# This Single Rule Underpins All of Physics

## One rule that replaces all of physics (0s)

- A single, simple rule underpins all of physics, encompassing every principle from classical mechanics to electromagnetism, quantum theory, and general relativity, as well as the fundamental particles that make up matter (19s).
- This rule can potentially replace all of physics, which may seem like a surprising or unconventional idea (24s).
- The rule may even explain the behavior of life itself, according to Derek (30s).
- The traditional way of thinking about the universe, using local pictures and differential equations, may be an outdated mindset (41s).
- It is possible that this classical mindset has things backwards, and a new perspective is needed to understand the universe (46s).

## The problem of fastest descent (51s)

- The problem of fastest descent involves finding the shape of a ramp that allows a mass to slide from point A to point B in the shortest time possible (51s).
- Common sense might suggest taking the shortest path, a straight line, but bending the ramp down at the beginning allows the mass to accelerate to a higher speed earlier and travel faster (1m12s).
- Galileo proposed that the arc of a circle is the fastest shape, but Johann Bernoulli set the problem as a challenge to mathematicians in 1696 to find the optimal solution (1m43s).
- Bernoulli's challenge was initially met with no submissions, but after extending the deadline, Gottfried Leibniz persuaded him to give foreigners a chance to participate (2m6s).
- Isaac Newton, who was no longer an active mathematician or physicist at the time, received Bernoulli's challenge and came up with a solution in one day, which he submitted to the journal Philosophical Transactions without signing it (2m36s).
- Bernoulli recognized Newton's solution and is alleged to have said, "I recognize the lion by his claw," indicating that he knew the solution was from Newton (3m13s).
- Although Newton dominated Bernoulli overall, Bernoulli's solution to the problem of fastest descent was actually more clever and creative (3m36s).
- Bernoulli's solution was inspired by the problem of how light travels from one place to another, which was contemplated by ancient philosophers such as Hero of Alexandria (3m46s).
- Hero of Alexandria realized that light always follows the shortest path in a single medium, but when it passes from one medium to another, it bends and refracts (3m56s).
- The guiding principle behind this phenomenon is Snell's Law, which states that the sine of the angle of incidence divided by the sine of the angle of refraction is equal to a constant, n, that depends on the nature of the two media (4m55s).

## Fermat's principle (5m9s)

- Pierre Fermat, a mathematician who worked on math in his free time, became interested in the question of why light obeys the principle of refraction, and he thought that Hero of Alexandria was on the right track but that it's not distance that is being minimized, but rather time (5m34s).
- To test this idea for refraction, Fermat would have had to work out every possible path light could take by varying the point where it intersects the boundary and compute the time for each, and then show that light takes the path for which the total travel time is the shortest (5m42s).
- Fermat initially didn't know how to solve the problem and worried that it would be complicated, but he eventually solved it five years later and showed that Snell's Law actually pops out as the minimizing path for light under certain conditions (6m18s).
- Fermat found that the constant n in Snell's Law is equal to the speed of light in the first medium divided by the speed of light in the second medium, which allows for a rewritten version of Snell's Law (6m30s).
- Fermat described his calculation as "the most extraordinary, the most unforeseen, and the happiest calculation" of his life, and his work showed that nature obeys an optimization principle, doing the best possible thing, in this case, light taking the shortest possible time (6m42s).
- Fermat's principle of least time can explain everything that was known about light at the time, and it was the first time that anyone showed that nature obeys an optimization principle (6m57s).

## Bernoulli's solution (7m17s)

- Bernoulli used Fermat's principle of least time to solve the problem of fastest descent by converting it from a mechanics problem to an optics problem, imagining a ray of light that speeds up as it enters less dense media (7m18s).
- By applying Snell's Law to each interface and making the layers thinner, Bernoulli obtained a continuous curve that models a falling object (7m31s).
- The speed of light in each layer is proportional to the distance from the top, similar to how the velocity of a falling object increases with the square root of the height from the top (8m27s).
- By plugging the expression for the speed of light into Snell's Law and analyzing a single interface, Bernoulli found that the sine of the angle divided by the square root of the height is constant across layers (8m53s).
- This constant ratio led Bernoulli to recognize the equation as that of a cycloid, also known as a brachistochrone curve, which is the path traced by a point attached to the rim of a rolling wheel (9m32s).
- The astonishing solution is that the fastest way to get from point A to point B is to follow an arc of a cycloid, not a circle (9m50s).
- The cycloid curve also has the property that no matter where the mass is released from, it always reaches the end at the same time, making it also known as the tautochrone curve (10m1s).
- Bernoulli's solution showed that two problems from separate fields of mathematics, optics and mechanics, have the same character, achieving more than he initially demanded (10m30s).

