Science Equation             Name Action       Equation S = ∫ t 1 t 2 L ⁢ d ⁢ t       Definition of terms S = action, L = T - V = kinetic energy minus potential energy (also known as the Lagrangian), t = time       Comments    This equation defines the action from the Lagrangian. Hamilton's principle of least action states that for a dynamical system, the action integral S is stationary under arbitrary small variations in position, which vanish at times t1 and t2 . Another way of saying this is that the action is extremised, that is at a maximum or minimum.    This principle can be made the starting point for the whole of classical mechanics. For a Lagrangian that depends on only position, velocity and time, the principle of least action leads to the equation of motion  ∂ L ∂ x - ⅆ ⅆ t ( ∂ L ∂ x ) = 0  also known as the Euler-Lagrange equation. Consider a 1 dimensional harmonic oscillator. Its Lagrangian is L = T - V = 1 2 m ⁢ x 2 - 1 2 k ⁢ x 2 . From the Euler-Lagrange equation, we obtain the equation of motion as   m x ¯ + k ⁢ x = 0 ,  the known equation for a harmonic oscillator.    The action used to be defined as twice the integral of the kinetic energy over a time interval. This definition is obsolete and should no longer be used.       References Classical Mechanics , T W B Kibble, Longman, 3rd edition, 1985         [ H O M E ] [ S C I E N C E ] [ M A T H S ] [ A S T R O N O M Y ] Download | Contact | About

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