Calculus of Variations

Derives Proposition 16.1. The integrand \(f(y,y')=\bigl[(1+y'^{2})/(2gy)\bigr]^{1/2}\) carries no explicit \(x\), so the Beltrami identity Equation (16.25) applies: \(f-y'\,\pp f/\pp y'\) is constant along any extremal. With

the combination collapses to

that is, \(y\left(1+y'^{2}\right)=2a\) with \(2a=1/(2gC^{2})\), a length. Hence \(y'=\sqrt{(2a-y)/y}\), and \(y\) never exceeds \(2a\). Substituting \(y=a(1-\cos\theta)=2a\sin^{2}(\theta/2)\) gives \(2a-y=2a\cos^{2}(\theta/2)\) and therefore \(y'=\cos(\theta/2)/\sin(\theta/2)\), so that

Integrating from \(\theta=0\), where \(x=y=0\), gives Equation (16.2).

Derives Proposition 16.4. By Theorem 16.43 the constrained extremals are the free extremals of \(U+\rho_{\ell}g\lambda L\), whose integrand, after division by the constant \(\rho_{\ell}g\), is \(F(y,y')=(y+\lambda)\sqrt{1+y'^{2}}\). It carries no explicit \(x\), so the Beltrami identity Equation (16.25) gives

a constant with the dimension of length. Writing \(u=y+\lambda\) this reads \(u=a\sqrt{1+u'^{2}}\), that is \(u'=\pm\sqrt{u^{2}/a^{2}-1}\), whose separated form \(\dd u/\sqrt{u^{2}-a^{2}}=\pm\dd x/a\) integrates to \(u=a\cosh\bigl((x-x_{c})/a\bigr)\). Restoring \(y=u-\lambda\) gives Equation (16.5). Expanding, \(a\cosh(z/a)=a+z^{2}/(2a)+z^{4}/(24a^{3})+\dots\), so the catenary agrees with a parabola only to second order about its vertex — which is why Galileo's conjecture survived casual inspection.

Derives Proposition 16.6. Parametrise the curve by arclength \(s\in[0,\ell]\), so that \(\dot x^{2}+\dot y^{2}=1\), and use the shoelace form of the area, \(A=\tfrac{1}{2}\oint\left(x\,\dd y-y\,\dd x\right)\). By Theorem 16.43 the constrained extremals are the free extremals of the integrand

The system Equation (16.30) gives, after imposing the arclength gauge \(\sqrt{\dot x^{2}+\dot y^{2}}=1\),

that is \(\lambda\ddot x=\dot y\) and \(\lambda\ddot y=-\dot x\). These are consistent with the gauge, since \(\dv{}{s}(\dot x^{2}+\dot y^{2}) =2\dot x\dot y/\lambda-2\dot y\dot x/\lambda=0\), and they give the signed curvature

constant. A plane curve of constant nonzero curvature is a circular arc, and a closed one of length \(\ell\) is the full circle of radius \(\abs{\lambda}=\ell/2\pi\).

Derives Proposition 16.8. Apply Equation (16.32) with two independent variables and the integrand \(f=W\equiv\sqrt{1+u_{x}^{2}+u_{y}^{2}}\), which does not depend on \(u\) itself. Since \(\pp f/\pp u_{x}=u_{x}/W\) and \(\pp f/\pp u_{y}=u_{y}/W\), the equation reads

Carrying out the differentiations, with \(W_{x}=(u_{x}u_{xx}+u_{y}u_{xy})/W\) and \(W_{y}=(u_{x}u_{xy}+u_{y}u_{yy})/W\), and multiplying through by \(W^{3}\),

which is Equation (16.7). The left-hand side of Equation (16.8) is \(2H\), twice the mean curvature of the graph in the sense of Differentiable Manifolds, Tensors, and Curvature, so the equation says \(H=0\).

Derives Proposition 16.9. The integrand \(f=y\sqrt{1+y'^{2}}\) carries no explicit \(x\), so Equation (16.25) gives \(f-y'f_{y'}=y/\sqrt{1+y'^{2}}=a\), a constant length. This is the equation solved in Proposition 16.4 with \(\lambda=0\), whose solution is Equation (16.9).

Derives Proposition 16.16. Fix \(\eta\ne0\) and set \(\varphi(\varepsilon)=J[y+\varepsilon\eta]\). Because \(\norm{\left(y+\varepsilon\eta\right)-y}_{1} =\abs{\varepsilon}\,\norm{\eta}_{1}\), every \(\varepsilon\) with \(\abs{\varepsilon}<\varepsilon_{0}/\norm{\eta}_{1}\) places \(y+\varepsilon\eta\) inside the weak \(\varepsilon_{0}\)-neighbourhood in which \(y\) extremises \(J\). Hence \(\varphi\) has a local extremum at \(\varepsilon=0\) and is differentiable there, so \(\varphi'(0)=\delta J[y;\eta]=0\) by Fermat's lemma (Lemma 7.33).

