The local time is 3 o'clock

There are some things in physics that we are so used to taking for granted that we don't even think about them. Consider, for example, the statement that classical physics is local - physical systems involve interactions that occur at the same points in space and at the same times. Does this have to be so? Let's look at a simple spring to see what happens to physics if we make things a little non-local.

The equation of motion for the position of a mass mm at the end of a spring with spring constant kk is

d2x(t)dt2=kmx(t)\frac{d^2 x(t)}{dt^2}=-\frac{k}{m} x(t)

if the other end of the spring is fixed in space. Consider solutions to this equation of the form x=eiEtx = e^{iEt} where EE is a constant (which is a priori possibly complex). Let NN be the number of possible values for EE that satisfy this equation.

We'll now make the spring equation slightly non-local in time. We do this by letting the acceleration of the mass be determined by not just x(t)x(t), but by a combination of the position at two different times, x(tΔt)x(t-\Delta t) and x(t+Δt)x(t+\Delta t), where Δt\Delta t is a constant. The new equation of motion is

d2x(t)dt2=k2m(x(t+Δt)+x(tΔt))\frac{d^2 x(t)}{dt^2}=-\frac{k}{2m}( x(t+\Delta t) + x(t-\Delta t))

so that as Δt0\Delta t \rightarrow 0 we return to the usual equation for a spring. This second equation with Δt0\Delta t \neq 0 also has solutions of the form x=eiEtx=e^{i E t} for certain values of EE. Let MM be the number of possible values of EE that satisfy this second equation of motion. What is N/MN/M?

Hint: don't make any extra assumptions about EE than you absolutely need to.


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