Our first example is the classical case of simple harmonic oscillations, namely a block sliding on a horizontal frictionless surface. The block is tied to a wall with a spring. If the spring is not compressed or stretched too far, the force on the block at a given position \( x \) is $$ F=-kx. $$ The negative sign means that the force acts to restore the object to an equilibrium position. Newton's equation of motion for this idealized system is then $$ m\frac{d^2x}{dt^2}=-kx, $$ or we could rephrase it as $$ \frac{d^2x}{dt^2}=-\frac{k}{m}x=-\omega_0^2x, \tag{43} $$ with the angular frequency \( \omega_0^2=k/m \).

The above differential equation has the advantage that it can be solved analytically with solutions on the form $$ x(t)=Acos(\omega_0t+\nu), $$ where \( A \) is the amplitude and \( \nu \) the phase constant. This provides in turn an important test for the numerical solution and the development of a program for more complicated cases which cannot be solved analytically.