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That might be a good follow-up/second question, just to keep this one focused. What you learn here might inform how you would want to tackle the prediction question
Hi everybody -- I've cleared out a few comments here today. Please use comments to seek clarification or suggest improvements to the question. If it does something else like provide an answer, post it as a proper answer below. Thanks!
There seems to be some confusion out there about a Lagrangian description of fluid dynamics, which is what I am talking about here (following a fluid particle's trajectory), and the Lagrangian of a fluid particle in the mechanical sense. What is written here in this answer, and in the question, is about going between a fixed-in-space view (Eulerian) and a following-fluid-particle view (Lagrangian) -- and again, this is not the same thing as the "Lagrangian" of some dynamical system. It's not my fault people re-use words!
