The Closest We’ve Come to a Theory of Everything

A single simple rule underpins all of physics, from classical mechanics to electromagnetism, quantum theory, general relativity, and the fundamental particles.

• Every principle in physics can be replaced by this single rule • May explain the behavior of life itself • Unified approach across all scientific domains

The principle represents a departure from traditional differential equation thinking about the universe, suggesting a fundamentally different way of understanding how nature works.

The explanation starts with a simple problem about the fastest descent, revealing how this ancient puzzle leads to modern physics.

Find the ramp shape that allows a mass to slide from point A to point B in the shortest time possible.

• Common sense suggests the shortest path is a straight line • Galileo proposed an arc of a circle, faster than any polygon • The question remains: what shape provides the optimal balance?

In June 1696, Johann Bernoulli challenged the world's best mathematicians to solve the problem, initially giving six months, then extending the deadline to reach international competitors like Newton.

Newton received the challenge on January 29, 1697, after work. Despite being irritated, he solved it overnight, something that took Bernoulli two weeks to accomplish.

Hero of Alexandria discovered that light follows the shortest path in a single medium, resulting in equal angles of incidence and reflection.

When light moves between different media (like air to water), it bends in a peculiar way without following the shortest path, following Snell's Law instead.

• Pierre Fermat hypothesized that light minimizes travel time, not distance • Proved that Snell's Law emerges from the principle of least time • Called this calculation 'the most extraordinary calculation' of his life

This was the first demonstration that nature obeys an optimization principle, showing that light takes the shortest possible time between two points.

Bernoulli converted the mechanics problem into an optics problem by imagining light moving through progressively less dense layers with increasing speed.

• Applied Snell's Law at each interface with continuously changing light speed • Recognized the resulting equation as describing a cycloid curve • Found that motion follows the arc traced by a point on a rolling wheel

The fastest descent path follows a brachistochrone (shortest time) curve, which is also a tautochrone (same time) curve, reaching the endpoint in equal time regardless of release point.

Bernoulli demonstrated that optical and mechanical problems from entirely separate mathematical fields share the same character, solving two important problems simultaneously.

Pierre Louis de Maupertuis proposed 'action' as a more fundamental quantity than time, defined as mass times velocity times distance.

• Increases with greater distance traveled • Increases with greater speed • Increases with greater mass • Total action is the sum for each segment of a journey

Maupertuis claimed that nature minimizes action, stating 'This action is the true expense of Nature, which she manages to make as small as possible.'

• Attacked and ridiculed by contemporaries including Samuel König and Voltaire • Accused of plagiarism and bad physics • Voltaire wrote a 32-page pamphlet mocking him • Caused extreme stress to Maupertuis near the end of his life

Leonhard Euler defended Maupertuis' principle and replaced the sum with an integral to handle continuously changing speed or direction.

• Total energy must be conserved • Energy must be the same for all paths being compared • These conditions were necessary insights Maupertuis had not realized

Euler developed a clunky but effective method to find paths that minimize action by varying every possible point along the path, which required mathematical innovation.

Euler was characterized as an astonishingly powerful mathematician and apparently good person, generous and empathetic, making his explanations understandable to others.

Joseph-Louis Lagrange was a shy, mostly self-taught 19-year-old working at the forefront of mathematics and collaborating with Euler's methods in 1754.

Euler praised Lagrange's work, saying he had 'extolled the theory to the highest summit of perfection,' which caused Euler 'the greatest joy.'

About five years after Maupertuis' death, Lagrange succeeded in providing a general proof of the principle of least action, completing the theoretical foundation.

Both Euler and Lagrange were proponents of the principle of least action and worked to provide rigorous mathematical foundations for this elegant concept.

• Consider infinite possible paths between two points • Find the path with least action by analogy to finding function minima • Assume a true path and add tiny variations to detect changes in action

At the path of least action, small variations produce no first-order change in action, similar to how a function at its minimum has a horizontal derivative.

The difference between trial path action and true path action equals zero to first order, allowing mathematical determination of the actual path taken by nature.

This approach works for all optimization problems, providing a systematic way to solve problems in mechanics, optics, and other fields of physics.

• Starting from Maupertuis' action as mass times velocity times distance • Euler converted this to an integral over distance • Replaced this with twice the kinetic energy integrated over time

Using energy conservation (total energy equals kinetic plus potential), the action was rewritten as kinetic minus potential energy integrated over time.

William Rowan Hamilton in 1834 wrote the principle as the integral of the Lagrangian (T minus V) over time, establishing what became known as Hamilton's Principle.

• Maupertuis' principle: integral over space with constant energy • Hamilton's Principle: integral over time with constant time between paths • Hamilton's version indicates how objects move, not just path shape

Using a thrown ball, consider all possible trajectories between start and end points and times, then find which one satisfies the principle of least action.

• Express kinetic and potential energy for the true path and trial paths • Apply variation method to find where action difference is zero • Use integration by parts to simplify the resulting equation

The path satisfying the principle of least action obeys the equation F equals ma, Newton's Second Law, showing these seemingly different approaches are equivalent.

Fermat's principle of least time is just a special case of the principle of least action, unifying light reflection, refraction, pendulums, planetary orbits, and stellar motion under one rule.

Euler and Lagrange showed how to make solving mechanics problems vastly simpler by using the Lagrangian instead of forces and vectors.

• Write down kinetic and potential energy • Plug values into the Euler-Lagrange differential equation • Solve to get equations of motion without calculating forces

• Works in multiple dimensions by solving the equation for each coordinate • Works with non-standard coordinate systems like polar or spherical coordinates • Simplifies complex problems like the double pendulum significantly

Anyone can crank through this method to get correct equations of motion without being a skilled physicist, making the approach democratizing and extremely powerful for problem-solving.

The principle is often called 'least action,' but it's more accurately called the 'principle of stationary action.'

• Setting a derivative to zero doesn't guarantee a minimum • The action can be a minimum, maximum, or saddle point • The condition is that the action is stationary (first-order variation is zero)

Very often the stationary point is a true minimum, but not always, which is why the more precise terminology is important.

Despite the terminology issue, the principle correctly describes the laws of motion whether or not the action is actually minimal.

Around the turn of the 20th century, action appeared as the key component in solving the UV catastrophe, one of atomic physics' biggest problems.

The breakthrough that led toward quantum theory brought action to prominence, not energy or force, suggesting action is the fundamental quantity.

It is spooky that this principle starting with a simple mechanics problem leads directly to quantum theory, hinting at something deep about reality's structure.

The full story of action's role in quantum mechanics and beyond will be explored in a separate video, continuing this journey of understanding nature's fundamental principles.

The principle of least action exemplifies how knowledge compounds and grows through steady progress one step at a time until it changes how we see the world completely.

• Learning a little every day compounds over time • Makes you smarter and a better problem-solver • Can be started right now through educational platforms

Major scientific discoveries follow the same pattern as personal learning: steady accumulation of knowledge builds toward transformative understanding.

Brilliant offers bite-sized lessons in physics, calculus, math, and programming that help build intuition and real understanding through hands-on problem-solving.