From Aristotle's philosophy to Feynman and quantum mechanics – one principle unifies the laws of physics: the principle of least action. Come hear about it in a lecture at the Givatayim Observatory, on Thursday, May 29.05 at 21:00 PM.
The lecture on the code of the universe will take place at the Givatayim Observatory at the Israeli Astronomical Society, Thursday, May 29.05 at 21:00 PM. To register, click here.
Does nature always choose the shortest path? From the philosophy of Aristotle, through the discoveries of Heron, Fermat, and Newton, to Richard Feynman's revolutionary interpretation of quantum mechanics, it turns out that nature operates according to a remarkably simple principle - the principle of least action. We invite you to dive into a fascinating journey following the elegant idea that conquered science, understand how the laws of physics merge into one unified principle, and discover how ancient thought changed the way we understand the universe.
״Nature does not work in vain." Thus said Aristotle to his students - everything has a purpose. The words of Aristotle, the great philosopher from ancient Greece, have resonated thousands of years ago, from the fourth century BC to the present day. The same thought, almost theological, that everything has a reason, is at the foundation of the scientific approach today. Nature has its own laws and only if we know them, we will be masters of the world. Sometimes this thought has led philosophers to base themselves on erroneous things. In 70 BC, the philosopher Heron of Alexandria claimed that light must move in a straight line just as an arrow moves straight at a constant speed towards a target. He tried to understand the reason for this - bodies moving in a straight line strive to move in a path that minimizes their travel time. This is the most elegant and simplest way to move from one place to another. He claimed that the force that moves bodies to move "Can't afford"Slow down. Of course, an arrow does not move in a straight line, or at a constant speed, and nothing can be deduced from this about the movement of light, but this perception led him to a very interesting conclusion - the reason that light is reflected at the same angle at which it strikes the surface stems from nature's "desire" to be economical. Although Euclid himself explained the phenomenon of reflection before him, Heron was a pioneer in his field. He proved that the angle of impact on the surface is equal to the angle of reflection only by assuming that light moves in a path that shortens the time of travel.
Heron drew inspiration for his revolutionary idea from Aristotle. Heron argued that if nature wanted to distort our vision, it would not use the principle of equal angles, that is, that the angle of incidence is equal to the angle of reflection. This argument reflects the fact that nature operates in the most economical and efficient way, that this phenomenon has a reason, that nothing is done without a purpose.
Equal to the angle of reflection
From reflection to refraction of light
Reflection from a mirror is not the only way in which light changes its path. For example, when a ray of light enters a transparent medium, it is refracted and deviates from its angle of incidence. The phenomenon of refraction has been known since ancient Greece and was first described by the astronomer Ptolemy. Ptolemy used to compile detailed tables of the angles of refraction in various materials, but he assumed—incorrectly—that the angle of refraction itself was directly proportional to the angle of incidence. The correct formula, now known as Snell's law, was only published in the 17th century, first by Wilbur Snell and a few years later by Descartes (historians of science claim that this law was discovered as early as the 10th century by the Arab mathematician Ibn Sahl and by the British astronomer Thomas Harriot). In the revised formula, the researchers showed that the sines of the angles are proportional to each other. To show this, Descartes assumed that light travels at different speeds as it passes from one medium to another. However, to make his formula fit observations, he assumed that the speed of light increased as the medium became denser. Descartes probably thought that light behaved like sound waves, traveling faster in a denser medium.
In 1657, about seven years after Descartes' death, a letter was sent to the French jurist Pierre de Fermat, and the "amateur" mathematician (it is now clear to everyone that he was much more than just an amateur) from another French philosopher named Marie Kiro de La Chaux-de-Maures, which included courageous arguments about the law of reflection of light: "For these reasons, in the phenomenon of reflection of light, we observe the equality of angles and this is observed on the shortest lines. This is not a phenomenon unique to light, because nature behaves in this way in all the movements it produces." He then writes, "If light behaves in this way in reflection, it will certainly behave in this way in refraction." To this, Fermat responded in two letters: "It is a physical principle that nature will choose to move along the simplest paths... There is nothing more likely than nature to choose to move along the simplest path, that is, the shortest path when time is not a factor or a path that minimizes the time of its movement."
