12.5 Hamilton’s principle, fields, and Noether

The calculus of variations reaches its purpose here. All of mechanics, and the field theories built on it, follow from a single variational statement: physical systems evolve along the path that makes an integral called the action stationary. Newton’s laws, conservation of energy and momentum, and the field equations of acoustics and electromagnetism are all Euler–Lagrange equations of one functional. This lesson assembles the pieces the chapter has built into that principle.

Hamilton’s principle

For a mechanical system described by a coordinate $q(t)$ , define the Lagrangian $L = T - U$ (kinetic minus potential energy) and the action

S[q] \;=\; \int_{t_1}^{t_2} L\big(q, \dot q, t\big)\; dt.

Hamilton’s principle states that the true motion between fixed endpoints $q(t_1)$ and $q(t_2)$ makes $S$ stationary, $\delta S = 0$ . This is precisely the variational problem of 12.1 with $x \to t$ , $y \to q$ , $F \to L$ , so its Euler–Lagrange equation is immediate:

\boxed{\;\frac{d}{dt}\!\left( \frac{\partial L}{\partial \dot q} \right) - \frac{\partial L}{\partial q} \;=\; 0.\;}

where

$S[q]$: the action — the functional made stationary J·s
$L = T - U$: the Lagrangian (kinetic minus potential energy) J
$q(t)$: generalised coordinate (the trajectory)
$\partial L/\partial \dot q$: the generalised momentum conjugate to q kg·m/s

▶ Euler–Lagrange for L = ½mq̇² − U(q) is Newton's law Derivation

Take the simplest mechanical Lagrangian, $L = \tfrac12 m\dot q^2 - U(q)$ . The two pieces of the Euler–Lagrange equation are

\frac{\partial L}{\partial \dot q} = m\dot q \quad\Longrightarrow\quad \frac{d}{dt}\frac{\partial L}{\partial \dot q} = m\ddot q, \qquad \frac{\partial L}{\partial q} = -\frac{dU}{dq}.

Euler–Lagrange, $\tfrac{d}{dt}(\partial L/\partial\dot q) - \partial L/\partial q = 0$ , therefore reads

m\ddot q + \frac{dU}{dq} = 0 \quad\Longrightarrow\quad m\ddot q = -\frac{dU}{dq} = F.

Newton’s second law — force equals mass times acceleration — recovered as the condition that the action is stationary. The Lagrangian $\tfrac12 m\dot q^2$ is the kinetic term of 12.1, and $-U$ the potential; their integral is the action the interactive below varies.

S = -3.121 J·s dS/dα = 2.961 ⟨T⟩ = 8.645 · ⟨U⟩ = 11.767 J

deformation α = 0.60 total time T = 1.0 s

Nudge the path off the true parabola and the action changes; at α = 0 the action is stationary (its slope vanishes). Nature's trajectory is the one that extremises ∫(T − U) dt — that is Hamilton's principle, and the Euler–Lagrange equation for it is Newton's second law.

The interactive throws a particle straight up and back under gravity. The dashed parabola is the true trajectory; deform it by $\alpha$ and watch the action change. At $\alpha = 0$ — the true path — the action curve is flat: $\delta S = 0$ . Every other path, higher or lower, changes $S$ . Nature’s trajectory is the stationary one, and its Euler–Lagrange equation is exactly $m\ddot q = -mg$ .

What the reformulation buys

Recasting $F = ma$ as $\delta S = 0$ buys concrete power beyond elegance (Physics 1.6 develops the mechanics in full):

Any coordinates. The Euler–Lagrange equation holds in whatever coordinates describe the system — angles, distances, normal modes — because a stationary action is stationary in every coordinate system. No need to resolve forces along awkward axes.
Constraints for free. A bead on a wire, a pendulum’s rigid rod: express the constraint in the coordinates and the constraint forces never appear, or appear as the multipliers of 12.4.
Several degrees of freedom. For coordinates $q_1, \ldots, q_n$ there is one Euler–Lagrange equation per coordinate, $\tfrac{d}{dt}(\partial L/\partial\dot q_i) - \partial L/\partial q_i = 0$ — the coupled equations of motion of the whole system, from one scalar $L$ .