## Maupertuis' principle (10m38s)

- Pierre Louis de Maupertuis, a student of Fermat, studied the behavior of light and particles around 40 years after Fermat and noticed that there are cases where the two behave very similarly (10m44s).
- Maupertuis proposed that Fermat's principle of least time might not be the most fundamental and that there could be a more foundational quantity being minimized, one that governs both light and particles (10m52s).
- In the 1740s, Maupertuis proposed a new quantity called the action, which is mass times velocity times distance, and suggested that the action is greater when something travels farther, faster, or is more massive (11m7s).
- The total action for a journey with multiple segments is the sum of the mass times velocity times distance for each segment (11m31s).
- Maupertuis claimed that the path a particle follows is the one that minimizes the action, and he provided a simple example of a 0.5 kilogram ball rolled over the ground and bouncing off a wall to illustrate this principle (12m6s).
- According to Maupertuis, the action is the "true expense of Nature, which she manages to make as small as possible," as stated in his writing in 1744 (12m19s).

## Maupertuis attacked and ridiculed (12m28s)

- Maupertuis' revolutionary idea was met with attack and ridicule from his peers, including his longtime friend and fellow physicist Samuel Konig, who claimed that Maupertuis' principal was wrong and that he had stolen it from Leibniz (12m38s).
- Voltaire, a close friend of Maupertuis, accused him of plagiarism, bad physics, and stupidity, and even wrote a 32-page pamphlet to mock him, possibly motivated by rumors of an affair between Maupertuis and Voltaire's lover (12m52s).
- Not everyone attacked Maupertuis, as some people simply ignored him, and he was not well-known or widely mentioned despite his significant contributions to physics (13m14s).
- The criticism and ridicule took a toll on Maupertuis, who was nearing the end of his life and had hoped that his principle of least action would be his lasting legacy (13m18s).
- Unfortunately, the criticism was somewhat justified, as Maupertuis had developed the principle of least action without a clear reason or rigorous mathematical basis, essentially picking it out of thin air (13m40s).
- The principle of least action, which states that nature minimizes the quantity of mass times velocity times distance, was not mathematically rigorous and lacked a clear justification for why nature should care about this quantity (13m56s).

## Euler & Lagrange to the rescue (14m1s)

- Leonhard Euler defended the principle of least action and replaced the sum with an integral to calculate the action while speed or direction changed continuously, allowing him to find the path of a particle around a central mass (14m9s).
- Euler used this method to find the path with the smallest action, which is similar to Fermat's problem, but with infinitely many variables to consider (14m25s).
- Euler invented a new method to handle this problem, which was clunky but worked, and he realized that the principle of least action only works if the total energy is conserved and the same for all paths considered (14m54s).
- Euler improved the mathematical rigor of the principle by finding two extra conditions and providing a specific example of it working (15m16s).
- Euler was a powerful mathematician who was generous and empathetic in his explanations, making his work still understandable today (15m27s).
- Joseph-Louis Lagrange, a shy 19-year-old mathematician, built upon Euler's work and shared his results with him, which led to a general proof of the principle of least action (15m59s).
- Lagrange and Euler were both proponents of the principle of least action, and Lagrange succeeded in providing a general proof around five years after sharing his results with Euler (16m30s).
- Lagrange's proof will be explained in a simplified way, using three steps: explaining the general approach, rewriting the principle into its modern form, and applying the math to a simple example (17m1s).

## The general approach to solving these problems (17m22s)

- The general approach to solving problems involves finding the path with the least action among infinite possible paths, which can be achieved by using a method similar to finding the minimum of a function, as realized by Euler and Lagrange (17m23s).
- To find the minimum of a function, the derivative is taken and set equal to 0, and the point where the slope is horizontal is considered the minimum, as a tiny step to the left or right would not change the value of the function (17m37s).
- Similarly, if a tiny bump or change is added to the path of least action, the action should not change, as it is at a special point, and any other path must have more action (17m51s).
- The counter to this is that the first-order terms, proportional to the deviation eta, should be 0, indicating that the difference in action will be 0, and any deviation in action will be proportional to eta squared or higher-order terms (18m24s).
- This concept can be visualized as being at the bottom of a bowl, where a tiny step away from the minimum would increase on one side and decrease on the other, making the coefficient of eta 0 (18m34s).
- The general approach to solving these problems involves writing the action of a trial path minus the action of the true path as equal to 0 to first order, which is a compact way of writing the principle of least action (19m15s).

