Copasetic Flow

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 Pages: 176-178 (Chapter 5, Example 5.2) Reading physics books, it often occurs to me that the authors must be aware of some patterns or 'rules of thumb' that the reader may not be privy to. Today's post expands a very truncated example from Marion and Thornton and hopefully clarifies it. This post also poses several questions in search of those patterns and rules mentioned above. After explaining the calculus of variations and the importance of Euler's equation Marion and Thornton follow up with a concrete example: the brachistochrone. The problem of the brachistochrone is to determine the path for a particle to move from point A to B under the influence of a constant force, (gravity for example), in the least amount of time. The 'least amount of time' phra...

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...