In 50 years of examining books of Physics, I've found none that provided such insight into Mechanics and its derivation by historic creators as Cornelius Lanczos' book, The Variational Principles of Mechanics, first published in 1949.

On p. 88, Lanczos takes up "d'Alembert's Principle", which reduces the physics of Dynamics to that of Statics.

Says Lanczos: "With a stroke of genius the eminent French nathematician and philosopher d'Alembert (1717-1783) succeeded in extending the applicability of the principle of virtual work from statics to dynamics. The simple but far-reaching idea of d'Alembert can be approached as follows. We start with the fundamental law of motion: 'mass times acceleration equal moving force' [vectors underlined]:

			mA = F (1)

and rewrite this equation in the form

			F _ mA = 0. (2)

We now define a vector I by the equation

			I = -mA. (3)

This vector I can be considered as a force, created by the motion. We call it the 'force of inertia'. With this concept the equation of Newton can be formulated as follows:

			F + I = 0. (4)

Apparently nothing is gained, since the intermediate step (3) gives merely a new name to the negative product of mass times acceleration. It is exactly this triviality which makes d'Alembert's principle such an ingenious invention and at the same time so open to distortion and misunderstanding."

Lanczos then goes on to explain this last comment. Here, I wish to use the above for my own derivation (below), which is an antitonic process as described in "Mother of All Processes?", this Website.

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MY DERIVATION OF d'ALEMBERT'S PRINCIPLE & NEWTON'S LAW Consider the antitone: J * K = CONSTANT. (5)

Take the logarithm of (5): log J + log K = 0. (6)

Relabel: F = log J, I = log K -> F + I = 0, d'Alembert's Principle. (7)

Relabel: I = ^-A. (8)

From (7), (8), F - mA = 0, or F = mA (9), Newton's Law (implicitly antitonic).

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HAMILTON'S PRINCIPLE IS ANTITONIC AND d'ALEMBERTIAN Hamilton's Principle has the form: T - V for kinetic energy, T, and potential energy, ^-V.

But the latter equals work: ^-V = W.

Hence, Hamilton's Principle takes the (antitonic, d'Alembertian) form: T + W.

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It follows that the Eulerian and Lagrangian have antitonic form.