This course is part 2 of the specialization Advanced Spacecraft Dynamics and Control. It assumes you have a strong foundation in spacecraft dynamics and control, including particle dynamics, rotating frame, rigid body kinematics and kinetics. The focus of the course is to understand key analytical mechanics methodologies to develop equations of motion in an algebraically efficient manner. The course starts by first developing D’Alembert’s principle and how the associated virtual work and virtual displacement concepts allows us to ignore non-working force terms. Unconstrained systems and holonomic constrains are investigated. Next Kane's equations and the virtual power form of D'Alembert's equations are briefly reviewed for particles.

Next Lagrange’s equations are developed which still assume a finite set of generalized coordinates, but can be applied to multiple rigid bodies as well. Lagrange multipliers are employed to apply Pfaffian constraints. Finally, Hamilton’s extended principle is developed to allow us to consider a dynamical system with flexible components. Here there are an infinite number of degrees of freedom. The course focuses on how to develop spacecraft related partial differential equations, but does not study numerically solving them. The course ends comparing the presented assumed mode methods to classical final element solutions. The material covered is taking from the book "Analytical Mechanics of Space Systems" available at https://arc.aiaa.org/doi/book/10.2514/4.105210.