Following the correspondence with Marie Curie, Fermat was able to explain the phenomenon of refraction using the argument that light moves in a path that minimizes its travel time and to reconstruct Snell's law from it, but in a modified form so that light moves more slowly in a dense medium, contrary to Descartes' approach.
Not everyone accepted Fermat's principle openly. Studies have shown that Fermat corresponded with a number of scientists who did not understand why nature would choose to minimize the time of movement. The idea was seen more as a philosophical view of nature than as a fundamental principle by which nature operates. On the other hand, some saw Fermat's idea as groundbreaking. Newton, for example, was the first to use minimization to calculate the most efficient geometric shape that would minimize air resistance. He discovered that a type of ellipsoid did the job and argued that this calculation might in the future help in building ships. Newton reached this conclusion with great genius and along the way invented mathematical tools that did not exist before, such as differential and integral calculus (integrals and derivatives). Newton was probably the first to develop an accurate mathematical calculation of this principle. Based on the mathematical tools he invented, Newton called this principle the "principle of variation" from the foreign word variation - change. That is, how a certain quantity changes as a result of a change in another parameter.
What is the fastest route between two points under the influence of gravity?
In the late 17th century, mathematicians were very curious about problems that required finding a maximum and a minimum. The interest was probably due to a certain aesthetic, theological ideas, and pure curiosity. A famous problem that occupied many was the brachistochrone problem written by the Swiss mathematician John Bernoulli. In 1696, in a letter he sent to Newton's archenemy, the German scientist and philosopher Gottfried Leibniz, he wrote: "Given two points, find the path that connects them so that a body of mass M will move in the shortest time." Bernoulli knew that under the influence of gravity, the body can accelerate. The maximum acceleration would be if it fell vertically to the ground, but the question is whether it is worth it to accelerate without moving horizontally to the point and then roll towards it, or to accelerate while moving at a slower pace? Leibniz answered this riddle in detail within a week – he called the curve he discovered the tachystoptota (the fastest curve) and added with a pun that he had solved this problem involuntarily. In response to Leibniz's letter, Bernoulli mentioned that the track was similar to a cycloid – the curve formed when a point is followed on a wheel that is rolling in a straight line at a time. In the letter, he also suggested another name for this track – brachistochrone – which means the shortest time in Greek. By this time, Bernoulli had already spread the problem to several famous people, and a year later, in 2, he published the brachistochrone problem and the solutions he received. Among the solvers was also Newton, who did not reveal his name, but Bernoulli knew it was him because, as he said, he recognized "the claws of the lion" (the lion in Newton's parable). Leibniz, however, preferred not to reveal his solution after he corresponded with other mathematicians who had similarly solved this problem. One of the solvers was Bernoulli himself, who used the same Fermat's principle to break the trajectory of a rolling body in such a way as to minimize its time of motion.
The cycloid is the fastest.
On the greatest fake news in scientific history
The metaphysical approach proposed by Aristotle was revived in the mid-18th century by the French scholar Pierre-Louis Maupertuis. Maupertuis was a student of Bernoulli, and is considered the first Newtonian in France and the entire European continent. His admiration for Newton was so great that in 1737 he led a scientific expedition to Lapland to prove Newton's claim that the Earth was oblate at the poles and not a perfect sphere - a claim that many of his contemporaries believed to be untrue. In light of his success, Maupertuis gained great scientific recognition and became very famous in the European scholarly community. Although he acknowledged that Newton's theory of gravity had competitors, he said that if God had been given the right to choose between all the competing theories of gravity, he would have chosen Newton's, because it was the most elegant and general. Like Newton, he was influenced by the theological and metaphysical worlds. In his letters, Maupertuis gave another interpretation of the principle of God's choice and used to repeat the famous theorem of Aristotle and Fermat that "nature works in the simplest way". In particular, he argued that "in nature, action is necessarily reduced to its smallest magnitude". The "action" as he called it, was developed by Maupertuis and is equal to the mass of the body times the velocity times the distance the body moves. By minimizing the action, Maupertuis was able to reproduce Snell's law in Descartes' version. Although he knew that this was a mistake, he considered it a great achievement. The main difference between Maupertuis and his predecessors is the magnitude that he minimizes. Maupertuis minimized the action, not the time for light to travel, and he also used the action to explain the motion of bodies colliding with each other. Of these successes, Maupertuis said: "The laws of motion and rest arise from this principle and are precisely observed in nature. This is a fact worthy of admiration and wonder. The movement of living beings or the growth of plants, all arise from this principle. [And anyone who recognizes this principle] understands how wonderful the universe is, it looks much more beautiful. Its author deserves great respect, especially after realizing that he wrote a small number of intelligently determined laws that are sufficient to describe all the movements in nature.
