Hamilton's principle

In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.

Mathematical formulation Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q1, q2, ..., qN) 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

S [ q ] = d e f ∫ t 1 t 2 L ( q ( t ) , q ˙ ( t ) , t ) d t {\displaystyle {\mathcal {S}}[\mathbf {q} ]\ {\stackrel {\mathrm {def} }{=}}\ \int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt}

where L ( q , q ˙ , t ) {\displaystyle L(\mathbf {q} ,{\dot {\mathbf {q} }},t)} 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 S {\displaystyle {\mathcal {S}}} . The action S {\displaystyle {\mathcal {S}}} 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

That is, the system takes a path in configuration space for which the action is stationary, with fixed boundary conditions at the beginning and the end of the path.

Derivation from Newton's laws of motion Although Hamilton's principle can be viewed as a postulate effectively replacing Newton's laws of motion, it can also be derived from Newton's laws. Starting with d'Alembert's principle:

∑ i ( F i − m i a i ) ⋅ δ r i = 0. {\displaystyle \sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}=0.}

Where i {\displaystyle i} is the index of a mass, F i {\displaystyle \mathbf {F} _{i}} is applied force (excluding constraint forces), a i {\displaystyle {\textbf {a}}_{i}} is acceleration of the mass, and δ r i {\displaystyle \delta \mathbf {r} _{i}} is the virtual displacement of the i {\displaystyle i} -th mass consistent with the constraints. The above equation holds true for all times t {\displaystyle t} so a definite integral with respect to t {\displaystyle t} must also be 0:

∫ t 1 t 2 ∑ i ( F i − m i r ¨ i ) ⋅ δ r i d t = 0. {\displaystyle \int _{t_{1}}^{t_{2}}\sum _{i}(\mathbf {F} _{i}-m_{i}{\ddot {\mathbf {r} }}_{i})\cdot \delta \mathbf {r} _{i}dt=0.}

δ r i {\displaystyle \delta \mathbf {r} _{i}} , the sum, and the integral can all be distributed:

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