Action

Remarks

Another standard quantity is the abbreviated action $S_0[q] = \int \sum_i p_i \, dq_i$, with $p_i = \partial L / \partial \dot q_i$, used in Maupertuis's principle. In physics the action is measured in joule-seconds, the unit of Planck's constant.

Example: a free particle of mass $m$ on the line, $L = \tfrac12 m\dot q^2$, moving from $q(0) = 0$ to $q(1) = a$. The path $q(t) = at$ has action $\int_0^1 \tfrac12 m a^2\, dt = \tfrac12 m a^2$. The path $q(t) = at^2$ has action $\int_0^1 \tfrac12 m (2at)^2\, dt = \tfrac23 m a^2$, which is larger.

Non-example: the Lagrangian $L$ itself is not the action; it is a function of one instant, while the action depends on a whole path.