# Analytical Mechanics for Spacecraft Dynamics

This course, part of the Advanced Spacecraft Dynamics and Control specialization, focuses on developing equations of motion for spacecraft dynamics using key analytical mechanics methodologies. It assumes a strong foundation in spacecraft dynamics and control, including particle dynamics, rotating frame, and rigid body kinematics and kinetics.

The course begins by exploring D’Alembert’s principle and the associated virtual work and virtual displacement concepts, allowing the neglect of non-working force terms. It then investigates unconstrained systems and holonomic constrains. The development of Lagrange’s equations, capable of being applied to multiple rigid bodies, is covered as well, along with the use of Lagrange multipliers to apply Pfaffian constraints. Finally, the course delves into Hamilton’s extended principle, allowing consideration of dynamical systems with flexible components, and the development of spacecraft-related partial differential equations.

By the end of the course, learners will be able to use virtual work methods to develop equations of motion for mechanical systems, understand how to use Lagrange multipliers to study constrained dynamical systems, and derive the equations of motion of a spacecraft with flexible sub-components.

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##### Course Modules

This course comprises modules on Generalized Methods of Analytical Mechanics, Energy Based Equations of Motion, and Variational Methods in Analytical Dynamics.

#### Generalized Methods of Analytical Mechanics

Welcome to the Course! This module introduces the course and offers a preview of what learners can expect. It then delves into the motivation for analytical mechanics and explores virtual displacements, virtual work, D’Alembert’s principle, and examples demonstrating these concepts. The module also covers torques acting on a rigid body, holonomic constraints, Pfaffian constraints, and general constrained optimization.

#### Energy Based Equations of Motion

This module focuses on energy-based equations of motion, starting with the derivation of basic Lagrange's equations and reviewing Lagrangian dynamics. It then explores Lagrange's equations with conservative forces, constrained Lagrange's equations, compact matrix form of Lagrange's equations, cyclic coordinates, and Routhian reduction. The development and examples of Boltzmann Hamel equations are also covered.

#### Variational Methods in Analytical Dynamics

The last module delves into variational methods in analytical dynamics, beginning with the motivation for these methods and variational calculus. It then explores Hamilton's principle function, principles, Hamilton's law of varying action, and reviews of Hamilton's extended principle. The module also covers hybrid coordinate definitions, reduction to a finite set of coordinates, and assumed modes methods, with examples and quizzes provided throughout.

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