The principle of least action is a fundamental principle in classical physics that governs the behavior of physical systems. It states that the path taken by a system between two points in spacetime is the one that minimizes the action.
The action, denoted by S, is a quantity defined in terms of the Lagrangian of the system. The Lagrangian, denoted by L, is a function that describes the difference between the kinetic and potential energies of the system. Mathematically, the action is given by the integral of the Lagrangian over time:
S = ∫ L dt
According to the principle of least action, the actual path taken by a system is the one that makes the action stationary. In other words, the variation of the action with respect to the path is zero. This principle is known as Hamilton's principle, after the physicist William Rowan Hamilton.
The principle of least action is a fundamental postulate of classical mechanics and does not have a deeper explanation within classical physics itself. It is considered a foundational principle from which the equations of motion, such as Newton's laws or the equations of motion in Hamiltonian mechanics, can be derived. It provides a concise and elegant formulation for understanding the dynamics of physical systems.
In the framework of quantum mechanics, the principle of least action is generalized to the principle of stationary action. In quantum physics, particles are described by wave functions, and the action is replaced by a quantity called the phase. The principle of stationary action, or the stationary phase principle, states that the actual path taken by a particle is the one that makes the phase stationary. This principle is a key concept in the formulation of quantum mechanics, particularly in the path integral formulation.
It is important to note that the principle of least action is a classical approximation and does not directly apply to all areas of modern physics, such as quantum field theory or general relativity. In those areas, different principles and mathematical frameworks are used to describe the behavior of physical systems.