Action Principle (Classical)

In the framework of modern physics development, starting from the action \( S \), physicists write down the action based on experiments, physical phenomena, constraints, etc. After writing down the action, they perform a variation \( \delta S \) (variation) and, under the condition of the principle of least action \( \delta S = 0 \), obtain the equations of motion that the physical system adheres to. Subsequently, physical quantities that satisfy the equations of motion are determined according to different physical systems. In classical mechanics, d'Alembert utilized the principle of virtual work to derive $$ {d \over dt}{\partial L \over \partial \dot{x}_i}-{\partial L \over \partial x_i}=0 $$ Later, Hamilton further clarified that the above equation satisfies $$ \int_a^b L(x_1,...,x_n,\dot{x_1},...,\dot{x_n},t)dt $$ which is the necessary result of applying the variational method to obtain the extremum \( \delta S = 0 \). In physics, the variational method is often used in the early development of new theories. Since physicists do not initially know the correct equations of motion, they guess the possible form of the action based on experimental results and physical experience. They then use the variational method to derive the equations of motion that describe the Lagrangian. Once the equations of motion are obtained, the variational method is no longer needed for subsequent problem-solving. For example, $$ \int_a^b L(x,\dot{x},t)dt $$ When applying variation, the Euler-Lagrange equation is obtained: $$ {d \over dt}{\partial L \over \partial \dot{x}}-{\partial L \over \partial x}=0 $$ Subsequent theoretical mechanics courses only need to solve the Euler-Lagrange equation. Thus, the variational method is not commonly covered in detail in textbooks related to mathematics, as it is only needed in the early stages of developing a theory. Therefore, the variational method will be briefly introduced here in a non-mathematically rigorous manner.

Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.