Tuesday, March 09, 2021

Feynman's puzzles answered by the principle of least action

Laissez-Faire principle of least action is a deep, philosophy-like wisdom of physics that isn't really taught and encouraged

We had discussions about many things including the Covid-19 strategies with a friend. Sweden has imposed almost no mandatory restrictions centrally. The country's curves (look e.g. at the deaths/avlidna in the lower right corner) look very smooth and organic. Covid-19 is largely over in Sweden and they had two almost identical waves. Each of them has killed some 6500 people. The second, winter-era wave had to arrive because the herd immunity obtained after the first, spring wave wasn't enough for the winter condition when the viruses spread more easily and \(R_0\) is higher.

"Do nothing" really is the best strategy, up to some details. We may go even further. It is a good idea to isolate the old and vulnerable people if you want to reduce the number of deaths but "prescribe nothing" is the best approach once again because the people will naturally shield themselves according to their fear which is generally correlated with their actual risks.

"Do nothing" is optimal in basically all of classical physics. What do I mean? I am talking about a class of problems that Richard Feynman has loved because they encode quite some part of the "real expertise of a theoretical physicist". He was surprised when he saw that many colleagues couldn't solve them.

Maximize your aging

OK, imagine that your rocket sits at the place \((x,y,z)\) up to time \(t\), can move somewhere between the time \(t\) and \(t'\), and at time \(t'\), it must already be at the point \((x',y',z')\). The task is to find a trajectory in the spacetime between the points \(X\) and \(X'\) such that the proper time shown on the clock inside the rocket is maximized (the astronauts' aging is as high as possible). Why does the proper time depend on the trajectory at all?

Well, it's because the special theory of relativity teaches us that the "rate of clocks" is reduced by the speed-dependent square root factor below\[ t_{\rm proper} = \Delta t \cdot \sqrt{ 1 - \frac{v^2}{c^2}}, \quad \Delta t = t'-t \] while the general theory of relativity teaches us that the speed of the clock is also reduced deeply in the gravitational potential\[ t_{\rm proper} = \Delta t \cdot \zav{ 1+\frac{\Phi}{c^2} } \] where \(\Phi\) is the gravitational potential (gravitational potential energy over mass) which is negative if you're close to a source of gravity (such as the center of the Earth) and goes to zero at infinity. Great. So to age maximally between \(X\) and \(X'\), you should better move to higher altitudes, outside the strong gravitational field, because the (second) gravitational red shift will slow down your aging less. However, you shouldn't fly there too quickly because speeds that approach the speed of light also slow down your aging. What is the compromise? How high should you fly and what is the precise trajectory?

This looks like an extremely difficult, artificial optimization problem. You have some factors coming from special relativity as well as general relativity and you're maximizing something (the proper time) that involves both of these factors. Someone could try to solve it by the brute force of algebra. He could even get the correct result – but he could completely misunderstand "why" he really had to get this result. Why? What is the result? Spoilers are everywhere.

OK, the right solution is that the jets should be completely turned off at all times between \(X\) and \(X'\): your rocket should move by a free fall (the initial direction and speed at \(X\) must be adjusted correctly to hit \(X'\)). In the context of the Earth, it means that it should move along the parabola (an upside down U-shaped trajectory) which the tennis balls naturally follow as well (with the right downward acceleration). Why is it the correct answer? Because the motion of objects in the gravitational field (the curved spacetime of general relativity) may be derived from the principle of least action \(\delta S = 0\) which is the minimization of the action\[ S = -mc^2 \cdot t_{\rm proper}. \] The proper time is nothing else than the length of the trajectory in the curved spacetime (expressed in units of time). We must multiply it by some energy-like quantity, \(-mc^2\) in this case, to get the right units of the action (which has the units of energy times time, just like Planck's constant, of course). The sign is negative because the minimization of the action is equivalent to the maximization of the proper time (the straight timelike lines are "maximally" long in the Minkowski space; that's different from the Euclidean space where the straight lines are the shortest ones).

So "do nothing with the jets" at all is the solution to the "maximization of the proper time" problem – but it is also the solution to the equations of motion with the gravitational force only. Why? Because these problems are exactly equivalent.

Saving a person who is drowning

Another exercise that Feynman promoted elsewhere and that is based on the same principle was the following:

You are getting suntanned on the beach, at some distance from the water, and someone (who is asexual and has no race or ethnicity or spine, to make it PC) is drowning. You forget to think whether such people deserve to be saved (that's also to make it PC) and react instinctively: you need to save the person as quickly as possible. What is the fastest way to get to the person when your speed is \(v_1\) on the sand and \(v_2\) in the water?

It is common sense that this path is composed of some paths in the sand and in the water and both parts are straight, otherwise you would be wasting time. But how are they connected? Because the speed in the water is probably much lower, the part in the water will be close to being perpendicular to the water-sand boundary (you really don't want to waste time in the slow water by picking tilted paths to the beach). But it won't be quite perpendicular... What is the angle?

Well, the angles will be such that the ratio of the sines of the two angles on the picture will be equal to the ratio of the speeds \(v_1/v_2\). That's exactly Snell's law of refraction because the two problems are exactly equivalent. You don't need to be "the light" to solve similar problems as "the light" is mindlessly solving all the time. And indeed, the light wants to minimize the time needed to travel along the path and the law of refraction may be derived from that principle.

