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$\DeclareMathOperator\SO{SO}$A priori: I apologize if this isn't up to Mathoverflow standards, I've had very little luck getting questions on this subject answered elsewhere. I'm looking for a physics ...

Let's start with a definition: Definition: A scalar k-th order differential equation on a smooth manifold $M$, is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $ for $\...