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Review: The Variational Principles of Mechanics

The Variational Principles of Mechanics, Cornelius Lanczos (University of Toronto Press, 1949). While sailing a little boat the other day, I thought of a new way to troll the Aristotelians. I love it when my hobbies converge like that, and if the second one sounds a little mean-spirited, well, remember that they deserve it.

Or alternatively you may have heard some of the dumb popularizations of quantum mechanics that refer to a particle “trying out all possible paths simultaneously.” It took me an embarrassingly long time to figure out that this is actually just philosophically-illiterate science journalist-speak for precisely this sort of extremal principle or final cause. There is a tremendous treasure of philosophical meaning behind the great theories of Euler and Lagrange, and of Hamilton and Jacobi, which is completely smothered in a purely formalistic treatment, although it cannot fail to be a source of the greatest intellectual enjoyment to every mathematically-minded person. It’s true that any laws of physics expressed in terms of finding the stationary point of an action principle will necessarily give rise to equations of motion that are coördinate invariant, deterministic, reversible, have the proper symmetries, all that good stuff.

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