Hamilton's Principle - Mathematical Formulation

Mathematical Formulation

Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q₁, q₂, ..., q_N) between two specified states q₁ = q(t₁) and q₂ = q(t₂) at two specified times t₁ and t₂ is a stationary point (a point where the variation is zero), of the action functional

$\mathcal{S} \ \stackrel{\mathrm{def}}{=}\ \int_{t_1}^{t_2} L(\mathbf{q}(t),\dot{\mathbf{q}}(t),t)\, dt$

where is the Lagrangian function for the system. In other words, any first-order perturbation of the true evolution results in (at most) second-order changes in . The action is a functional, i.e., something that takes as its input a function and returns a single number, a scalar. In terms of functional analysis, Hamilton's principle states that the true evolution of a physical system is a solution of the functional equation

Hamilton's principle

$\frac{\delta \mathcal{S}}{\delta \mathbf{q}(t)}=0$

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