A note on “least” action
Preview note.
“I believe in the principle of least action.”
The familiar name is the principle of least action. More precisely, the physical path makes the action stationary:
\[\delta S = 0.\]For a trajectory \(q(t)\), the action is
\[S[q] = \int_{t_1}^{t_2} L\bigl(q(t), \dot q(t), t\bigr)\,\mathrm{d}t.\]A vanishing first variation does not necessarily imply a minimum. “Stationary action” is the more general statement — and a useful reminder to look beyond the name.