Derives Proposition 16.17. The map \((\varepsilon,x)\longmapsto f\left(x,y+\varepsilon\eta,y'+\varepsilon\eta'\right)\) is \(C^{1}\) on the compact set \([-1,1]\times[a,b]\), so its \(\varepsilon\)-derivative is continuous there and differentiation under the integral sign is legitimate. Carrying it out and using the chain rule (Proposition 7.31),

which integrates to Equation (16.15).

Derives Lemma 16.18. Suppose \(g(x_{0})\ne0\) for some \(x_{0}\); replacing \(g\) by \(-g\) if necessary, assume \(g(x_{0})>0\), and by continuity \(x_{0}\) may be taken interior. By continuity again (Definition 7.20) there is \(\delta>0\) with \([x_{0}-\delta,x_{0}+\delta]\subset(a,b)\) and \(g>\tfrac12g(x_{0})\) on that interval. Put

This \(\eta\) is of class \(C^{1}\) — both it and \(\eta'=4(x-x_{0})\left[(x-x_{0})^{2}-\delta^{2}\right]\) vanish at \(x=x_{0}\pm\delta\) — and lies in \(C^{1}_{0}[a,b]\). It is strictly positive on the open interval, so

contradicting Equation (16.16).

Derives Lemma 16.19. Let \(c=\left(b-a\right)^{-1}\int_{a}^{b}h\,\dd x\) and define

Then \(\eta(a)=0\), and \(\eta(b)=\int_{a}^{b}h\,\dd x-c(b-a)=0\) by the choice of \(c\), so \(\eta\in C^{1}_{0}[a,b]\); it is of class \(C^{1}\) with \(\eta'=h-c\) by the fundamental theorem of calculus (Theorem 7.42). Substituting into Equation (16.18),

the last integral vanishing by the definition of \(c\). A continuous nonnegative integrand with zero integral vanishes identically, so \(h\equiv c\).

Derives Lemma 16.20. Set \(A(x)=\int_{a}^{x}M(t)\,\dd t\), which is \(C^{1}\) with \(A'=M\) (Theorem 7.42). Integrating by parts (Equation (7.28)) and using \(\eta(a)=\eta(b)=0\),

So Equation (16.19) says \(\int_{a}^{b}\left(N-A\right)\eta'\,\dd x=0\) for every admissible \(\eta\), whence \(N-A\) is a constant by Lemma 16.19. Thus \(N=A+\text{const}\) is of class \(C^{1}\) with \(N'=A'=M\).

Derives Theorem 16.22. By Proposition 16.16 and Proposition 16.17,

Both coefficients are continuous in \(x\), because \(f\) is \(C^{2}\) and \(y\in C^{1}\). Lemma 16.20 with \(M=\pp f/\pp y\) and \(N=\pp f/\pp y'\) then says that \(N\) is \(C^{1}\) with \(N'=M\), which is Equation (16.20).

Derives Corollary 16.23. Integrate Equation (16.20) from \(a\) to \(x\), using Theorem 7.43 on the \(C^{1}\) function \(\pp f/\pp y'\).

Derives Proposition 16.26. Write \(\Psi_{x}\), \(\Psi_{u}\), \(\Psi_{xu}\), \(\Psi_{uu}\) for the partials of \(\Psi\) and note that \(y'=\Psi_{x}+\Psi_{u}u'\), whence \(\pp y'/\pp u'=\Psi_{u}\) and \(\pp y'/\pp u=\Psi_{xu}+\Psi_{uu}u'=\dd\Psi_{u}/\dd x\), the total derivative along the curve. The chain rule then gives

and therefore

Subtracting, the two \(\pp f/\pp y'\cdot\dd\Psi_{u}/\dd x\) terms cancel and Equation (16.23) remains. Since \(\Psi_{u}\ne0\), one side vanishes exactly when the other does.

Derives Proposition 16.28. Immediate from Equation (16.20): the first term vanishes, so the total derivative of \(\pp f/\pp y'\) is zero, and a \(C^{1}\) function with vanishing derivative on an interval is constant (Corollary 7.36).

Derives Theorem 16.29. Differentiate the left-hand side along the curve, using the chain rule and \(\pp f/\pp x=0\):

the bracket vanishing by Equation (16.20).

Derives Theorem 16.31. Variations \(\eta\) now need only satisfy \(\eta(a)=0\). Those that also vanish at \(b\) form the class of Theorem 16.22, so Equation (16.20) holds and \(\pp f/\pp y'\) is \(C^{1}\). For a general admissible \(\eta\), integrating Equation (16.15) by parts now keeps the boundary term:

the integral vanishing by Equation (16.20) and the term at \(a\) by \(\eta(a)=0\). Choosing \(\eta\) with \(\eta(b)=1\) gives Equation (16.26).

