The principle of least action is a fundamental concept in physics that describes the behavior of systems over time. It states that the path followed by a physical system between two points in spacetime is the one for which the action is minimized or, more accurately, stationary. The action is defined as the integral of a Lagrangian function along the path taken by the system.
In classical mechanics, the principle of least action is derived from the principle of stationary action, which states that the true path taken by a system is the one for which the variation of the action with respect to small changes in the path is zero. This principle is a consequence of the underlying symmetries and invariances of the system.
Fundamental physics, such as quantum mechanics and quantum field theory, also incorporates the principle of least action but with some modifications. In quantum mechanics, the principle is generalized through the path integral formulation, which involves summing over all possible paths of a system rather than finding a single path of least action. Each path is assigned a phase factor based on its action, and the probability of observing a particular outcome is determined by the interference of these phases.
In the context of quantum field theory, the principle of least action is expressed through the variational principle of the action for fields. This principle leads to the equations of motion for the fields, such as the famous Schrödinger equation or the Klein-Gordon equation.
In summary, the principle of least action is a fundamental principle in physics that describes the behavior of physical systems. It is derived from the principle of stationary action and is applicable to both classical and quantum mechanical systems, providing a mathematical framework for understanding their dynamics.