## A method for computing extremal maximum-range thrust-limited rocket trajectories with application to lunar transport |

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a,tf amounts of fuel angular momentum APPENDIX Athena control and trajectories differential equations dynamic equations effective computational equations 14 Ernest extremal maximum range Figure final to initial fixed range fs(t ft/sec function gravitational acceleration gravitational constant gravitational field identity matrix initial lunar thrust-weight initial mass ratios initial thrust-weight ratio initial-value algorithm Keplerian gravity field Landing phase Langley Research Center launch and soft Launch phase lunar surface lunar thrust-weight ratio m(tf m/sec matrix matrix transpose maximum range trajectories minimal minimum fuel extremals MINIMUM FUEL PROBLEMS NASA NASA TN necessary conditions Needham obtain solutions Optimal control Optimal control theory Performing Organization points of launch polar coordinate Pontryagin maximum principle presented r(tf radial velocity range accrued ratio of 5.0 rmax rocket vehicle soft landing surface range technique thrust angle THRUST-LIMITED ROCKET TRAJECTORIES Tmax two-point boundary-value problem value of mf/m0 Variation of altitude vector velocity vertical direction