Derives Theorem 16.32. Embed the extremal in a one-parameter family of admissible curves \(y_{\varepsilon}\) with terminal abscissa \(b_{\varepsilon}=b+\varepsilon \beta+O(\varepsilon^{2})\) and terminal ordinate \(y_{\varepsilon}(b_{\varepsilon})=\psi(b_{\varepsilon})\), and write \(\eta\) for the vertical variation \(\left.\pp y_{\varepsilon}/\pp\varepsilon\right|_{0}\) at fixed \(x\), so \(\eta(a)=0\). Differentiating \(J_{\varepsilon}=\int_{a}^{b_{\varepsilon}}f\,\dd x\) at \(\varepsilon=0\), and using \(\dd\left(\int_{a}^{b_{\varepsilon}}\right)/\dd\varepsilon =\beta f(b)\) for the moving limit,

the integral collapsing as in Equation (16.27). The endpoint constraint relates \(\beta\) and \(\eta(b)\): differentiating \(y_{\varepsilon}(b_{\varepsilon})=\psi(b_{\varepsilon})\) gives \(\eta(b)+\beta\,y'(b)=\beta\,\psi'(b)\), that is \(\eta(b)=\beta\left(\psi'-y'\right)_{b}\). Substituting and dividing by the arbitrary \(\beta\ne0\) yields Equation (16.28).

Derives Corollary 16.33. The first condition follows from Equation (16.21), which holds on each smooth arc with the same constant \(C\) — the derivation of Lemma 16.20 never used differentiability of \(y\) — so \(\pp f/\pp y'\) equals a continuous function of \(x\) throughout \([a,b]\).

For the second, regard the corner \(\left(x_{0},y_{0}\right)\) as a common free endpoint of the two arcs, and vary it. Repeating the computation of Theorem 16.32 for the left arc, whose terminal point moves by \(\left(\beta,\gamma\right)\) with \(\gamma=\left.\pp y_{\varepsilon}(x_{0})/\pp\varepsilon\right|_{0} +\beta y'(x_{0}^{-})\), contributes \(\beta f|_{x_{0}^{-}} +\left(\pp f/\pp y'\right)_{x_{0}^{-}} \left(\gamma-\beta y'(x_{0}^{-})\right)\) to \(\delta J\); the right arc, for which the same point is an initial point, contributes the same expression at \(x_{0}^{+}\) with the opposite sign. Stationarity for all \(\left(\beta,\gamma\right)\) forces both brackets to match: the coefficient of \(\gamma\) gives the first condition again, and the coefficient of \(\beta\) gives \(\left[f-y'\,\pp f/\pp y'\right]\) continuous, the second.

Derives Theorem 16.35. Fix \(i\) and vary only \(y^{i}\), holding the other \(n-1\) functions fixed: \(J\) restricted to that one-parameter family is a functional of the type of Equation (16.11), and Theorem 16.22 gives the \(i\)-th equation. Repeating for each \(i\) gives the system, and no further information is available from simultaneous variations, since the first variation is linear in \(\left(\eta^{1},\dots,\eta^{n}\right)\).

Derives Theorem 16.36. Let \(\eta^{A}\in C^{1}(\overline{\Omega})\) vanish on \(\pp\Omega\) and put \(\varphi_{\varepsilon}=\varphi+\varepsilon\eta\). Differentiating under the integral sign as in Proposition 16.17,

Write \(\pi_{A}^{\mu}=\pp\Lag/\pp(\pp_{\mu}\varphi^{A})\), which is \(C^{1}\) because \(\Lag\) is \(C^{2}\) and \(\varphi\in C^{2}\). Then \(\pi_{A}^{\mu}\,\pp_{\mu}\eta^{A} =\pp_{\mu}\left(\pi_{A}^{\mu}\eta^{A}\right) -\eta^{A}\,\pp_{\mu}\pi_{A}^{\mu}\), and the divergence theorem (Theorem 7.101) turns the first piece into a surface integral over \(\pp\Omega\), where \(\eta^{A}=0\). Hence

Stationarity makes this vanish for every such \(\eta\). The bracket is continuous, and the multidimensional form of Lemma 16.18 — proved by the same argument, with the bump Equation (16.17) replaced by its product over the \(D\) coordinates on a small cube inside \(\Omega\) — forces it to vanish identically, one component \(A\) at a time.

