Here’s a (relatively) easy-to-understand explanation of the closest thing we have to a theory of everything in physics, based on a single simple rule called the principle of least action:
One Rule to Explain Everything in Physics
Physicists have found that almost everything in the universe — from how planets move to how light bends, from classical physics to quantum theory — can be explained by just one fundamental idea: nature always “chooses” the path that makes a certain quantity called “action” stationary (usually minimized).
How Did We Get Here? The Story Behind the Principle of Least Action
1. The Problem of Fastest Descent (Brachistochrone Problem):
Imagine you want to slide a ball down a ramp from point A to point B as fast as possible. Simple guess: a straight line is the fastest. But actually, a curved path lets the ball pick up speed sooner and arrive faster. The question was: what exact shape is the fastest?
2. Johann Bernoulli’s Challenge:
Bernoulli challenged famous mathematicians, including Newton, to solve this. After some clever work involving light behavior, Bernoulli found the solution: the fastest path is a cycloid curve (like a point on a rolling wheel).
3. What Does Light Have to Do with It? Fermat’s Principle of Least Time:
Fermat had earlier suggested that light travels the route that takes the least time, even if it’s not the shortest distance (like when it bends passing from air to water). This idea of nature selecting the “best” path underlies the brachistochrone problem’s solution too.
4. Maupertuis and the Idea of “Action”:
Maupertuis took it further by guessing there is a deeper rule: nature minimizes a quantity called action, which combines mass, velocity, and distance traveled. For particles and light alike, nature’s behavior corresponds to minimizing this action.
Though revolutionary, Maupertuis’ idea was initially not well-received because it wasn’t mathematically rigorous or clearly justified.
5. Euler and Lagrange Bring Mathematical Power:
Euler improved the math by turning sums into integrals, handling continuous paths instead of discrete steps, and showed this “least action” principle could describe planetary orbits too.
Lagrange later gave a general proof of the principle, connecting it to equations of motion and Newton’s laws.
What Is Action and Why Is This Principle Special?
– Action is an integral over time of the difference between kinetic energy (energy of motion) and potential energy (energy from position or forces), called the Lagrangian.
– The principle states that the actual path taken by a system makes the action “stationary”—meaning if you were to wiggle the path a little, action wouldn’t change to first order.
– This condition turns out to be equivalent to known physics laws, like Newton’s Second Law (F=ma), and works for light, particles, planets, and even fields.
Why Not Just Use Newton’s Laws?
Using Newton’s Law directly involves forces and vectors and can be complicated, especially in complex systems (like double pendulums). The principle of least action, through the Lagrangian, uses energies (scalars) and integrals that work with any coordinate system and can simplify many physics problems.
How Can The Principle of Least Action Be Applied?
An example demonstrating the principle of least action is projectile motion, such as throwing a ball into the air.
Instead of using forces directly (Newton’s Second Law), you use the principle of least action by focusing only on the ball’s kinetic energy (energy of motion) and potential energy (energy due to height).
The action is defined as the integral over time of the difference between kinetic and potential energy (called the Lagrangian).
Setup
- y(t) = height of the ball at time t (meters, m)
- m = mass of the ball (kilograms, kg)
- g = acceleration due to gravity (9.8 m/s²)
Energy Definitions
Kinetic energy:
T = (1/2) m (dy/dt)²
where dy/dt is the velocity (meters per second, m/s).
Potential energy:
V = m g y
Units: joules (J).
Lagrangian
L = T – V = (1/2) m (dy/dt)² – m g y
Euler-Lagrange Equation
The path that minimizes the action satisfies:
∂L/∂y – d/dt (∂L/∂(dy/dt)) = 0
Calculating each term:
- ∂L/∂y = – m g
- ∂L/∂(dy/dt) = m (dy/dt)
- d/dt (∂L/∂(dy/dt)) = m (d²y/dt²)
Plugging in gives:
– m g – m (d²y/dt²) = 0
⇒ m (d²y/dt²) = – m g
Example with Numbers
For a ball of mass 0.5 kg, the equation becomes:
0.5 (d²y/dt²) = – 0.5 × 9.8
⇒ (d²y/dt²) = -9.8 m/s²
Solving this differential equation yields the parabolic trajectory:
y(t) = y₀ + v₀ t – (1/2) g t²
This example shows how the principle of least action provides the trajectory without explicitly calculating forces.