Maupertuis argued that the principle of least action – the idea that nature always chooses a path that minimizes action – applies to everything, not just light or inanimate objects. It is important to understand how far ahead of its time this concept was: in the past, each scientist or philosopher explained a phenomenon with the help of a unique principle tailored specifically to its dimensions. Over the years, as the number of phenomena discovered increased, various branches of physics emerged – from thermodynamics to electrodynamics, the physics of waves and light, chaos, astrophysics, and more. All of these relied on their own set of laws, with no clear connection between them. Years later, with the discovery of quantum mechanics and the deepening of our knowledge of the world, we understood that all of physics relies on a limited number of laws or principles. This is precisely what Maupertuis intended, and he did succeed in doing so. He came up with the principle that unites the various branches of physics, which is the principle of least action.
Some did not like the ideas that Maupertuis promoted, even envied him. One of Maupertuis's most prominent opponents was Voltaire, a well-known French philosopher and thinker from the 18th century. Even before they became enemies, Voltaire and Maupertuis collaborated in spreading Newton's ideas. After Voltaire discovered that Maupertuis was having an affair with his beloved mathematician Emilie, tension arose between them. Voltaire used the argument of Jonathan König, a researcher whom Maupertuis himself had appointed when he was president of the Academy of Sciences in Berlin. König claimed that Maupertuis had copied the idea of the principle of least action from none other than Leibniz. König claimed plagiarism based on a letter he allegedly found, in which Leibniz corresponds with mathematician Jacob Hermann about the idea of least action. However, a number of historians of science have challenged this fact in recent years, claiming that Leibniz's letter does not exist, that König never claimed plagiarism, and that the whole story is in fact "the most successful fake news in modern history." Although historians dismiss the plagiarism claim, Leibniz did work on his own version of the operation. Although he formulated an inaccurate expression, it helped him reconstruct Descartes' formula for the refraction of light. Contrary to Maupertuis, he argued that this was the correct formula and that light did indeed travel faster in a denser medium.
The ongoing conflict between König and Maupertuis swept through the scientific community in Europe. News of the forgery reached the King of Prussia, who came to Voltaire's defense. At the height of the conflict, the Academy of Sciences in Berlin decided to establish a commission of inquiry to put an end to the ongoing conflict and get to the root of the truth. The person who headed the commission was none other than the great mathematician Leonhard Euler. After completing the extensive investigation, the commission he headed came to the conclusion that such a letter had never been found and that Maupertuis was the first to discover the principle of least action. Historians of science believe that König probably presented a certain piece of the alleged letter, but over the years it was discovered that it mentioned terms that were not common in Leibniz's time.
On the principle of minimal action
Euler not only believed in the righteousness of Maupertuis's method, he also thought his idea was brilliant. After the case was closed, he set about formulating the principle of action in precise mathematical terms. He used the mathematics he had developed to show that the orbits of celestial bodies could be calculated. In fact, the orbit that Euler showed did not necessarily minimize the action, but was a saddle point of the function and at the same time extended the physical idea. The modern form of the principle of minimal action came thanks to the French-Italian mathematician Lagrange and looks like this:
The letter S represents the action. It is a function that "eats" a path and "emit" a number. More precisely, -S is called a functional because its input is actually a function (the path is described as a function in space and time). The principle of minimal action says that the path that minimizes -S is the path that the body will "choose" to move on. To calculate the action cell, we need to generalize (to tangerize, or sum) an expression called the Lagrangian along the curve. In general, the Lagrangian, L, encodes within it the kinetic energy of the body minus its potential energy. Without going into too much detail, these are elements that encode what the moving body "feels" in its environment and how much kinetic energy it has.
