1 American Institute of Aeronautics and Astronautics Applications of Calculus of Variations to Aircraft and Spacecraft Path Planning Andrea L’Afflitto * and Cornel Sultan † Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University The problem of finding the trajectory for an aerospace vehicle moving between two fixed points is investigated using Calculus of Variations (CV), which can provide necessary and sufficient conditions for trajectories that optimize a cost index. Within the framework of a systematic study aimed at tackling analytically the trajectory generation problem, this paper presents applications of the CV to find trajectories that optimize kinetic energy, energy consumption, and fuel consumption considering several environmental conditions. Nomenclature a = acceleration of the vehicle a c = acceleration of the vehicle due to control forces f = generic function f x = derivative of f with respect to the second component f r = derivative of f with respect to the third component g = gravitational acceleration J = optimization/cost/performance index k 1 , k 2 … = integration constants r = position vector t = time t 1 , t 2 = initial and final time u = control vector v = velocity vector v 1 , v 2 = initial and final velocity x = state vector x 1 , x 2 = initial and final state y, z = generic piecewise smooth (PWS) functions  1 ,  2 … = multipliers μ = gravitational parameter I. Introduction alculus of variations is the branch of mathematics concerned with finding extrema of functionals. Although the first systematic studies in this field are dated back to the eighteenth century, a new interest raised in the last forty years leading to great advances in the area of optimization 1,2 . The problem of finding the optimal trajectory for a spacecraft or aircraft moving between two given fixed positions can be tackled by the theory of the Simplest Problem of Calculus of Variations (SPCV), also known as the Fixed End Problem. With the advent of the computer era several approaches to this problem, which involve selecting arbitrary parameterizations for the trajectories and searching for purely numerical solutions, have been attempted 2 . The main disadvantage of these techniques is that the selected parameterizations often have no relevance to the performance index of interest. Calculus of Variations * Ph.D. Student, Department of Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall, Blacksburg, VA 24061, andrea.lafflitto@vt.edu , AIAA Space Logistics TC Associate Member, AIAA Senior Member. † Assistant Professor, Department of Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall, Blacksburg, VA 24061, csultan@vt.edu , AIAA Senior Member. C