Derives Theorem 16.38. Admissible variations \(\eta\) satisfy \(\eta^{(j)}(a)=\eta^{(j)}(b)=0\) for \(j=0,\dots,m-1\). As in Proposition 16.17,

Integrate the \(k\)-th term by parts \(k\) times (Equation (7.28)). Each integration produces a boundary term carrying a factor \(\eta^{(j)}\) with \(j\le k-1\le m-1\), all of which vanish, and contributes one factor of \(-1\); after \(k\) steps the term is \(\left(-1\right)^{k}\int\eta\,\dd^{k} \left(\pp f/\pp y^{(k)}\right)/\dd x^{k}\). Summing over \(k\),

and Lemma 16.18 gives Equation (16.35).

Derives Proposition 16.39. Introduce \(Q_{1}=q\), \(Q_{2}=\dot q\) and the Ostrogradsky momenta

Nondegeneracy lets \(P_{2}=\pp\Lag/\pp\ddot q\) be inverted for \(\ddot q=\mathcal{F}\left(Q_{1},Q_{2},P_{2}\right)\). The Beltrami constant of Theorem 16.29, extended to the higher-derivative case by the same computation, is

Only the first term contains \(P_{1}\), and it contains it linearly, so for fixed \(\left(Q_{1},Q_{2},P_{2}\right)\) with \(Q_{2}\ne0\) the value of \(\Ham\) runs over all of \(\R\) as \(P_{1}\) does.

Derives Theorem 16.41. Admissible variations \(\eta^{i}\in C^{1}_{0}[a,b]\) are no longer arbitrary: differentiating Equation (16.37) along the family \(y+\varepsilon\eta\) at \(\varepsilon=0\) gives the \(m\) pointwise linear conditions

Stationarity gives, after the integration by parts of Theorem 16.22,

for every \(\eta\) obeying Equation (16.39). By the rank hypothesis the indices may be relabelled so that the leading \(m\times m\) block \(\pp g^{\alpha}/\pp y^{\beta}\), \(\beta=1,\dots,m\), is invertible at every \(x\); call \(y^{1},\dots,y^{m}\) dependent and \(y^{m+1},\dots,y^{n}\) free. Define \(\lambda_{\alpha}(x)\) as the unique solution of the linear system

which is continuous in \(x\) because the block and the \(E_{\beta}\) are. With this choice the first \(m\) of Equation (16.38) hold by construction, since \(g^{\alpha}\) does not contain \(y'^{i}\) and hence \(\pp F/\pp y'^{i}=\pp f/\pp y'^{i}\). Substituting back into Equation (16.40) and using Equation (16.39) to eliminate the dependent \(\eta^{\beta}\),

now for arbitrary \(\eta^{k}\in C^{1}_{0}[a,b]\), since the dependent components can always be recovered from Equation (16.39). Lemma 16.18, applied one free index at a time, gives the remaining \(n-m\) equations.

Derives Theorem 16.43. Because \(y\) is not an extremal of \(K\), Lemma 16.20 supplies some \(\eta_{2}\in C^{1}_{0}[a,b]\) with \(\delta K[y;\eta_{2}]\ne0\). Fix an arbitrary \(\eta_{1}\in C^{1}_{0}[a,b]\) and consider the two-parameter family \(y_{\varepsilon}=y+\varepsilon_{1}\eta_{1}+\varepsilon_{2}\eta_{2}\), setting

Both are \(C^{1}\) near the origin, with \(\pp\Phi/\pp\varepsilon_{i}\big|_{0}=\delta J[y;\eta_{i}]\) and similarly for \(\Psi\), by the argument of Proposition 16.17. Since \(\Psi(0,0)=\ell\) and \(\pp\Psi/\pp\varepsilon_{2}\big|_{0}=\delta K[y;\eta_{2}]\ne0\), the implicit function theorem (Real Analysis) provides a \(C^{1}\) function \(\varepsilon_{2}\left(\varepsilon_{1}\right)\) with \(\varepsilon_{2}(0)=0\) and \(\Psi\left(\varepsilon_{1},\varepsilon_{2}(\varepsilon_{1})\right) =\ell\) for small \(\varepsilon_{1}\). Every curve of this one-parameter family is admissible and lies in a weak neighbourhood of \(y\), so \(\varepsilon_{1}\longmapsto \Phi\left(\varepsilon_{1},\varepsilon_{2}(\varepsilon_{1})\right)\) has an extremum at \(0\):

the second identity coming from differentiating the constraint. Setting \(\lambda=-\delta J[y;\eta_{2}]/\delta K[y;\eta_{2}]\), a constant independent of \(\eta_{1}\), the display becomes \(\delta J[y;\eta_{1}]+\lambda\,\delta K[y;\eta_{1}]=0\), that is \(\delta\left(J+\lambda K\right)[y;\eta_{1}]=0\) for every \(\eta_{1}\in C^{1}_{0}[a,b]\). Lemma 16.20 then yields Equation (16.41).

