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Action (physics)
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=== Simple example === For a trajectory of a ball moving in the air on Earth the '''action''' is defined between two points in time, <math>t_1</math> and <math>t_2</math> as the kinetic energy (KE) minus the potential energy (PE), integrated over time.<ref name=FeynmanII>{{Cite web |title=The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action |url=https://www.feynmanlectures.caltech.edu/II_19.html |access-date=2023-11-03 |website=www.feynmanlectures.caltech.edu}}</ref> :<math>S = \int_{t_1}^{t_2} \left( KE(t) - PE(t)\right) dt</math> The action balances kinetic against potential energy.<ref name=FeynmanII/> The kinetic energy of a ball of mass <math>m</math> is <math>(1/2)mv^2</math> where <math>v</math> is the velocity of the ball; the potential energy is <math>mgx</math> where <math>g</math> is the acceleration due to gravity. Then the action between <math>t_1</math> and <math>t_2</math> is :<math>S = \int_{t_1}^{t_2} \left(\frac{1}{2}m v^2(t) - mg x(t) \right) dt</math> The action value depends upon the trajectory taken by the ball through <math>x(t)</math> and <math>v(t)</math>. This makes the action an input to the powerful [[stationary-action principle]] for [[classical mechanics|classical]] and for [[quantum mechanics]]. Newton's equations of motion for the ball 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 Newton's laws are difficult to apply. Replace the ball with an electron: classical mechanics fails but stationary action continues to work.<ref name=FeynmanII/> The energy difference in the simple action definition, kinetic minus potential energy, is generalized and called [[Lagrangian (physics)#The Lagrangian|the Lagrangian]] for more complex cases.
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