User:Johnjbarton/sandbox/action
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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.[1] For a trajectory of a baseball moving in the air on Earth the action is defined between two points in time, and as the kinetic energy minus the potential energy, integrated over time.[2]
The action balances kinetic against potential energy.[2] The kinetic energy of a baseball of mass is where the velocity of the ball is written as the derivative of its position on the trajectory; the potential energy is where is the gravitational constant. Then the action between and is
The action value depends upon the trajectory taken by the baseball through . 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.[2]