Derives Proposition 16.50. Differentiating twice under the integral sign, as in Proposition 16.17,

The cross term is a total derivative up to one integration by parts: \(2\eta\eta'=\left(\eta^{2}\right)'\), so by Equation (7.28) and \(\eta(a)=\eta(b)=0\),

which absorbed into the first term gives Equation (16.45) with the coefficients Equation (16.46).

Derives Theorem 16.51. At a weak minimum \(\varepsilon\longmapsto J[y+\varepsilon\eta]\) has a minimum at \(\varepsilon=0\), so \(\delta^{2}J[y;\eta]\ge0\) for every \(\eta\in C^{1}_{0}[a,b]\). Suppose instead that \(P(x_{0})<0\) at some \(x_{0}\), which by continuity may be taken interior, and choose \(\delta>0\) and \(k>0\) with \([x_{0}-\delta,x_{0}+\delta]\subset(a,b)\) and \(P\le-k\) there. Take the localized variation

which lies in \(C^{1}_{0}[a,b]\) because \(\eta\) and \(\eta'\) both vanish at \(x_{0}\pm\delta\). Substituting \(t=(x-x_{0}+\delta)/2\delta\),

Hence, writing \(M=\max_{[a,b]}\abs{Q}\),

which is negative for \(\delta\) small enough — a contradiction. So \(P\ge0\) everywhere.

Derives Theorem 16.55. Suppose \(c\in(a,b)\) is conjugate to \(a\), with \(u\) the solution of Equation (16.49) having \(u(a)=u(c)=0\) and \(u'(a)=1\); then \(u\) does not vanish identically, and \(u'(c)\ne0\), since a second-order linear solution vanishing together with its derivative at a point vanishes identically. Define the broken variation

which is continuous and piecewise \(C^{1}\) with \(\eta(a)=\eta(b)=0\). Multiplying Equation (16.49) by \(u\) and integrating by parts over \([a,c]\),

because \(u\) vanishes at both ends. Hence \(\delta^{2}J[y;\eta]=0\).

Because \(y\) is a weak minimum, \(\delta^{2}J[y;\cdot]\ge0\) on \(C^{1}_{0}[a,b]\), and hence on the larger class of piecewise-\(C^{1}\) variations vanishing at the endpoints: a corner may be rounded off inside an interval of length \(\varepsilon\), which changes \(\int\left(P\eta'^{2}+Q\eta^{2}\right)\dd x\) by \(O(\varepsilon)\), so the quadratic form is nonnegative on the closure by continuity. So \(\eta\) minimises \(\delta^{2}J\) over the wider class. A minimiser must satisfy the Euler–Lagrange equation Equation (16.49) on each smooth arc, and by the first Weierstrass–Erdmann condition of Corollary 16.33 — here the continuity of \(\pp/\pp\eta'\left(P\eta'^{2}+Q\eta^{2}\right) =2P\eta'\) — the product \(P\eta'\) must be continuous at \(c\). But \(P(c)\eta'(c^{-})=P(c)u'(c)\ne0\) while \(P(c)\eta'(c^{+})=0\), a contradiction.

Derives Theorem 16.56. Set \(W=-P\,w'/w\), which is \(C^{1}\) on \([a,b]\) because \(w\) has no zero there. Differentiating and using Equation (16.49) for \(w\),

For any \(\eta\in C^{1}_{0}[a,b]\) the quantity \(W\eta^{2}\) vanishes at both endpoints, so \(\int_{a}^{b}\left(W\eta^{2}\right)'\dd x=0\) and this may be added to Equation (16.45) free of charge:

the completion of the square being exact precisely because Equation (16.50) gives \(Q+W'=W^{2}/P\). The integrand is nonnegative, so \(\delta^{2}J\ge0\), and it vanishes only if \(\eta'=-W\eta/P\) throughout. That linear equation with \(\eta(a)=0\) has only the zero solution, so \(\delta^{2}J[y;\eta]>0\) unless \(\eta\equiv0\).

Derives Theorem 16.60. Fix an interior \(x_{0}\), write \(y_{0}=y(x_{0})\), fix \(q\in\R\) and fix \(\Delta>0\) small enough that \([x_{0},x_{0}+\Delta]\subset(a,b)\). For \(0\le h<\Delta\) define the comparison curve \(Y_{h}\): equal to \(y\) outside \(\left(x_{0},x_{0}+\Delta\right)\); the segment of slope \(q\) from \(\left(x_{0},y_{0}\right)\) on \([x_{0},x_{0}+h]\); and on \([x_{0}+h,x_{0}+\Delta]\) the straight segment rejoining \(\left(x_{0}+\Delta,y(x_{0}+\Delta)\right)\), whose slope is

Each \(Y_{h}\) is admissible and piecewise \(C^{1}\), and \(\norm{Y_{h}-y}_{0}\longrightarrow0\) as \(h\to0\) at fixed \(\Delta\), so for small \(h\) it lies in the strong neighbourhood on which \(y\) minimises. Hence \(G(h)\equiv J[Y_{h}]-J[y]\) satisfies \(G(0)=0\) and \(G(h)\ge0\), so \(G'(0^{+})\ge0\).

