An Invariant Formulation for the Minimum Induced Drag Conditions of Non-planar Wing Systems Demasi Luciano * , Dipace Antonio † , Monegato Giovanni ‡ , and Cavallaro Rauno § Under the hypotheses of linear potential flow and rigid wake aligned with the freestream, a configuration-invariant analytical formulation for the induced drag minimization of single- wing non-planar systems is presented. Following a variational approach, the resulting Euler-Lagrange integral equation in the unknown circulation distribution is obtained. The kernel presents a singularity of the first order and an efficient computational method, ideal for the early conceptual phases of the design, is proposed. Munk’s theorem on the normalwash and its relation with the geometry of the wing under optimal conditions is naturally obtained with the present method. Moreover, Munk’s con- stant of proportionality, not provided in his original work, is demonstrated to be the ratio between the freestream velocity and the optimal aerodynamic efficiency. The Augmented Munk’s Minimum Induced Drag Theorem is then formulated. Additional induced drag theorems are demonstrated following the derivations of this invariant procedure. Several non-planar wing systems are proposed and analyzed and the optimal induced drag and circulation are provided. The conjecture regarding the equality of the optimum induced drag of a quasi-closed C-wing with the induced drag of the corresponding closed system is also verified for several configurations including the boxwing. Nomenclature V ∞ ,ρ ∞ Freestream velocity and density L Lift D ind Induced drag Γ Circulation γ (η)dη Vortex filament ℓ Lifting line η,ξ Arc length abscissas a semi-length of the lifting line u,u v Parameters used to represent the lifting line y,z Cartesian coordinates identifying a point on ℓ v n Normalwash Y Kernel of the integral equation λ Lagrange multiplier E Aerodynamic efficiency ε Optimal aerodynamic efficiency ratio α Incidence b w Semi-wingspan * Associate Professor, Department of Aerospace Engineering, San Diego State University, San Diego (USA), Member AIAA, Email: ldemasi@mail.sdsu.edu † PhD student, Computational Science Research Center, San Diego State University, PhD student at the Dipartimento di Ingegneria Aerospaziale, Universit`a di Pisa. Member AIAA. ‡ Professor, Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino (Italy) § PhD Candidate, Department of Aerospace Engineering, San Diego State University and Department of Structural Engineering, University of California San Diego, San Diego (USA), Member AIAA 1 of 33 American Institute of Aeronautics and Astronautics 52nd Aerospace Sciences Meeting 13-17 January 2014, National Harbor, Maryland AIAA 2014-0901 Copyright © 2014 by Luciano Demasi, Antonio Dipace, Giovanni Monegato, and Rauno Cavallaro. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. AIAA SciTech