Thanks Daniel, that's a lot of good material to think about!

The view of paths of least action as geodesics has a ring of elegance.

I'm also interested in your reference to Dan Piponi's explanation (previous message), because it is self-contained and sounds basic.

He uses motion through discrete time as model to explain the meaning of action. Okay. Then differentiation gets replaced with the matrix for the finite difference operator, $D$.

Then he says:

> Notice how apart from the edges, $D$ is minus its own transpose. Ultimately, _this_ is the minus sign that appears in the definition of the Lagrangian.

It sounds promising, but can someone explain what "apart from the edges, $D$ is minus its own transpose" means?

To me, $D$ doesn't look at all like its own transpose, even "in the middle" of the matrix:

* The diagonals of $D$ and of $D^T$ are identical -- they are filled with -1 values.

* On the off-diagonal, where $D$ has a 1, $D^T$ has a zero.