Hamilton's principle

Remarks

This is the case of mechanics of the stationary-action principle. For the proof, the first variation of $S$ in a direction $h$ with $h(t_0) = h(t_1) = 0$ is $\int_{t_0}^{t_1} \sum_i \left( m_i \dot r_i \cdot \dot h_i - \nabla_{r_i} V \cdot h_i \right) dt$. An integration by parts turns it into $-\int_{t_0}^{t_1} \sum_i \left( m_i \ddot r_i + \nabla_{r_i} V \right) \cdot h_i \, dt$, which vanishes for all such $h$ exactly when Newton's equations hold, by the fundamental lemma of the calculus of variations. So Newton's equations are the Euler–Lagrange equations of $L = T - V$. Example: a body falling near the Earth's surface, with $V(r) = m g z$, gives $\ddot z = -g$ and $\ddot x = \ddot y = 0$. The endpoint times are fixed; Maupertuis's principle instead fixes the energy and lets the time vary. For a system with ideal holonomic constraints, the same equivalence holds in generalized coordinates on the constraint manifold, and the constraint forces drop out. William Rowan Hamilton formulated the principle in 1834 and 1835.

Going deeper

For a student

The principle makes mechanics independent of coordinates: $T - V$ can be written in polar, spherical or any other coordinates, and the Euler–Lagrange equations in those coordinates are the equations of motion. It is the starting point of Lagrangian and Hamiltonian mechanics, and through the Legendre transform it leads to Hamilton's canonical equations. It also connects mechanics with the calculus of variations, so that tools such as the Jacobi condition decide when the action is in fact a minimum.