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principle of stationary action

A general integral variational principle of classical mechanics (cf. Variational principles of classical mechanics), established by W. Hamilton  for holonomic systems restricted by ideal stationary constraints, and generalized by M.V. Ostrogradski  to non-stationary geometrical constraints. According to this principle, in a real motion of the system acted upon by potential forces,

$$S=\int\limits_{t_0}^{t_1}(T-U)dt=\int\limits_{t_0}^{t_1}Ldt$$

has a stationary value as compared with near, kinetically-possible, motions, with initial and final positions of the system and times of motion identical with those for the real motion. Here, $T$ is the kinetic energy, $U$ is the potential energy and $L=T-U$ is the Lagrange function of the system. In certain cases the true motion corresponds not only to a stationary point of the functional $S$, but corresponds to its smallest value. For this reason the Hamilton–Ostrogradski principle is sometimes called the principle of least action. In the case of non-potential active forces $F_v$ the condition of stationary action, $\delta S=0$, is replaced by the condition

$$\int\limits_{t_0}^{t_1}\left(\delta T+\sum_vF_v\cdot\delta r_v\right)dt=0.$$

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