## Writing the principle into its modern form (19m32s)

- Maupertuis' action is rewritten in its modern form, starting with the sum of mass times velocity times distance, which is then changed into an integral of mass times velocity integrated over distance (19m36s).
- The velocity is equal to ds over dt, which can be rearranged to get ds equals vdt, and plugging this in results in an integral of mv squared over time, which is twice the kinetic energy (19m56s).
- The total energy must be conserved, and total energy is just kinetic plus potential, so the integral can be rewritten as T equals E minus V (20m6s).
- The variation of T plus E minus V integrated over time is equal to 0, and this integral can be split into two, with the energy term being constant and integrable over time (20m16s).
- The variation of E times t can be written as E times the variation of t plus t times the variation of E, but the energy of different paths has to be the same, so the variation between them is 0 and this term drops out (20m40s).
- The variation of the integral is equal to minus the energy times the variation of time, which can be made 0 by only considering paths that have the same travel time (20m56s).
- Maupertuis' principle has changed into another form where the variation of kinetic energy minus potential energy integrated over time is equal to 0 (21m27s).
- This new form is equivalent to the principle of least action, which can be written as the integral of Ldt, where L is the Lagrangian, or T minus V, the kinetic minus potential energy (22m4s).
- The first person to write the principle of least action in this form was William Rowan Hamilton in 1834, and it is now known as Hamilton's Principle (22m9s).
- Hamilton's Principle is the modern way of writing the principle of least action and is used in almost every physics book, as it tells you how objects move from one place to another rather than just giving you the shape of the path (22m35s).
- Two important differences between Maupertuis' principle and Hamilton's Principle are that the action is now an integral over time instead of space, and with Hamilton's Principle, you now need a start and end point, and also a start and end time (22m51s).
- The third difference is that with Maupertuis' principle, you need to keep the energy of different paths the same, but the time can vary, while with Hamilton's Principle, the energies can differ but the time has to be the same between paths (23m10s).

## Why the principle works (23m17s)

- To understand why the principle of least action works, a simple example of throwing a ball straight up in the air is used, where the ball goes from a start point to an end point in a certain amount of time (23m21s).
- The height of the ball is represented as y of t, and infinitely many possible trajectories can connect the start and end points, with the condition that all paths must have the same start and end point and the same amount of time elapsed between them (23m53s).
- To find the real trajectory, small variations are made to the true path, y of t, by adding a small change, eta of t, at every time step, resulting in a new trial path, q of t (24m18s).
- The difference in action between the true path and the new trial path is computed, and the equation is solved to find that the action is minimal for the path that satisfies a specific differential equation (25m22s).
- The differential equation is simplified to show that the path that satisfies the principle of least action is the one that obeys F equals ma, which is equivalent to Newton's Second Law (25m57s).
- The principle of least action is a unifying principle that covers more than just mechanics, and it is equivalent to Fermat's principle of least time, which is a special case of the principle of least action (26m18s).
- The principle can be used to describe various phenomena, including light reflection and refraction, pendulum clocks, planetary orbits, and star orbits (26m32s).
- The variation of the action being 0 is a simple rule that unifies various fields of physics, and it was recognized by Euler as a significant development in the application of the principle of least action (26m43s).

## Another way to do mechanics (27m8s)

- The principle of least action provides an alternative method to solve mechanics problems, using energies and scalars instead of forces and vectors, in addition to the traditional method of using Newton's Second Law (27m8s).
- The principle of least action can be simplified using the Lagrangian, which is defined as T minus V, where T is the kinetic energy and V is the potential energy (27m44s).
- By using the Lagrangian, the principle of least action can be reduced to a differential equation, making it easier to solve mechanics problems (27m54s).
- To solve a mechanics problem using the Lagrangian approach, one only needs to write down the kinetic and potential energy and plug it into the Euler-Lagrange equation (28m1s).
- The Lagrangian approach is powerful and can be used to solve problems in multiple dimensions by solving the Euler-Lagrange equation for each coordinate (28m37s).
- The Lagrangian approach also allows for the use of non-traditional coordinate systems, such as polar coordinates, which can be more suitable for certain problems (28m45s).
- The Lagrangian approach can be used to solve complex problems, such as the double pendulum, which is difficult to solve using forces and vectors (29m4s).
- The principle of least action is sometimes referred to as the principle of stationary action, as it does not always result in a minimum, but rather a stationary point (29m35s).
- The principle of stationary action is a more accurate description of the principle, as it acknowledges that the resulting action may not always be a minimum (29m58s).

## A “spooky” breakthrough (30m19s)

- Action is a fundamental concept that extends beyond classical mechanics and played a key role in solving the UV catastrophe, a significant problem in atomic physics around the turn of the 20th century (30m28s).
- The breakthrough that led to the development of quantum theory involved action, not energy or force, which is a significant aspect of the principle of least action (30m40s).
- The story of the principle of least action is an example of how knowledge compounds over time through steady progress, leading to a complete change in our understanding of the world (31m2s).
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