leastaction

The principle of least action is a fundamental concept in physics that states that the path taken by a physical system between two configurations is the one that minimizes a quantity called the action. The action is a mathematical function that depends on the system's state and how it changes over time. It is often defined as the integral of the Lagrangian over time. The Lagrangian itself is typically the difference between the kinetic and potential energy of the system.

This principle can be applied to a wide range of physical phenomena, from the motion of a single particle to the behavior of fields. In classical mechanics, applying the principle of least action leads to the derivation of Newton's laws of motion. In electromagnetism, it is used to derive Maxwell's equations.

The principle of least action is a powerful variational principle. Instead of focusing on forces acting at each instant, it considers the entire trajectory of the system. The path that minimizes the action is called the extremal path. In many cases, this extremal path is indeed a minimum, but it could also be a maximum or a saddle point of the action.

The concept is closely related to Noether's theorem, which establishes a connection between symmetries of the action and conserved quantities. For example, the symmetry of the action with respect to time translations leads to the conservation of energy. The principle of least action provides a unified framework for understanding many of the fundamental laws of physics.