Of course, there are infinite paths that we need to estimate, but to find the minimum, there is no need to calculate infinite terms. In high school, you already learned that to find the minimum of a function, you need to differentiate it and set the derivative to zero. This is more or less how Euler and Lagrange formulated the way to find the minimum of the action. It turns out that from this principle alone, the minimum point accurately reproduces the equations of motion of bodies that Newton formulated. The beauty of minimizing the action is in its simplicity and the fact that it explains Newton's laws from a single principle.
The quantum interpretation of the principle of least action
This is how science actually works. As we deepen our knowledge of the world, more connections are discovered and the number of principles required to explain phenomena is reduced. About two hundred years after the revolution of Euler and Lagrange, a new mechanics took shape – quantum mechanics. This is a mechanics that was initially formulated to explain the behavior of particles in nature, but today, about a hundred years later, we know that it is actually the mechanics of nature, and from which everything stems. There will be more extensive articles on this blog about quantum mechanics, but to summarize briefly, from its very beginning, the new mechanics offered a new interpretation of the behavior of particles – instead of discussing the question of the exact location of particles at a given time, it is more correct to ask what the probability is that a particle will be in a certain place at a given time. Because of the inherent uncertainty built into these mechanics, there is no way to know absolutely where things are, and they will not be located precisely until we measure them, for example by taking a photograph or measuring an electric current.
In the mid-twentieth century, the principle of least action also found its way into quantum mechanics. The person who introduced the principle into quantum mechanics was the Jewish physicist and Nobel Prize winner Richard Feynman, who gave it a new interpretation by incorporating the probabilistic nature of the universe. Feynman argued that the path that minimizes the action is actually the most likely path that a particle will move on, but not necessarily the only one. In other words, for every path there is a probability that the particle will move on it, but the path with the highest probability is the path that minimizes the action (more precisely, it is not always a true minimum but sometimes a saddle point, as Euler showed). The probability that a particle will move on other paths also depends on the size of the action: the more “distorted” a particular path appears to be compared to the classical path, the less likely it is to be chosen.
Since the probability of moving in non-classical paths is different from zero, they may affect the measured quantities in experiments. In his books, Feynman uses this principle to explain interesting retardation patterns such as in the famous two-slit experiment (which we will discuss in other posts), but of course it can be used to predict any other observable phenomenon.
Now, in a single equation, we can heuristically write down all possible histories of the universe. This is a powerful equation that in principle we really cannot calculate its outcome precisely, but in unique cases we can approximate it and predict effects that were not observed before the discovery of quantum mechanics and the extension of the principle of least action.
The wonderful equation before you summarizes all the possible states that the universe can be in. In this equation, the action appears in the power of an exponent because this is the way to relate a low action to a high probability. The smaller the action, the greater the exponent to the minus power of the action. The action in this equation depends on all sorts of parameters, which I began to denote with the Greek letter phi. Without going into too much detail, phi mathematically describes certain particles in nature. The action that describes the universe obviously depends on all sorts of types of particles, which is why I added three dots after it. We summarized the probabilities (or more accurately, the probability density) in a single block that we call the Z distribution function (inspired by statistical physics). With -Z, you can calculate everything, but because in most cases the distribution function is difficult to calculate, physicists focus on particular cases, in which the distribution function is simpler.
Now it's time to look at the big picture. Thanks to one principle, we can actually calculate, in principle, any measurable quantity in the universe. We no longer need secondary laws, the principle of least action covers everything. This is a claim whose power should not be underestimated and it is the one that expresses Aristotle's view in the clearest form. At the same time, the calculation can be very complicated and so we do not throw away everything we have learned so far. These can simplify the calculations without trying to sum up all the possible histories of the universe.
On the Standard Model and Beyond
In some cases, where particles collide with each other, such as at the Large Hadron Collider at Cernen, physicists can use the principle of least action to predict what will happen after the collision. As a reminder, to use this principle we need an action, or to be more specific, a Lagrangian that encodes the behavior of particles. That Lagrangian was developed over decades until around the 2012s, and its latest component was verified in XNUMX with the discovery of the Higgs boson (which was called the “God particle” in the media, but of course there is no God element in it). This Lagrangian is called the Lagrangian of the Standard Model and it encodes the properties of particles in nature – from the matter particles called quarks and leptons to the particles that mediate the forces in nature – the gluons, photons, W and Z.
