Functionals and the space of paths

Plain-language version: Comparing paths

A functional is a rule that assigns one number to a whole function. The calculus of variations is the study of such rules, and Hamilton's principle is a statement about one of them. This page fixes the objects the rest of the mathematics track works with: the admissible set, the functional itself, and what it means for two paths to be close.

The admissible set

A true path and several comparison paths between fixed endpoints.

Fix two points $a<b$ on the real line and two values $y_a$, $y_b$. Consider the set

$𝒜=\{y\in C^2[a,b]:y(a)=y_a,y(b)=y_b\}\mathrm{(}\mathrm{1}\mathrm{)}$

of twice continuously differentiable functions joining the two endpoints. Call $𝒜$ the admissible set and each of its members an admissible path.

Two features of $𝒜$ matter and are easy to miss.

First, $𝒜$ is not a vector space. The sum of two admissible paths takes the value $2y_a$ at $a$, so it is not admissible. This is why the variation defined on the next page is built from a separate space of differences rather than from $𝒜$ itself.

Second, the smoothness class is a choice, not a fact. Taking $C^2$ makes the Euler-Lagrange equation a genuine second-order differential equation. Weaker classes admit solutions with corners, and the brachistochrone with a broken extremal is the standard reason to care. The Weierstrass theory on the Weierstrass excess function works in the wider class.

The space of variations

Set

$𝒱=\{\eta \in C^2[a,b]:\eta (a)=\eta (b)=0\}\mathrm{(}\mathrm{2}\mathrm{)}$

This is a vector space. Its members vanish at both endpoints, so for any admissible $y$ and any real $\epsilon$ the path

$y_{\epsilon }=y+\epsilon \eta \mathrm{(}\mathrm{3}\mathrm{)}$

is again admissible. Equation (3) is the whole device. It turns a question about a set of paths into a question about a single real variable $\epsilon$, and every derivation in this track runs through it.

The relation between $𝒜$ and $𝒱$ is that $𝒜$ is an affine set: pick any one admissible path as an origin and every other admissible path is that one plus a member of $𝒱$.

The functional

Let $L(x,y,y^{\prime })$ be a function of three real arguments, continuous with continuous partial derivatives up to second order. Define

$J[y]={\int }_a^{b}L(x,y(x),y^{\prime }(x))dx\mathrm{(}\mathrm{4}\mathrm{)}$

for $y\in 𝒜$. The map $J$ takes a path and returns a number. $L$ is the integrand, or, when the independent variable is time and the functional is action, the Lagrangian.

Note what the three slots of $L$ mean. Inside the integral, $y$ and $y^{\prime }$ are not independent — the second is the derivative of the first. But $L$ is defined as a function of three free arguments, and its partial derivatives $\partial L/\partial y$ and $\partial L/\partial y^{\prime }$ are taken with all three held independent. This distinction is the commonest source of error in the subject. When $\partial L/\partial y^{\prime }$ is written below, the slot is differentiated, not the function $y^{\prime }(x)$.

Distance between paths

"Nearby path" needs a metric, and there is more than one reasonable choice. Two are used throughout this track.

The weak norm is

$\Vert \eta {\Vert }_1=\underset{[a,b]}{max}|\eta (x)|+\underset{[a,b]}{max}|{\eta }^{\prime }(x)|\mathrm{(}\mathrm{5}\mathrm{)}$

and the strong norm is

$\Vert \eta {\Vert }_0=\underset{[a,b]}{max}|\eta (x)|\mathrm{(}\mathrm{6}\mathrm{)}$

A path is close to $y$ in the strong sense if it stays close in position. It is close in the weak sense only if it also stays close in slope.

Every $\Vert \cdot {\Vert }_1$-neighbourhood is contained in a $\Vert \cdot {\Vert }_0$-neighbourhood, so a strong neighbourhood is the larger set and a strong minimum is the stronger claim. The naming is unfortunate and standard; it is worth reading twice.

The distinction is not decoration. A path can minimise $J$ against every weakly nearby competitor and lose to a strongly nearby one that wanders close in position while oscillating wildly in slope. Sorting out which claim holds is the business of the Weierstrass excess function.

Extrema and stationarity

$y\in 𝒜$ is a weak local minimum of $J$ if there is $\delta >0$ with $J[y+\eta ]\ge J[y]$ for every $\eta \in 𝒱$ satisfying $\Vert \eta {\Vert }_1<\delta$. Replacing $\Vert \cdot {\Vert }_1$ by $\Vert \cdot {\Vert }_0$ gives a strong local minimum, and reversing the inequality gives the maxima.

Stationary is a different and weaker condition, defined on the next page. A stationary path need be neither a minimum nor a maximum. Hamilton was explicit about this from the start: reviewing what Lagrange and others had drawn from the vanishing of the variation, he called it the well known law of least, or as it might be better called, of stationary action.

The gap between "stationary" and "minimum" is the subject of the second variation and the Jacobi condition and conjugate points, and it is the reason this volume does not call the subject the principle of least action.

Notation

This volume writes $L$ for a general integrand and reserves $T-V$ for the mechanical Lagrangian, introduced on Central force and the Kepler problem. Hamilton's own $V$ is the characteristic function, an entirely different object, treated on Action as a function of its endpoints. The concordance on Notation tabulates the collision.

Next

the first variation differentiates (4) with respect to $\epsilon$ along the family (3).