Monthly Archives: March 2013

THE PRINCIPLE OF LEAST ACTION

First Principle of the Natural World

 

It’s with a certain sense of awe that I introduce a new page on this website, to be devoted to the Principle of Least Action. I’ve written about this principle, on this website (see the article here) and elsewhere, in various ways and contexts, but it appears now for the first time as the centerpiece around which writings on this theme will be gathered.

The principle itself, though simple, requires careful statement, before we do that, however, we must pause to take the measure of situation. Even as leading physicists and mathematicians have embraced Least Action as the single moving principle of the entire natural world, most others, even at the highest levels of education and professionalism in fields other than mathematics or the physical sciences, regard it as of no interest to themselves, or more likely, have never heard of it all.

The issue is acute: our old conceptions speak of mechanism, with even the most subtle of natural bodies composed ultimately of inert parts moved by impressed forces, according to equations knowable only to experts. What a different picture Least Action paints! Wholeness is, in truth primary, with causality flowing from whole to part, not from part to whole. Nature is everywhere self-moving, and throughout, life is real. In short, the era of Newton is behind us, and once again, nature lives! We cannot know yet, what the consequences of embracing this truth might be: but the time to begin exploring this question is surely now, before we have altogether destroyed this planet–the living system of which we are all organic parts, and on which our lives depend. It’s the mission of this web page to explore the concept of least action, and some of the many ways it may affect our institutions and or lives. Such a trajectory of thought, which I would call dialectical, has been the theme of the ongoing blog commentary on  Newton/Maxwwell/Marx. Between Newton and Least Action, we may be living in the acute stress-field of a dialectical advance of human understanding.

As a firm mathematical foundation for further discussions of Least Action, here is an elegant sequence of steps leading from Newton to Least Action, following closely the argument given by Cornelius Lanczos in his Variational Principles of Mechanics. I have distinguished seven steps in this argument, adding a few notes by way of commentary.

Note that this does not propose to prove the truth of Least Action (though some rugged Newtonians such as Kelvin might take it in this sense!). Rather it demonstrates the formal equivalence of Newton’s laws and the principle of least action. Lanczos points out that the argument is reversible: can be traversed in either direction. The two equations are formally equivalent, but speak of completely different worlds.

We should note that they are not of equal explanatory power. Least Action serves as foundation for General Relativity and Quantum Mechanics, areas in which Newton is powerless.