Only the interval \(\left(x_{0},x_{0}+\Delta\right)\) contributes. Writing \(\ell_{h}(x)=y_{0}+qh+s_{h}\left(x-x_{0}-h\right)\) for the rejoining segment,

Differentiating at \(h=0\), where \(\ell_{0}(x)=y_{0}+s_{\Delta}(x-x_{0})\), the two moving limits give \(f\left(x_{0},y_{0},q\right)-f\left(x_{0},y_{0},s_{\Delta}\right)\), while

so that

the partial derivatives being evaluated at \(\left(x,\ell_{0}(x),s_{\Delta}\right)\). Now let \(\Delta\to0\). The first part of the integral is \(O(\Delta)\); the second tends to \(\pp f/\pp y'\left(x_{0},y_{0},y'(x_{0})\right)\), since \(\Delta^{-1}\int_{x_{0}}^{x_{0}+\Delta}\) of a continuous function tends to its value at \(x_{0}\) and \(s_{\Delta}\to y'(x_{0})\) by the mean value theorem (Theorem 7.35). Writing \(p=y'(x_{0})\), the limit of Equation (16.53) is \(f(x_{0},y_{0},q)-f(x_{0},y_{0},p)-(q-p)\,\pp f/\pp y'(x_{0},y_{0},p) =E\left(x_{0},y_{0},p,q\right)\). Since \(G'(0^{+})\ge0\) for every \(\Delta\), the limit is nonnegative, which is Equation (16.52) at \(x_{0}\); continuity extends it to the endpoints.

Derives Theorem 16.63. Write the integrand as \(\mathcal{P}\,\dd x+\mathcal{Q}\,\dd y\) with

both \(C^{1}\) on \(R\). Path independence on a simply connected region is equivalent to \(\pp\mathcal{P}/\pp y=\pp\mathcal{Q}/\pp x\) (Proposition 7.104). Abbreviating \(f_{y}\), \(f_{y'}\), \(f_{xy'}\), \(f_{yy'}\), \(f_{y'y'}\) for the partial derivatives evaluated at \(\left(x,y,p(x,y)\right)\), the chain rule gives

Each member of the field satisfies Equation (16.20), and along it \(\dd/\dd x=\pp_{x}+p\,\pp_{y}+\left(\pp p/\pp x +p\,\pp p/\pp y\right)\pp_{y'}\), so the Euler–Lagrange equation reads

Subtracting the two displays,

by Equation (16.56). Finally, on a member of the field \(\dd y=p\,\dd x\), so the two \(f_{y'}\) terms in Equation (16.55) cancel and the integrand reduces to \(f\left(x,y,p\right)\dd x=f\left(x,y,y'\right)\dd x\).

Derives Theorem 16.64. Let \(Y\) be admissible with graph in \(R\). By Theorem 16.63, \(I\) takes the same value on the graph of \(Y\) as on the graph of \(y^{*}\), and on the latter it equals \(J[y^{*}]\). Evaluating \(I\) along the graph of \(Y\), where \(\dd y=Y'\,\dd x\),

with \(p=p\left(x,Y(x)\right)\). Subtracting from \(J[Y]=\int_{a}^{b}f\left(x,Y,Y'\right)\dd x\) and comparing with Equation (16.51),

by Equation (16.57).

Derives Corollary 16.65. Theorem 16.56 gives \(\delta^{2}J[y^{*};\eta]>0\) for every admissible \(\eta\ne0\), and that by itself is not enough: the infimum of \(\delta^{2}J[y^{*};\cdot]\) over the sphere \(\norm{\eta}_{1}=1\) is zero. A smoothed tent of width \(\delta\) and unit maximum slope near an interior point has \(\delta^{2}J=O(\delta)\) while its \(C^{1}\) norm tends to \(1\), so no positive lower bound survives the passage to the limit and the positivity must be upgraded to coercivity first.