Where did the Standard Model come from? How did physicists know how to encode the Lagrangian of nature's particles? Here comes another principle that I should write about separately: the deep connection between symmetry and conservation laws. This principle states that if a certain symmetry is expressed in nature, there is a measurable quantity that does not change over time. Symmetry, for example, can be expressed in shifts in space or time - in other words, the results of an experiment will not change if it is conducted in a different location or on a different day. Reciprocally, if an experiment shows that a physical quantity remains constant over time, we can conclude from this that there is symmetry in the system - a symmetry responsible for the conservation of quantity. In experiments conducted in particle accelerators, quantities that do not change are also measured, such as charge, momentum, energy, and more. These conserved quantities are directly related to the symmetries that the Lagrangian must express. For example, in the theory of relativity, Einstein taught us that the Lorentz symmetry, which is expressed in spacetime itself, entails conservation of energy and momentum. Therefore, we would expect the Lagrangian of the Standard Model to also have such a symmetry. In fact, the Standard Model contains the most basic mathematical elements that quantum theory can maintain while requiring the existence of the symmetries that we observe in nature.
Today we know that the Lagrangian of the Standard Model is incomplete. Gravity is not included in it, and dark matter and energy are also missing. Ever since these were discovered, physicists have been trying to find fundamental explanations for dark phenomena. A natural solution for dark matter assumes that it is a particle with some mass that comes into contact with other particles. How do we test this assumption? First, we start by building the model – adding new terms to the Lagrangian of the Standard Model, which do not destroy the properties of the known particles, and correspond with the physical properties of dark matter. However, the number of possibilities for building a consistent model that includes dark matter is enormous, and most of the experiments conducted so far have failed to provide clear clues that would decide between them. Thus, between the abundance of theories and the paucity of findings, we are left – literally – in the dark.
Despite the difficulties, the very fact that the principle of least action provides a flexible framework within which the Standard Model can be extended and explanations for phenomena that are not understood are evidence of the principle's power. It is possible that in the future we will need additional principles to solve all the remaining puzzles about the universe, but there is no doubt that the principle of least action is here to stay for many years to come.
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The images are from Wikipedia. The historical facts were taken from the book “The Principle of Least Action – History and Physics” by Alberto Rojo and Anatoly Bloch.
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The Biological Envelope – Man and the Evolutionary Process
The biological shell we call “human” or “Homo sapiens” is the product of a deterministic and evolutionary process, whose sole purpose is to survive and reproduce. Everything that has developed around that shell – consciousness, concepts, society and culture – are temporary tools created to serve that same basic purpose.
Stage One: The Body and the Biological Basis
1. The human body as a functional tool:
• The biological body is designed to serve only two main purposes: to preserve itself (survival) and to pass on its genes to the next generation (reproduction).
• Every biological system in the body works to support this process, and it has no meaning or purpose beyond that.
2. Man as part of nature:
• Like any other organism, man is subject to the basic laws of nature: the strongest survive.
• Man's success in controlling his environment is an expression of evolutionary power dynamics, and does not indicate true "advancement" or "uniqueness."
Stage Two: Consciousness and Concepts
1. Consciousness as an evolutionary tool:
• Consciousness is a tool that evolved out of the need to optimize the functioning of the body.
• It allows a person to:
• Process his environment.
• Plan ahead.
• Deal with threats and find solutions that promote its survival.
2. Concepts are organizational tools:
• Concepts – such as “emotion,” “meaning,” “morality,” and “belief” – are not natural phenomena.
They are products of consciousness, created to organize and regulate existence.
• They help a person understand their world.
• They create social structures that enable cooperation and stability.
3. Society as a product of concepts:
• Society is a shared system built around these concepts.
Her role:
• Preserve the existence of biological envelopes.
• Enable collective behavior that increases the chances of survival and reproduction.
Step Three: Recognizing Temporality and Progress
1. The Earth and the human body are temporary:
• Our biological body is adapted to the Earth and its conditions, but both are temporary.
Physical changes (such as climate change or space travel) will require solutions that are independent of the biological body.
2. “Forwardness” as an existential necessity:
• “Foresight” – thinking about the future – is a tool that evolved from consciousness.
Its purpose is to prepare the envelope and consciousness for a new reality in which the current conditions will no longer exist.
