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Perhaps the basic idea of modern physics is the principle of action (action principle). Any field of physics is governed by this rule. We will begin a journey, starting with the scene of light passing through water and ending with a discussion of quantum mechanics.

Light and Fermat's principle consider that a beam of light starts from point An and crosses the water to point B. What is the path of this beam of light? We can assume that it moves in a straight line to minimize the distance. In reality, however, light does not travel in this way. Instead, it is refracted at the boundary between air and water.

But why? In 1662, Pierre de Fermat found the answer. He explained that the path of light from one point to another minimizes its travel time.

Fermat's principle shows that the path of light between two points always minimizes its propagation time.

According to this principle and the different speeds of light in different media, we can derive Snell's law of refraction mathematically:

The principle of Snell's law of refraction We have seen before that light always moves along the shortest path. In 1788, Joseph Louis Lagrange expanded the idea. He suggested that any particle moving in space would minimize its action.

The principle of action points out that particles in motion always move along the spatial path that minimizes their range of action.

But what exactly is its "function"? In physics, action is a scalar that describes the changes of physical systems over time. The action is important because the equation of motion of the system can be derived from the principle of static action (the principle of minimum action). More formally, action is a mathematical functional that takes the trajectory (path) of the system as a parameter and a real number as a result. Usually, different paths have different actions. The role of mechanical system S, for example, when a ball is thrown into the air, has such a precise mathematical definition.

First of all, L is called the Lagrangian of the system, which is the difference between the kinetic energy T and the potential energy V.

The Lagrangian is the difference between the kinetic energy and the potential energy of the system. We write it as L (t) to show that it is a function of time. In other words, when an object moves in space, it always changes with time. Since the action is an integral of time, it is the area surrounded by the curve L (t) between time t = tweak 1 and t = taper 2.

Lagrangian has no physical meaning in reality. It is only useful in mathematics.

How do we find a path with the smallest S in space? The results show that the path of minimum action also satisfies an equation called Euler-Lagrange equation.

The Euler-Lagrange equation doesn't have to worry too much about mathematical details. We only need to remember to replace the Lagrangian of the system into the Euler-Lagrange equation to get its equation of motion. These equations of motion define the path to minimize the action.

Going back to the example of throwing the ball, its Lagrangian (the kinetic energy of the ball minus its gravitational potential energy) is inserted into the Euler-Lagrange equation and its equation of motion is obtained. These equations of motion describe the parabolic path, assuming that there is no air resistance.

The infinite quantum path of the ball emitted at the initial angle of theta, but what happens when these objects are very small? For example, a single electron, in these cases, we must turn to quantum mechanics. In 1948, Richard Feynman defined quantum mechanics as a path integral using the principle of action.

The Feynman path integral formula of quantum mechanics we already know that a particle in motion has a unique path that minimizes its "action" from one point in space to another. In addition, this means that we know the exact position of the particles in space at any given time. However, this is impossible in quantum mechanics. In the quantum world, we cannot know the exact path of particles from A to B. Instead, we should ask

What is the probability of particles from A to B?

The answer to this question is obtained by calculating the path integral of the quantum system. In this process, we need to consider the infinitely possible path from A to B.

A small part of the infinitely possible path of a quantum particle from point A to point B.

Each path is assigned a probability amplitude. The final probability of particles from A to B is obtained by adding up the probabilistic amplitudes of each path, no matter how absurd the path is.

The principle of action comes into play when we realize that a more reasonable path has a smaller effect S and a larger probability amplitude. On the contrary, unreasonable paths have greater effects and smaller amplitudes. Since most of the possible paths are unreasonable, their contribution to the total probability is very small. So,

The most reasonable path with the minimum action S has the greatest contribution to the probability of particles from A to B.

This is the basic idea of Feynman's path integral.

The (minimum) principle of action can be used to derive Newtonian, Lagrange and Hamiltonian equations of motion, and even general relativity. This principle is still at the center of modern physics and mathematics, and has been applied to thermodynamics, fluid mechanics, relativity, quantum mechanics, particle physics and string theory.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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