# The local time is 3 o'clock

**Classical Mechanics**Level 5

The equation of motion for the position of a mass \(m\) at the end of a spring with spring constant \(k\) is

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

if the other end of the spring is fixed in space. Consider solutions to this equation of the form \(x = e^{iEt}\) where \(E\) is a constant (which is a priori possibly complex). Let \(N\) be the number of possible values for \(E\) 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 \(x(t)\), but by a combination of the position at two different times, \(x(t-\Delta t)\) and \(x(t+\Delta t)\), where \(\Delta t\) is a constant. The new equation of motion is

\(\frac{d^2 x(t)}{dt^2}=-\frac{k}{2m}( x(t+\Delta t) + x(t-\Delta t))\)

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

Hint: don't make any extra assumptions about \(E\) than you absolutely need to.