Symmetry and conservation: Noether’s theorem

The two first integrals of 12.2 are the first glimpse of a general law. In the language of mechanics:

If $L$ does not depend on a coordinate $q_i$ (a cyclic coordinate), its conjugate momentum $p_i = \partial L/\partial\dot q_i$ is conserved. Translational symmetry $\Rightarrow$ conserved momentum.
If $L$ does not depend explicitly on time, the energy $H = \sum_i \dot q_i\, p_i - L$ is conserved (the Beltrami identity). Time-translation symmetry $\Rightarrow$ conserved energy.

Noether’s theorem is the statement that these are instances of one principle: every continuous symmetry of the action corresponds to a conserved quantity, and every conservation law arises from a symmetry. Rotational symmetry gives angular momentum; the gauge symmetry of electromagnetism gives conservation of charge. Conservation laws are not separate postulates but shadows of the sameness of physics under continuous change — a result proved once, at the level of the action, and inherited by every theory written variationally.

From coordinates to fields

The final generalisation replaces the coordinate $q(t)$ by a field $\phi(\mathbf r, t)$ defined at every point of space. The action integrates a Lagrangian density $\mathcal L$ over space and time, $S = \int dt\int d^3r\;\mathcal L(\phi, \partial_t\phi, \nabla\phi)$ , and stationarity gives the field Euler–Lagrange equation

\partial_t\!\left(\frac{\partial \mathcal L}{\partial(\partial_t\phi)}\right) + \nabla\cdot\!\left(\frac{\partial \mathcal L}{\partial(\nabla\phi)}\right) - \frac{\partial \mathcal L}{\partial\phi} = 0.

This is the equation applied in Sound 4.8 to the acoustic Lagrangian density to produce the wave equation, and it is the template for every classical field theory. Noether’s theorem carries over too, now yielding conserved currents — the acoustic energy density and intensity, the electromagnetic energy and Poynting flux — as the conserved quantities partnered to time- and space-translation symmetry of the field.

What we use this for

The calculus of variations is the mathematical foundation under a large fraction of the bookshelf:

Lagrangian mechanics (Physics 1.6) — the whole reformulation of dynamics, generalised coordinates, and normal modes rests on Hamilton’s principle.
The wave equation from an action (Sound 4.8) — route 4 to the acoustic wave equation, and the energy and momentum currents Noether hands back to Sound Ch 5.
Fermat’s principle and ray acoustics (Sound 7.2) — least-time optics and the geometric limit of wave propagation.
Geodesics and general relativity — free fall as the stationary path $\delta\int ds = 0$ in curved spacetime.
Elasticity, minimal surfaces, and equilibrium shapes — a bent beam, a soap film, a hanging chain: each an energy minimiser solved by Euler–Lagrange.

⏳ The history — Least action, from Maupertuis to Hamilton

The idea that nature minimises something is old and was long entangled with metaphysics. Pierre-Louis Maupertuis announced a principle of least action in 1744, convinced it revealed divine economy in the universe — a claim that drew ridicule from Voltaire and a priority dispute with Euler, who had the sounder mathematics. Lagrange put the mechanics on rigorous footing in the Mécanique analytique (1788). The modern, assumption-free form is due to William Rowan Hamilton, whose 1834–35 papers stated the principle as the stationarity (not necessarily minimum) of the action integral and unified optics and mechanics under it. Stripped of metaphysics, Hamilton’s principle became the organising statement of theoretical physics: not that nature is thrifty, but that its laws are the Euler–Lagrange equations of an action.

That closes the chapter, and with it the variational thread that runs from a bead on a wire to the field equations of acoustics. The calculus of variations is the one idea — extremise a functional — from which Newton’s law, Snell’s law, the catenary, the wave equation, and the conservation laws all descend.