This chapter presents several of the most important concepts from analytical dynamics. We derive Lagrangeâ€™s equation and how it can be used for the derivation of governing equations of motion. It is, especially, useful for the derivation of the equations of motion for systems, discrete or continuous, with more than one degree-of-freedom, where the Newtonian free body diagrams become more difficult to apply. We also derive Hamiltonâ€™s principle, an integral energy formulation, also applicable to both discrete and continuous systems, and see how it is related to Lagrangeâ€™s equation. Hamiltonâ€™s principleÂ is, especially, relevant to the work in Chaps. 4 and 5.