Review: The Variational Principles of Mechanics

35 points by telotortium on 2024-05-14 | 11 comments

Automated Summary

This article reviews 'The Variational Principles of Mechanics' by Cornelius Lanczos and uses the opportunity to discuss the history of scientific thought, particularly focusing on the concept of causality. The review touches upon Aristotelian physics, which posits that final causes (the ends or purposes of things) are more fundamental than efficient causes (the immediate sources of motion), and contrasts it with Newtonian mechanics, which focuses on efficient causes and allows for quantitative predictions. The review highlights the importance of Lagrangian mechanics, the subject of the reviewed book and Lanczos' own favorite physics textbook, as it proves that final causes play a crucial role in the universe's functioning, a point that Einstein, a friend of Lanczos, appreciated. The review delves into the brachistochrone problem as an example of the Euler-Lagrange equations and concludes by reflecting on the philosophical implications of the Lagrangian approach in understanding the universe.

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nathan_compton on 2024-05-15

The only really good way of understanding the variational principal in my experience as a physicist who has chewed on it informally since getting out of grad school is to recognize that energy, potential or kinetic, comes after the variational principal, not before it. In school we are taught to write down the kinetic and potential energies and then to form a lagrangian and turn the crank, but it is in fact the lagrangian formalism which determines which expression we write down for both terms. All the physics before, including the characterization of kinetic and potential energy as concepts, is fumbling towards that idea.

Really, if you look at Hamiltonian Mechanics this is more clear, since most of the ideas in Hamiltonian mechanics flow from the basic idea that p generates q AND either that paths in state space don't cross and/or that time evolution is unitary (depending on whether you want classical or quantum mechanics to shake out). These are more or less unavoidable assumptions which we impose or our efforts to understand reality and from these basic ideas we simply fiddle around with expressions until we find the mechanics that match our observations and from which we can sometimes identify objects which behave like the potential and kinetic energies we learn about in grade school.

Another way to appreciate this is the difficulty of coming up with a good totally fundamental definition of energy independent of a physical system. Energy is a handy way of thinking about how things will move. That is all.

mjburgess on 2024-05-15

Energy is an accounting system for tracking motion over time, to ensure past/future motion lines up.

So by imposing dTotalEnergy/dt = 0, you do indeed get the apparent teleological conspiracy of nature, to ensure that the end 'can be no greater than the beginning'.

Perhaps the rehabilitation of these ancient greek causal maxims lies in seeing them as attempts to phrase principles of conservation.

Since Aristotle would have only thought that these principles apply to the essential basic constitutive properties of nature, later found to be mass/charge/etc., these do succeed.

mjburgess on 2024-05-15

It's important to contextualize Aristotle, farther of science, against Plato and Protagoras, fathers of philosophy. All of the worst impulses of a priori theorizing were in Plato and Protagoras -- from pure blind objectivity to pure blind subjectivity. Aristotle was working hard to defend a study of the world from both of them, delivering ingenuous arguments for realism which are no worse than any given since.

His philosophy is more-or-less a modern scientific realism: the world exists, it has essential properties, knowing the world is knowing these properties, these must be discovered, in general we do not know the accidental from the essential, etc.

GranularRecipe on 2024-05-15

Arguing against Aristotelian physics in 20th / 21st century is like pounding a rotten strawman. Does anyone know a modern (influential) Aristotelian philosopher who holds views the writer is criticising?

Jensson on 2024-05-15

People who didn't study physics in school tend to hold Aristotelian views or similar. Getting people to stop having Aristotelian views is one of the biggest and hardest things you do in early physics/science classes for kids, and it fails for a large number of those kids so they keep them their entire lives as they don't study more.

You wont find them among stem people like most people here on HN, but there are a lot of them out there, even in high end professions like medicine etc.

omnicognate on 2024-05-15

Sounds like you've only read the opening of the article. You could read the last sentence to see that what lies in between isn't "arguing against Aristotelian physics". (It isn't a book review either, contrary to the title.)

GranularRecipe on 2024-05-15

Yes, I was referring to the opening of the article which ridicules (trolls) Aristotelian physics. No, I did not only read the beginning. I also skimmed over the rest.

omnicognate on 2024-05-15

Then you'll be aware that he goes on to defend an Aristotelian, teleological view (which he describes, in accordance with your comment, as only nowadays held by a few "cranks"), so I'm not sure what prompted the question?

GranularRecipe on 2024-05-15

Let me expound my arguments more verbosely:

The author set himself an ambitious goal of webbing an intricate story about Western physics from Aristotle to Lagrange. Good story-telling starts with something familiar, seemingly truthful to the reader, that is yet (partially) wrong or incomplete (unknown to the reader). And through the story, the reader gains wisdom by being surprised of something new that (partially) falsifies their current body of knowledge.

The author starts with something that the modern-day reader knows to be wrong, and ends with a dubious conclusion (that somehow the variational principle validates Aristotelian teleology).

A good story would start with the Newtonian mechanics, the prevailing view among those who did not study physics at the university. The author does not need to heed my amateur advice: He can attack strawmen, i.e. valid yet superfluous, off-target criticisms.

octed on 2024-05-15

HN has been surprisingly good today. Seen a couple of very high quality posts (at leasts posts that are to my liking), this one included. I thank OP for posting this.

tech_ken on 2024-05-15

> Anyway, I wanted to learn about this stuff, and I’m a firm believer that you can’t learn anything in physics without getting your hands dirty and solving some integrals.

If there's a better reason to remain entirely ignorant of all physics I can't think of it