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Where is the ultimate law? Is it in the integral formalism or in the differential formalism?

This is a layman.  I have finished reading  a book entitled "Computing with quantum cats from Colossus to qubits" written by John Gribbin.  In the book, the author introduces a notion of principle of least action.

Referring to my physics dictionary(Bai-fookan, 1996, 3ed ed. in Japanese), it is written that an action integral is an integral of Lagrangian function by time variable from t0 to t1 along some path.  It is written further that the Hamilton's principle dictates that an action integral must be stationary along the real occuring path that the physical system under consideration takes.  This is the Hamilton's principle.  And furthermore, it is written that "actually by choosing appropriate functional form for the Lagrangian function, many laws like the ones of relativistic theory, ones  of electromagnetism and so on as well as the ones of classical mechanism, can all be derived by the same Hamilton's principle."

In the book, John Gribbin wrote that R.P.Feynman proved that by applying the principle of least action quantum mechanically, one can derive the Schroedinger equation.  Oh, I wonder which is the supreme law?  Is it the Lagrangian plus Hamilton's principle or the Schroedinger equation itself?

Thanks for reading, Sirs

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