Calculus of variations

Extremising functionals — the Euler–Lagrange equation and the principle of least action.

Ordinary calculus finds the number $x$ that minimises a function $f(x)$: set $f'(x) = 0$ and solve. The calculus of variations asks the harder and deeper question — find the function $y(x)$ that minimises a quantity depending on the whole function at once. What shape does a hanging chain take? Along what path does a bead slide down fastest? What curve encloses the most area for a given perimeter? Each answer is a function selected from an infinite-dimensional space of candidates by a single stationarity condition, and that condition — the Euler–Lagrange equation — turns every such problem into a differential equation.

This is the mathematics beneath the variational formulation of physics. Fermat’s principle (light takes the fastest path), Hamilton’s principle (a system follows the path of stationary action), the shape of soap films, the geodesics of general relativity, and the field equations of acoustics and electromagnetism are all Euler–Lagrange equations for an appropriate functional. The fourth route to the wave equation and the whole of Lagrangian mechanics rest on the machinery this chapter builds. Get the variational idea and a large part of theoretical physics stops being a list of equations and becomes one principle applied over and over.

• 12.1 Functionals and the first variation — what a functional is; extremising over a space of functions; the perturbation $y + \varepsilon\eta$ and the first variation $\delta J$; the stationarity condition $\delta J = 0$.
• 12.2 The Euler–Lagrange equation — the fundamental lemma; deriving $\dfrac{\partial F}{\partial y} - \dfrac{d}{dx}\dfrac{\partial F}{\partial y'} = 0$ from $\delta J = 0$; the shortest path as the first worked example.
• 12.3 Classic problems — the brachistochrone and its cycloid, the catenary, geodesics on a surface, and Fermat’s principle for light.
• 12.4 Constraints, Lagrange multipliers, and isoperimetric problems — extremising subject to a constraint; the multiplier method; Dido’s problem and the isoperimetric inequality.
• 12.5 Hamilton’s principle, fields, and Noether — action, the Lagrangian, Euler–Lagrange as Newton’s law; functionals of several functions and of fields; Noether’s theorem tying symmetry to conservation.

The chapter extends single-variable calculus and produces, as its Euler–Lagrange equations, the ODEs and PDEs the rest of the bookshelf solves.