3. Beyond biology and concepts:
• In the future, when biology reaches its limits, a new system – biotechnological or cognitive – will be required that can continue existence under changing conditions.
• Concepts and consciousness, as we know them today, will dissolve or undergo fundamental change.
Step Four: The Conclusion
1. Only an evolutionary process:
• Man, like every organism, is the product of an evolutionary process in which every system and every tool was created to serve the same purpose: to survive and reproduce.
• Culture, society, and concepts are nothing more than temporary tools that aid the process.
2. There is no real “introduction”:
• Human control over nature and the environment is not evidence of “uniqueness” or “progress,” but rather a direct continuation of the basic laws of nature: the strongest survive.
3. The future beyond biology:
• As consciousness advances, it will recognize that biology is a temporary phase, and that the concepts created to serve it will lose their meaning.
• The future belongs to new systems – technological, cognitive, or other – that can exist without the biological envelope.
The main thought:
Humans are a link in a long process in which every biological structure is designed to survive and reproduce.
Consciousness, concepts, society, and thinking about the future are tools created out of that need, and have no meaning beyond that.
Human existence, like any evolutionary process, is not an end in itself but rather part of the greater process of nature.
How does it feel? Does it manage to combine the two answers in a way that reflects your ideas clearly and sharply?
The biological envelope – humans as part of a natural and universal process:
In order to put ourselves in our real place, we must recognize that human society, with all its concepts, structures, and consciousness, is nothing more than a link in the evolutionary chain of nature. Although human culture sees itself as advanced and developed, it is in fact a primitive society in essence, driven – just like any other organism – by the need to survive and reproduce.
Survival of the fittest – a universal principle:
1. Man as part of nature:
• Every biological structure – every organism, including humans – is driven by one basic action: to survive and reproduce.
• What is called “culture” or “progress” is nothing more than a product of that basic urge, wrapped in concepts designed to regulate and organize existence.
2. Dynamics of control:
• Human control over animals and nature as a whole is nothing more than an expression of “survival of the fittest.”
• This control, which we consider an achievement, is nothing more than part of an evolutionary process in which physical, mental, or social strength allows an organism to survive.
• The force that drives humans is no different from the force that drives animals or other species – it just manifests itself in a more complex way.
Human culture – primitive complexity:
1. Concepts as tools for control:
• Concepts like “society,” “meaning,” or “justice” are created out of a need to manage collective life.
• They do not free us from our basic nature, but only disguise it.
Ultimately, even the highest complexities of culture stem from basic needs: control, survival, and culture.
2. The Misguided View of Primitiveness:
• When we look at other societies (such as tribal societies) and call them “primitive,” we ignore the fact that human behavior itself has remained primitive at its core – the control of power and the preservation of the species.
• Our modernity is only a mask, covering up the basic instincts that still drive us.
Man facing his future:
1. Existence is not above nature:
• Man is not special or “superior” in nature – he is just another organism that has found unique ways to survive.
• His success in controlling the Earth does not guarantee his long-term survival, especially in light of the changing environment and universe.
2. Biology as a temporary process:
• If the biological envelope fails to adapt to the physical changes of the Earth and the universe, human existence will end – like every other species that has not survived.
• The concepts created to regulate life today will lose their relevance, just as they evolved out of temporary necessity.
3. The future as a result of the dynamics of survival:
• The possible transition to technology, artificial intelligence, or non-biological systems is a direct continuation of the drive to survive.
• It has no “purpose” other than continuity – just as there is no purpose to current biological existence beyond its existence.
The conclusion – man in the evolutionary process:
Man is nothing more than an organism that has undergone evolutionary processes that led to temporary control over nature.
• Our biological envelope is a link in a long process, in which the strongest survive and continue to evolve.
• Culture, consciousness, and concepts are temporary tools created to serve the same basic purpose: to survive and reproduce.
• Ultimately, humans – like any other organism – are subject to the same simple laws of nature: to survive and pass on their genes.
The article probably presents the historical facts correctly. It is assumed that the article was written following a video by Vertsium on YouTube and/or other films that appeared on the subject. In these films, people confused mathematics with reality. It is recommended to see the answer in the video:
https://youtu.be/XcY3ZtgYis0?si=7nUsZjj8v3ADAr6Z
Thanks for the enlightening article.