It’s More Properly the Principle of Stationary Action and Not Least Action
In the Principle of Stationary Action, “balanced” means that if you imagine slightly changing the path a system takes, the overall action—an integral measuring a compromise between kinetic energy (energy of movement) and potential energy (energy of position)—does not change to first order; the increases and decreases in these energies counteract each other perfectly. This “balancing” means the system’s actual trajectory is exquisitely tuned so that small deviations neither lower nor raise the action immediately, placing it at a stationary point (a minimum, maximum, or saddle). It’s like walking along the top of a ridge where any small step sideways doesn’t immediately make you go higher or lower—your path is optimally poised, not necessarily minimizing effort in the everyday sense, but minimizing or balancing the integral “cost” of the entire journey according to physical laws.
“Change to first order” means looking at the very first, immediate effect of a small variation. Imagine you slightly adjust the path a system takes—if the action (a calculated quantity) changes only by a very tiny amount that is proportional to how big your adjustment is, that’s a first-order change. In the Principle of Stationary Action, the actual path is where this first-order change is exactly zero, meaning any small shift doesn’t cause an immediate increase or decrease in action. This is why the action is called “stationary” — it’s like being at a flat point (a peak, trough, or saddle) where the slope right at that point is zero, balancing increases and decreases perfectly so no tiny move makes it better or worse right away.
In other words, the universe runs on the Principle of Stationary Action, which means that out of all possible paths a system could take between two points, it follows the one where the total action—a measure combining kinetic and potential energy over time—is perfectly balanced so that tiny changes in the path don’t cause any immediate increase or decrease in this action. This balance, or “stationarity,” means the system’s path is optimally poised at a point where the first (immediate) change in action is zero, like walking on a flat ridge where small steps don’t raise or lower you. Unlike the simple idea that “a body at rest remains at rest,” this principle governs how systems evolve dynamically in time by selecting paths that satisfy this subtle, stable equilibrium condition.
Quantum Theory and Beyond
Surprisingly, action plays a central role in quantum mechanics and modern physics, hinting that this principle is even more fundamental than just classical motion. It underpins attempts to unify all forces and particles under one “theory of everything.”
Application to Life
To live the best possible life by the principle of least action, embrace simplicity and efficiency in your choices—seek paths and habits that require the least wasted effort while still achieving your goals, much like nature selects the path of least action to minimize “cost” in its processes. This means prioritizing what truly matters, focusing energy on meaningful actions, and avoiding unnecessary complications or distractions. By aligning your decisions to minimize resistance and maximize flow, you create a life that naturally follows the most harmonious and fulfilling trajectory, just as a ball thrown in the air follows the path that requires the least “action” to reach its destination.
The principle of least action does not mean being as lazy as possible. Instead, it means nature—and by analogy, you—seek the most harmonious, balanced, and effective path to achieve your goals, minimizing unnecessary effort or detours but not avoiding meaningful action. It’s about optimizing your energy and choices so you move efficiently and purposefully, like how a thrown ball follows the smoothest and most elegant path to its destination, not the easiest or laziest one. So living by this principle encourages thoughtful, deliberate, and well-directed effort rather than laziness.
In Summary
– Nature follows a simple overarching rule: it picks the path where the action is stationary (usually minimal).
– This idea connects widely different phenomena — motion of planets, paths of light, behavior of particles — into one framework.
– Developed over centuries, it’s the closest we have to a universal law describing everything in physics.
If you think of physics as a huge puzzle, the principle of least action is one of the key pieces that fits many parts together — providing a deep, elegant way to understand the universe with a single principle.