Step 1: a coercive lower bound. Let \(w\) be the zero-free Jacobi solution supplied by the strengthened Jacobi condition, and let \(W=-P\,w'/w\) and \(\mu(x)=\exp\int_{a}^{x}\left(W/P\right)\dd s\), so that \(\mu>0\) is \(C^{1}\) on \([a,b]\). The completion of the square in the proof of Theorem 16.56 is the exact identity

valid for every \(\eta\in C^{1}_{0}[a,b]\). Since \(\eta(a)=0\) we have \(\mu(x)\eta(x)=\int_{a}^{x}\mu\,v\,\dd s\), so the Cauchy–Schwarz inequality bounds \(\max\abs{\eta}\) by a constant depending only on \(\mu\) and \(b-a\) times \(\left(\int_{a}^{b}v^{2}\right)^{1/2}\); hence \(\int\eta^{2}\le\kappa_{2}\int v^{2}\), and \(\eta'=v-W\eta/P\) then gives \(\int\eta'^{2}\le\kappa_{3}\int v^{2}\) with \(\kappa_{3}\) likewise independent of \(\eta\). With \(P\ge P_{\min}>0\) this is the coercivity estimate

Step 2: the remainder. For \(\eta\in C^{1}_{0}[a,b]\), Taylor's theorem (Theorem 7.38) applied to \(f\) in its last two arguments, the vanishing of the first variation, and the integration by parts of Proposition 16.50 give

where \(\omega(t)\to0\) as \(t\to0\): the second derivatives of \(f\) are uniformly continuous on a compact neighbourhood of the graph of \(y^{*}\) because \(f\) is \(C^{3}\), and \(2\abs{\eta\eta'}\le\eta^{2}+\eta'^{2}\). This is the step the norm on the sphere could not supply — the remainder is now measured against \(\int(\eta^{2}+\eta'^{2})\), the same quantity Equation (16.59) controls from below.

Step 3. Because \(\eta(a)=0\), Cauchy–Schwarz also gives \(\int\eta^{2}\le\left(b-a\right)^{2}\int\eta'^{2}\). Writing \(\kappa=1+\left(b-a\right)^{2}\) and combining the two steps,

Choose \(\varepsilon>0\) with \(\kappa\,\omega(\varepsilon)<c/2\). For every \(\eta\ne0\) with \(\norm{\eta}_{1}<\varepsilon\) the bracket is positive and \(\int\eta'^{2}>0\), since an \(\eta\in C^{1}_{0}[a,b]\) with \(\eta'\equiv0\) vanishes identically. So \(y^{*}\) is a strict weak local minimum.

Derives Proposition 16.68. The integrand is nonnegative, so \(J\ge0\). For \(\varepsilon>0\) put

which lies in \(\mathcal{C}\), being smooth and odd with \(y_{\varepsilon}(\pm1)=\pm1\). Its derivative is \(y_{\varepsilon}'=\varepsilon \bigl[\arctan(1/\varepsilon)\bigr]^{-1} \left(\varepsilon^{2}+x^{2}\right)^{-1}\), so, using \(x^{2}\le\varepsilon^{2}+x^{2}\),

which tends to \(0\) as \(\varepsilon\to0^{+}\) because \(\arctan(1/\varepsilon)\to\pi/2\). Hence the infimum is \(0\).

Suppose now \(J[y]=0\) for some \(y\in\mathcal{C}\). The integrand \(x^{2}y'^{2}\) is continuous and nonnegative with vanishing integral, so it vanishes identically; hence \(y'(x)=0\) for every \(x\ne0\), and \(y\) is constant on \([-1,0)\) and on \((0,1]\). Continuity at \(0\) forces the two constants to agree, so \(y\) is constant on \([-1,1]\), contradicting \(y(-1)=-1\ne1=y(1)\).

Derives Theorem 16.70. Let \(m=\inf_{X}J<+\infty\) and pick a minimising sequence \((u_{n})\), so \(J[u_{n}]\longrightarrow m\). The sequence \(\left(J[u_{n}]\right)\) is bounded above, so by coercivity \(\left(\norm{u_{n}}\right)\) is bounded: otherwise a subsequence would have \(\norm{u_{n_{k}}}\to\infty\) and hence \(J[u_{n_{k}}]\to+\infty\). In a reflexive Banach space a bounded sequence has a weakly convergent subsequence — this is the compactness input, the Eberlein–Smulian theorem, whose Hilbert-space case is the one used in this book (Hilbert Spaces) — so \(u_{n_{k}}\rightharpoonup u\) for some \(u\in X\). Weak lower semicontinuity then gives

and \(J[u]\ge m\) by the definition of the infimum, so \(J[u]=m>-\infty\).

Derives Lemma 16.72. Convexity of a \(C^{1}\) function means its graph lies above every tangent:

Apply this pointwise with \(q_{1}=u'(x)\) and \(q_{2}=u_{n}'(x)\) and integrate:

The growth hypothesis puts \(\pp f/\pp q\left(\cdot,u'\right)\) in \(L^{r'}(a,b)\) with \(r'=r/(r-1)\), which is the dual of \(L^{r}\), so the last integral tends to \(0\) by the definition of weak convergence. Taking \(\liminf\) of both sides gives the claim.

