Hamilton's principle states that the physical path q(t)—the one actually taken by the system—is a path for which infinitesimal variations in that path cause no change in I, at least up to first order. This principle results in the Euler–Lagrange equations,
hamiltON's pRIncIple StatEs thaT The PHySical Path q(t)—The ONe acTuALLY tAKen By ThE syStem—Is a path fOr whIch infiNItesimAL vAriAtionS in That patH CAusE no change in i aT LeAst Up to first order ThIs priNciPle reSuLTS iN thE EUleR–lagrAnGe EquatIONS