The first variation

Plain-language version: Comparing paths

The first variation of a functional is the derivative of that functional along a one-parameter family of paths, evaluated where the family passes through the path of interest. It is the exact analogue of the first derivative in ordinary calculus, and it plays the same role: it vanishes at an extremum, and its vanishing is necessary but not sufficient.

The variation defined

A true path and several comparison paths between fixed endpoints.

Take $J$, $𝒜$ and $𝒱$ as on Functionals and the space of paths. Fix $y\in 𝒜$ and $\eta \in 𝒱$, and define the real function

$\Phi (\epsilon )=J[y+\epsilon \eta ]={\int }_a^{b}L(x,y+\epsilon \eta ,y^{\prime }+\epsilon {\eta }^{\prime })dx\mathrm{(}\mathrm{1}\mathrm{)}$

$\Phi$ is an ordinary function of one real variable. The first variation of $J$ at $y$ in the direction $\eta$ is

$\delta J[y;\eta ]={\Phi }^{\prime }(0)={\frac{d}{d\epsilon }J[y+\epsilon \eta ]|}_{\epsilon =0}\mathrm{(}\mathrm{2}\mathrm{)}$

$y$ is stationary for $J$ if $\delta J[y;\eta ]=0$ for every $\eta \in 𝒱$.

The definition is a reduction, and that is its whole value. A question about an infinite-dimensional set of competing paths becomes a question about the derivative of one real function at one point.

Computing it

A one-parameter family of paths through the true path.

Differentiate (1) under the integral sign. The hypotheses on $L$ from the previous page license this: $L$ and its first partials are continuous on a compact set, so the difference quotients converge uniformly.

${\Phi }^{\prime }(\epsilon )={\int }_a^b[\frac{\partial L}{\partial y}\eta +\frac{\partial L}{\partial y^{\prime }}{\eta }^{\prime }]dx\mathrm{(}\mathrm{3}\mathrm{)}$

where both partials are evaluated at $(x,y+\epsilon \eta ,y^{\prime }+\epsilon {\eta }^{\prime })$. Setting $\epsilon =0$,

$\delta J[y;\eta ]={\int }_a^b[L_y\eta +L_{y^{\prime }}{\eta }^{\prime }]dx\mathrm{(}\mathrm{4}\mathrm{)}$

with $L_y$ and $L_{y^{\prime }}$ now evaluated along $y$ itself. Equation (4) is the first variation in its raw form. Both $\eta$ and ${\eta }^{\prime }$ appear, which is the obstacle: they are not independent, so no conclusion about the integrand follows yet.

Removing the derivative of the variation

The boundary term left by integration by parts.

Integrate the second term by parts:

${\int }_a^b{L}_{y^{\prime }}{\eta }^{\prime }dx=[L_{y^{\prime }}\eta {]}_a^b-{\int }_a^b\frac{d}{dx}(L_{y^{\prime }})\eta dx\mathrm{(}\mathrm{5}\mathrm{)}$

The boundary term vanishes, because $\eta \in 𝒱$ and therefore $\eta (a)=\eta (b)=0$. This is the only place the fixed endpoints are used, and it is where they earn their keep. Substituting into (4),

$\delta J[y;\eta ]={\int }_a^b[L_y-\frac{d}{dx}{L}_{y^{\prime }}]\eta dx\mathrm{(}\mathrm{6}\mathrm{)}$

Only $\eta$ now appears, and it appears multiplied by an expression that does not involve $\eta$ at all. Write

$E(x)=L_y-\frac{d}{dx}{L}_{y^{\prime }}\mathrm{(}\mathrm{7}\mathrm{)}$

so that $\delta J[y;\eta ]={\int }_a^{b}E(x)\eta (x)dx$. Noether calls $E$ the Lagrange expression and it recurs under that name on Noether's first theorem.

What has and has not been shown

Stationarity says (6) vanishes for every admissible $\eta$. It does not yet say $E\equiv 0$. An integral can vanish because its integrand vanishes, or because positive and negative contributions cancel. Ruling out the second possibility is a separate theorem, and it is the subject of the fundamental lemma.

Two further cautions.

Stationary is not extremal. ${\Phi }^{\prime }(0)=0$ constrains $\Phi$ only to first order. Whether $y$ minimises, maximises, or does neither depends on ${\Phi }^{″}(0)$ and beyond, treated on the second variation.

The family is a probe, not a physical path. The paths $y+\epsilon \eta$ for $\epsilon \ne 0$ generally violate the equations of motion. They are comparison curves, and Hamilton described them in exactly those terms: geometrically imaginable, but dynamically impossible motion.

The $\delta$ notation

The classical literature writes $\delta y$ for $\epsilon \eta$ and $\delta J$ for the first variation, and manipulates $\delta$ as an operator commuting with $d/dx$ and with $\int$. Hamilton's papers are written this way throughout; his fundamental formula reads

$\Sigma .m(x^{″}\delta x+y^{″}\delta y+z^{″}\delta z)=\delta U\mathrm{(}\mathrm{8}\mathrm{)}$

The operator notation is compact and correct, and this volume uses it in the history track where the sources use it. The $\epsilon$-family notation is used in the mathematics track because it makes the hypotheses visible: it is clear what is differentiated, with respect to what, and where the result is evaluated.

Variations at free endpoints

If one or both endpoint values are not prescribed, $𝒱$ is enlarged and the boundary term in (5) no longer vanishes. It then supplies an extra condition on the extremal, called a natural boundary condition. That case is worked on Boundary terms and natural boundary conditions.

Next

the fundamental lemma proves that a continuous $E$ whose integral against every admissible $\eta$ vanishes must itself vanish identically.