A slippery slope

I was thinking about some of the related issues while I was close to this bridge an hour ago. You see some inclined plane. Nature has proven the fearmongers who promoted the ideas about the "once in 500 years drought" to be politically brainwashed crackpots and liars (there is absolutely no scientific reason to expect the changes of the CO2 concentration and similar man-made activities to make Czechia unusually dry in the 2020s) so there's a lot of mud everywhere. If you look at the opposite side of the river, you see an inclined plane, the angle is some 45 degrees, and I had to avoid the horizontal plateau near the river because it's muddy (and full of puddles). But the inclined plane itself is somewhat slippery as well because the rocks are covered by no adhesive ducks. What is the optimal trajectory not to slip?

Well, you surely know why I am asking this question. It's another application of the "principle of the least action". You actually want to shoot yourself quickly enough, upwards, and move naturally along the same parabola (including the same speed at every moment) that a bowling ball would follow on the inclined plane. So you imagine a bowling ball that moves on the inclined plane and avoids the mud, you figure out what the speed of the ball is, and you follow (and I followed) the precisely same trajectory that the idealized ball would follow.

Why is it the right solution (as I can confirm experimentally)? Because it maximizes the proper time (or the action) with the extra condition that your shoes are on the surface of the inclined plane. In the language of the forces, the only forces that act on the bowling ball are the gravitational force; and a force perpendicular to the inclined plane (which makes sure that the ball doesn't fly above the rocks but especially that it doesn't penetrate to them, either). So this ideal trajectory – the two conditions are guaranteed by the minimization of the action; and by the extra constraint defined by the inclined plane – is exactly the trajectory that removes the two components of the force that are parallel to the inclined plane, and these two components are exactly the components that require friction and that will fail if you slip (if the friction force required for such an unsafe trajectory is higher than the materials can achieve).

To avoid mud under the bridge on the Farmers' Square in Pilsen-Lobzy, you need to move along the same natural path that a bowling ball would pick. (Intelligent readers will be able to generalize this statement from Pilsen-Lobzy to other places; from me to other people; from bowling balls to other objects, and perhaps they will be able to generalize the idea to other forces, fields, and completely different physical systems, too.) I won't annoy you with additional examples showing the clever ways to walk in Nature (like being a harmonic oscillator that jups on two sides of a cylindrical tube which has water in the middle, I tried it on Saturday in one-kilometer-long water tunnels).

The message

These problems are cool because something extremely clever and important is sufficient for a solution and necessary for an elegant solution to the problems. The principle of least action is, in some sense, enough to determine the evolution according to almost any law of classical (non-quantum) physics. Some good physicists may fail to see the connection but physicists who like to "think about Nature deeply" should see it.

The real issue is that "the optimization problems" may be seen as something that is built "on top of physics", a task for economists and managers. But in reality, Nature (according to classical physics) is solving optimization problems at all times. When you're asked about a similar optimization problem, you – a person who thinks as a physicist – should compare it with the other optimization problems that you know. And indeed, you should know the "extremization of the action" problems because they capture the laws of classical physics. And if you just look carefully, you will see that the puzzles are completely equivalent to some of the most elementary and canonical problems in physics.

Someone may succeed, someone may fail to solve the problems. But I think that the principles such as the principle of least action aren't really emphasized by the postmodern education system; and the fact that they may actually be equivalent to some truly realistic problems you want to solve is being obfuscated even if the principles are presented.

The totally incorrect philosophy "the schoolkid should discover everything herself" (which is just one aspect of the generally collapsing standards and requirements of schools across the Western civilization) may be partly blamed for this evolution. Why? It's because the principle of least action is exactly one of the deep ideas that almost no kid will discover herself – and even if she does, she won't quite appreciate the great power and universality of this principle and various ways to interpret and explain the principle.

It doesn't help to the thriving of the principle in 2021 that there is no known derivation of "white men and fossil fuels must be suppressed" from the principle of least action. To make things worse, all the co-fathers of the principle such as Euler, Lagrange, Hamilton, Jacobi, Morse, Gauss, and Hertz, were white men. With such a racist, sexist pedigree, the principle may soon be banned at "universities" altogether. ;-)

A good "universal thinker or renaissance man" should master exactly ideas that include the principle of least action or that are similar to it in their "deeply philosophical depth and universality". In reality, instead of such men with deep thoughts and multi-dimensional knowledge and understanding of the world, our postmodern civilization produces "data scientists" who are typically totally ignorant about these matters. They only know the ideas that a kid of the average intelligence may invent after a reasonable time and the full formulation of the laws of classical physics in terms of optimization problems doesn't belong to this category. So these "data scientists" end up misevaluating what is important, what is not; what is easy, what is not; what is universal, what is not; what is almost certain, what is not. And they consequently produce ludicrously wrong if not absolutely idiotic recommendations. And indeed, the Covid-19 lockdowns are a hugely consequential, global example of the antiscientific policymaking claimed to be scientific by the people who are really neither scientists, nor philosophers, nor fair-and-balanced, nor intelligent.

P.S.: The minimization of the action is a way to derive the Euler-Lagrange equations, basically the "universal evolution equations in classical physics". But even this, general enough interpretation of the "importance of optimization" fails to describe the importance of optimization in the whole "realm of thinking". The laws of economics (and perhaps ecology) are really derived from the maximization of individuals' utility functions, too. Lots of things and processes in the world, at many levels, work as solvers to optimization problems. Gottfried Leibniz opined that our world was "the best one among the possible worlds" which had a flavor of appreciating the omnipresence of the optimization problems. Almost all the progressive ideology is pretty much in a fundamental conflict with all these principles. The comrades of assorted flavors of the Marxist crimson fight to sling mud on everything – on the way how capitalism, the nation states, democracies, the civilization, and Nature work – in order to replace the laws by theirs which they claim to be more just. But their laws are self-evidently toxic garbage and are pretty much guaranteed to be garbage because they greatly deviate from the natural laws that maximize the well-being.

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