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The Calculus of Variations and Hamilton's Principle from the Top Down

This installment of “It’s Obvious. Not!” looks at: Book: “Classical Dynamics of Particles and Systems” Edition: third Authors: Jerry B. Marion and Stephen T. Thornton Publisher: Harcourt Brace Jovanovich Page: 172-177 Chapter Five in the third edition is titled "The Calculus of Variations" The book does a great job of giving a very detailed bottom-up derivation of Euler's equation and a second form of Euler's equation. I had a much easier time with the material once I figured out that Euler's equation was actually the goal of the derivation and how Euler's equation is used. Since the top-down view made things simple for me, I decided to post it here for other top-down thinkers. Chapter five is simply building a set of tools to be used in chapter six with regard to Hamilton's Principle and Lagrangian mechanics. So, maybe the first question should be, 'Why are Hamilton's Principle and Langrangian mechanics important?' Newtonian Mechanics esse...