Expressing a Conserved Quantity in Terms of a Derivative to Create a Variational Principle in Physics

In this note we wish to examine two cases in which variational principles seem to follow from expressing conserved quantities in terms of derivatives. Given conservation in a system, conserved quantities may be expressed in terms of “derivatives” which add to zero creating the variational principle. The first case is Fermat’s Least Time Principle for light which is equivalent to conservation of momentum in the direction of an invariant spatial variable for two ray problems (reflection/refraction, reflection from a moving mirror). In the case of a particle with mass Maurpertuis’ principle for particles with rest mass (1) seems to behave in the same manner. In fact, one may create two different variational principles. The first deals with using d/dx acting on sqrt(xx+yy) to create sin(theta) which appears in (px) = p sin(theta) if theta is measured from the y axis. The second approach is more direct and involves multiply (px) by x i.e. ultimately creating A = -Et+ (px)x + (py)y + (pz)z. In a Maxwell-Boltzmann (MB) gas this same idea seems to appear again. One may use a variational principle i.e. maximize entropy with respect to the constraint Sum over i ei n(i)=E or use time reversal of an elastic collision i.e. e1+e2 = e3+e4 and n(e1)n(e2) = n(e3)n(e4). Taking ln of the latter and equating to the former multiplied by a constant yields the MB distribution. Thus one associates ln(n(ei)) with -ei/T where ei is a conserved quantity. From this conserved quantity one may mathematically create a variational principle i.e. the maximization of entropy. Like with -Et+px one may associate ln(n(ei)) with -ei/T i.e.take conserved quantity and multiply it by the variable n(ei) so that d/d n(ei) returns ei. We argue that ultimately in Fermat’s principle and that of an MB gas, one associates a function with a conserved quantity namely psin(theta)=px (conserved quantity) or ln(n(ei)) = - ei/T where ei is part of a conservation equation. As a result, it may seem as if there are two independent ways to obtain physical solutions for the same problem. For example in (2) it states that Snell’s law may be obtained either using conservation of momentum in the x direction or Fermat’s principle. For an MB gas one may maximize entropy subject to a constraint or use reaction balance. We argue that these approaches are not independent. Given a conserved quantity, one may mathematically create a corresponding variational principle at least in these two cases