Explain the agreement between quadratic and linearized lift models

Determine why the numerical optimum projection angles obtained from the quadratic lift law agree remarkably well with the exact optimum projection angle derived from the linearized equations of motion when drag is absent.

Background

The paper compares two descriptions of projectile motion in the absence of drag: an exact solution of the linearized equations, in which the lift parameter is treated as constant, and numerical solutions of the full equations with a quadratic lift law whose magnitude depends on the projectile speed. Despite their different velocity dependences, the numerical results from the quadratic model closely match the exact linearized solution across the range shown in Figure 1.

The paper reports this agreement as an unresolved issue rather than deriving an explanation for it. Understanding the source and validity of the agreement would clarify when the simpler linearized model can reliably represent the optimum projection angle for projectiles subject to lift.