User:Johnjbarton/sandbox/action

Simple example
Introductory physics often begins with Newton's laws of motion, relating force and motion; action is part of a completely equivalent alternative approach with practical and educational advantages. For a trajectory of a baseball moving in the air on Earth the action is defined between two points in time, $$t_1$$ and $$t_2$$ as the kinetic energy minus the potential energy, integrated over time.
 * $$S = \int_{t_1}^{t_2} \left( KE(t) - PE(t)\right) dt$$

The action balances kinetic against potential energy. The kinetic energy of a baseball of mass $$m$$ is $$(1/2)m(dx/dt)^2$$ where the velocity of the ball is written as the derivative of its position on the trajectory; the potential energy is $$mgx$$ where $$g$$ is the gravitational constant. Then the action between $$t_1$$ and $$t_2$$ is
 * $$S = \int_{t_1}^{t_2} \left(\frac{1}{2}m \left( \frac{dx(t)}{dt}\right)^2 - mg x(t) \right) dt$$

The action value depends upon the trajectory taken by the baseball through $$x(t)$$. This makes the action an input to the powerful stationary-action principle for classical and for quantum mechanics. Newton's equations of motion for the baseball can be derived from the action using the stationary-action principle, but the advantages of action-based mechanics only begin to appear in cases where the Newton's laws are difficult to apply. Replace the baseball by an electron: classical mechanics fails but stationary action continues to work.

Planck's quantum of action
Planck's constant, written as $$h$$ or $$\hbar$$ when including a factor of $$1/2\pi$$, is also called the quantum of action. Like action, this constant has unit of energy times time. It figures in all significant quantum equations, like the uncertainty principle and the de Broglie wavelength. Whenever the value of the action approaches Planck's constant, quantum effects are significant.