First integral and solution for the vertical coordinate

If we multiply the differential equation for the vertical coordinate z t by the vertical velocity z t and integrate, we obtain 

The right-hand sided is a cubic polynomial in w[t] which is shown in Fig. 3, plotted as a function of w for W= —1/2 (corresponding to an initial angular displacement from the downward vertical of 60°) and three values of K. 

Since the first integral must be non-negative, the motion is restricted to the region above the horizontal axis. The three values of w for which the first integral is zero are given symbolically by 

The value of the third zero is never less than 1, so that the motion is restricted to the interval hounded hy the first two zeros. If the two zeros are equal, which occurs 

This cubic can he transformed to canonical form (with the quadratic term missing) hy means of the linear transformation 

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