Application of the Weierstrass condition in rocket trajectory optimization

The Weierstrass condition is applied to the variable-thrust subarc in an optimal chemical or electric rocket trajectory. For the chemical rocket, there are two controls, one singular and one nonsingular. It is shown that the Weierstrass condition cannot be applied to the singular control. In its place, the generalized Legendre-Clebsch condition proves that the particular variable-thrust subarc known as the Lawden spiral is not minimizing. For an electric rocket operating at maximum power, both controls are nonsingular. The application of the Weierstrass condition is straightforward and shows that the variable-thrust subarc is minimizing.