
results in the usual form for the kinetic energy ΔK(a,b) = 1/2.
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 i.e. 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. Evidence for this is for instance the flawed particle model of light (see my page about the Photoeffect
) and also discrepancies when trying to theoretically predict the light output of radiating gases (see https://www.plasmaphysics.org.uk/research/#A5
The momentum (change) of a particle can be defined by the integral of the force acting on it over time (which is also called 'impulse'), i.e.
(4) Δp(ta,tb) = ta∫tbdt F(t)
which using F
immediately integrates to
(5) Δp(ta,tb) = mv(tb) - mv(ta) .
Now in a closed system the sum of all internal forces must be zero, i.e. for a two particle system one has Newtons' third law
(6) F1,2= - F2,1 .
With Eqs.(4) and (5) this means however that in a two body system
(7) Δp1(ta,tb) = -Δp2(ta,tb)
(8) m1[v1(tb) - v1(ta)] - m2[v2(tb) - v2(ta)] = 0 ,
which is the law of momentum conservation.
However, again it is obvious that this conclusion does only hold for particles in their mutual static force fields as otherwise Eqs.(4) and (6) could not be applied. It is therefore theoretically unjustified and inconsistent to generalize (as it is often done) the momentum conservation law to other non-classical phenomena in physics like the interaction of light with atoms (see my page about the Photoeffect