§ DICTIONARY · CONCEPT
Stationary action
The precise formulation of least action: δS = 0 for every first-order variation of the path vanishing at the endpoints.
§ 01
Definition
Calling the principle 'least' action is a useful slogan but technically inaccurate. What the physical path actually satisfies is stationarity: if you perturb the path by any small, endpoint-preserving variation δq(t), the first-order change in S vanishes.
The second-order change decides whether the extremum is a minimum (usually), a saddle (occasionally, for long time intervals near a conjugate point), or, rarely, a maximum. What matters for physics is only the first-order condition, because that is what is equivalent — via the calculus of variations — to the Euler-Lagrange equation, and hence to Newton's second law.