Optimal (minimum-fuel) power-limited spacecraft trajectories are considered. The equations of motion and the necessary conditions for an optimal trajectory are combined into a single fourth-order differential equation. The component equations are derived in inertial cartesian coordinates for an arbitrary, time-invariant gravitational potential function. The associated variational equations are also derived. In addition, expressions for several integrals of the motion are provided. These integrals provide a check on the accuracy of the numerical integration that is required to solve the boundary value problem. Illustrative examples discussed are the two-body problem and the gravitational field of an oblate spheroid. A formulation of the optimal trajectory problem in a rotating coordinate frame is also derived. Illustrative examples discussed are the restricted three-body problem and the Hill-Clohessy-Wiltshire linear model. Numerical examples for the two-body problem are presented.