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.
I think it’s worth noting that, like with a lot of things, calculus of variations requires defining a set of boundary conditions on which to derive the Euler Lagrange equations. Donald E Kirk’s Optimal Control theory has the best exploration of the consequences of this for various derived optimization problems. But the Newtonian style formulation is a differential equation, where the boundary conditions are NOT inherent in the expression of the equations of motion, they are ad hoc and specific to the particular problem.
What I’m trying to say is that the teleological approach of Lagrangian mechanics already assumes a fixed beginning and end, where the Newtonian formulation does not. Lots of physical laws do this, Maxwell’s equations have an integral and a differential form, where the integral equations (a global, teleological form) have the boundary conditions embedded in them, and the differential ones (a local, more obviously causal form) do not. So it is circular reasoning to look at the Lagrangian, global formulation and discover that this implies a fixed endpoint, this is in fact an assumption of the method, not a consequence.
*which kinda stinks, I don't have enough income to support all the substacks I read, and I hate subscribing 'cause it means more stuff in my email inbox. Idea for substack, charge me ~$100 a year and I get to 'support' ~10 substacks, and also not get any emails, maybe one summary email/ week
Those journalists you criticize actually have it right. The universe does try all paths for the particles. Are you familiar with Feynman's reformulation of Quantum physics? The extreme classical paths (variational minimums or maximuns) appear from interference from *all possible paths.
Perhaps you have seen the QFT explanation for why there is no teleology, and you mention it in the footnote, but I didn't see a response. Is your point that there are situations where the optimization is not explained by cancellation?
Every possible path has an amplitude(a unit complex number). But the amplitude rotates fast as a function of the path, so that there is cancellation except at the critical path where the derivative of the action is 0, so nearby paths reinforce rather than cancel the amplitude. Since probability of the path is proportional to |amplitude|^2, the paths around the critical paths are most likely to be traversed.
I didn't expect to like this review so much! Thanks a lot for the link to Ted Chiang's story. I was already a fan of "Arrival", but didn't know it was based on a short story. While it's a pity that Villeneuve (probably had to) left out the Lagrangian part, I do think the movie has gained immensely in poignancy and power compared to the story, and that is a rare achievement.
One of my persistent annoyances is that universities teach two kinds of physics courses - the Newtonian kind, that they teach to engineering students and other non-majors, and the Lagrangian kind, that they teach to physics students.
I think it’s worth noting that, like with a lot of things, calculus of variations requires defining a set of boundary conditions on which to derive the Euler Lagrange equations. Donald E Kirk’s Optimal Control theory has the best exploration of the consequences of this for various derived optimization problems. But the Newtonian style formulation is a differential equation, where the boundary conditions are NOT inherent in the expression of the equations of motion, they are ad hoc and specific to the particular problem.
What I’m trying to say is that the teleological approach of Lagrangian mechanics already assumes a fixed beginning and end, where the Newtonian formulation does not. Lots of physical laws do this, Maxwell’s equations have an integral and a differential form, where the integral equations (a global, teleological form) have the boundary conditions embedded in them, and the differential ones (a local, more obviously causal form) do not. So it is circular reasoning to look at the Lagrangian, global formulation and discover that this implies a fixed endpoint, this is in fact an assumption of the method, not a consequence.
OMG Conan and physics. I'm now an unpaid subscriber*, so a shout out to the least action principle in the Feynman lectures. https://www.feynmanlectures.caltech.edu/II_19.html
*which kinda stinks, I don't have enough income to support all the substacks I read, and I hate subscribing 'cause it means more stuff in my email inbox. Idea for substack, charge me ~$100 a year and I get to 'support' ~10 substacks, and also not get any emails, maybe one summary email/ week
Those journalists you criticize actually have it right. The universe does try all paths for the particles. Are you familiar with Feynman's reformulation of Quantum physics? The extreme classical paths (variational minimums or maximuns) appear from interference from *all possible paths.
Perhaps you have seen the QFT explanation for why there is no teleology, and you mention it in the footnote, but I didn't see a response. Is your point that there are situations where the optimization is not explained by cancellation?
Every possible path has an amplitude(a unit complex number). But the amplitude rotates fast as a function of the path, so that there is cancellation except at the critical path where the derivative of the action is 0, so nearby paths reinforce rather than cancel the amplitude. Since probability of the path is proportional to |amplitude|^2, the paths around the critical paths are most likely to be traversed.
I didn't expect to like this review so much! Thanks a lot for the link to Ted Chiang's story. I was already a fan of "Arrival", but didn't know it was based on a short story. While it's a pity that Villeneuve (probably had to) left out the Lagrangian part, I do think the movie has gained immensely in poignancy and power compared to the story, and that is a rare achievement.
One of my persistent annoyances is that universities teach two kinds of physics courses - the Newtonian kind, that they teach to engineering students and other non-majors, and the Lagrangian kind, that they teach to physics students.
And they're not at all alike!