Hypothesis: ---------- | ALL LAWS OF NATURE ARE ONLY APPROXIMATELY TRUE Just take it or leave it. I'm not going to defend this as a new kind of dogma. Instead, we are going to explore some consequences of the proposition, in a very specific mathematical sense. And that's it. As an application of the above Hypothesis, consider the classical _ideal_ harmonic oscillator, which is described analytically by the following differential equation and boundary conditions: d^2x/dt^2 + oo^2.x = 0 ; x(0) = 0 ; dx/dt(0) = oo ; x = x(t) x = space coordinate, t = time, oo = (angular) frequency. The "exact", but I would rather say ANALYTICAL, solution of the above equation is well known. And we call it the _ideal_ solution: x(t) = sin(oo.t) But according to our Hypothesis, the _true_ equations are not "exact"; they are _approximations_ of the above ideal differential equation: x(t + dt) = x(t) + dt.x'(t) + dt^2/2.x''(t) + dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (1) x(t - dt) = x(t) - dt.x'(t) + dt^2/2.x''(t) - dt^3/6.x'''(t) + dt^4/24.x''''(t) + .. (2) - 2.x(t) = - 2.x(t) (3) ------------------------------------------------------------------ + x(t + dt) - 2.x(t) + x(t - dt) = dt^2.x''(t) + dt^4/12.x''''(t) .. So the TRUE equation of motion is, with dt = finite and uncertain: x(t + dt) - 2.x(t) + x(t - dt) ------------------------------ + oo^2.x(t) = dt^2 = h.o.t. + dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 + oo^2.x(t) = 0 Where h.o.t. = higher order terms, to be neglected. The equations of motion are solved numerically as follows. Start with: x(0) = 0 ; (x(dt) - x(0))/dt = oo ==> x(dt) = x(0) + oo.dt x(t + dt) - 2.x(t) + x(t - dt) ------------------------------ + oo^2.x(t) = 0 ==> dt^2 x(t + dt) = 2.x(t) - x(t - dt) - (dt.oo)^2.x(t) ==> x(0.dt) = 0 x(1.dt) = x(0) + oo.dt x(2.dt) = 2.x(1.dt) - x(0.dt) - (dt.oo)^2 x(1.dt) x(3.dt) = 2.x(2.dt) - x(1.dt) - (dt.oo)^2 x(2.dt) x(4.dt) = 2.x(3.dt) - x(2.dt) - (dt.oo)^2 x(3.dt) ......... Etcetera And similar equations if the time intervals (dt) have unequal sizes. In our numerical experiments, we have chosen oo = 1 and dt = 0.1 . So far so good. But there is an analytical approach as well. Solve: dt^2/12.d^4x(t)/dx^4 + d^2x(t)/dt^2 + oo^2.x(t) = 0 Characteristic equation: dt^2/12.L^4 + L^2 + oo^2 = 0 Substitute L^2 = M ==> dt^2/12.M^2 + M + oo^2 = 0 A quadratic equation. The solution can be written as: M = oo^2 / [ - 1/2 +/- 1/2.sqrt(1 - {dt.oo}^2/3) ] ==> L = +/- i.oo / sqrt(1/2 + 1/2*sqrt(1 - sqr(dt*oo)/3)) , +/- i.oo / sqrt(1/2 - 1/2*sqrt(1 - sqr(dt*oo)/3)) The latter frequencies are of order the discretization (dt) and hence will be neglected. The former frequencies imply a small correction on the (no longer) "exact" frequency (oo). Thus the "true" solution is a sine with frequency oo' = oo / sqrt(1/2 + 1/2*sqrt(1 - sqr(dt*oo)/3)) Meaning that, due to the _discrete_ nature of the differentials (dt) , it oscillates somewhat faster: as x(t) = sin(oo'.t) with oo' > oo . After a time lapse t = N.2.pi/oo the analytical solution x(t) = 0 . After t = (N.2.pi + pi/2)/oo' the true numerical solution x(t) = 1 . At the same time it holds that N.2.pi/oo = (N.2.pi + pi/2)/oo' ==> N.oo' = (N + 1/4).oo' ==> N = (1/4)/(oo'/oo - 1) So we can calculate the first moment at which the two curves differ a magnitude = 1 . But different time steps cause different divergences from the _ideal_ x(t) = sin(oo.t) curve. Therefore, in fact, we have a whole statistical _ensemble_ of true x(t) = sin(oo'.t) curves and their deviations from the ideal curve become larger as time proceeds. Needless to say that our numerical experiments are a confirmation of the above theory. We find that the ideal and true curves differ by an amount 1 in x after 37636 time steps dt = 0.1 . But, in reality, (dt) is supposed to be a stochastic variable with some mean and a spread, meaning that there is a bundle of solutions instead of an exact one. Now think about the "grand" consequences. Does the above imply that: TIME IS IRREVERSIBLE ? Han de Bruijn