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Ioannis1404
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In classical mechanics to establish the Euler-Lagrange equations of motion of a particle we "minimize" the action, that is the integral of the Lagrangian, prescribing as the integral limits the initial and final positions of the particle. Usually, for a problem in mechanics we do not know the initial and final positions but the initial position q and initial velocity q'. How can we "minimize" the action under the conditions of given initial position and initial velocity (instead of initial and final positins) to get the Euler-Lagrange equations?