Derives Theorem 16.76. Write \(B[u,v]=\int_{a}^{b}\left(p\,u'v'+q\,uv\right)\dd x\) for the energy form, so that \(\mathcal{R}[y]=B[y,y]/\avg{y,y}_{r}\), and fix \(K>0\) with \(q+Kr\ge0\) on \([a,b]\), which is possible because \(q\) is continuous and \(r>0\) on a compact interval. Put \(B_{K}[u,v]=B[u,v]+K\avg{u,v}_{r}\). Since \(p>0\) and \(q+Kr\ge0\),

Expand \(y=\sum_{n\ge1}c_{n}y_{n}\) with \(c_{n}=\avg{y,y_{n}}_{r}\); by completeness the denominator of Equation (16.65) is \(\sum_{n}c_{n}^{2}\).

The one integration by parts needed is against the eigenfunctions, not against \(y\): for \(y\in C^{1}_{0}[a,b]\) and \(y_{n}\in C^{2}[a,b]\), Equation (7.28) with \(y(a)=y(b)=0\) moves the derivative onto \(y_{n}\),

by Equation (16.66). Only \(y_{n}\) is differentiated twice, so nothing beyond \(y\in C^{1}_{0}[a,b]\) is used — which matters, because the trial functions of Proposition 16.79 are only \(C^{1}\). Hence \(B_{K}[y,y_{n}]=\left(\lambda_{n}+K\right)c_{n}\), and for the partial sums \(S_{N}=\sum_{n\le N}c_{n}y_{n}\) bilinearity and orthonormality give \(B_{K}[y,S_{N}]=B_{K}[S_{N},S_{N}] =\sum_{n\le N}\left(\lambda_{n}+K\right)c_{n}^{2}\). Therefore

by Equation (16.68). Taking \(u=y_{1}\) there, together with Equation (16.69) for \(y=y_{1}\), shows \(B_{K}[y_{1},y_{1}]=\lambda_{1}+K\ge0\), so every term of the sum is nonnegative and \(N\to\infty\) is legitimate: \(B_{K}[y,y]\ge\sum_{n}\left(\lambda_{n}+K\right)c_{n}^{2} \ge\left(\lambda_{1}+K\right)\sum_{n}c_{n}^{2}\). Subtracting \(K\avg{y,y}_{r}=K\sum_{n}c_{n}^{2}\) from both sides leaves \(B[y,y]\ge\lambda_{1}\avg{y,y}_{r}\), that is

If \(\mathcal{R}[y]=\lambda_{1}\) then both inequalities in Equation (16.70) are equalities, and the second forces \(c_{n}=0\) whenever \(\lambda_{n}>\lambda_{1}\); the lowest eigenvalue of a regular Sturm–Liouville problem is simple, so \(y\) agrees with a multiple of \(y_{1}\) in the weighted inner product and, both being continuous, pointwise. Taking \(y=y_{1}\) in Equation (16.69) gives \(\mathcal{R}[y_{1}]=\lambda_{1}\), so the value is attained.

Derives Corollary 16.78. The inequality is Equation (16.67). For the error statement, write \(\chi=y_{1}+\epsilon\) with \(\avg{\epsilon,y_{1}}_{r}=0\); then \(c_{1}=1\) and \(\sum_{n\ge2}c_{n}^{2}=\norm{\epsilon}_{r}^{2}\), so Equation (16.71) gives

whose numerator is a quadratic form in \(\epsilon\): the error is second order in the deviation. It must be measured in the energy norm of Remark 16.77 and not in \(\norm{\cdot}_{r}\), since \(\lambda_{n}-\lambda_{1}\) is unbounded and a deviation small in the weighted \(L^{2}\) norm may have a large derivative.

Derives Proposition 16.79. Substituting the expansion into Equation (16.65) gives \(\mathcal{R}=\left(c_{j}K_{jk}c_{k}\right) /\left(c_{j}M_{jk}c_{k}\right)\), a ratio of quadratic forms in the \(N\) numbers \(c_{k}\). Stationarity with respect to \(c_{j}\) requires

which, after multiplying by \(\tfrac12 c_{l}M_{lm}c_{m}\) and recognising \(\mathcal{R}=\lambda\), is Equation (16.72). The bound \(\lambda_{1}\le\lambda^{(N)}\) is Corollary 16.78 applied to the minimising member of the trial family, and enlarging the family can only lower the minimum because the smaller family is a subset of the larger.

Derives Theorem 16.84. Write \(\bar y^{i}(\bar x)\) for the transformed curve as a function of the transformed independent variable, and compare it with the original curve at the same value of \(x\). From Equation (16.74), \(\bar y^{i}(\bar x)=y^{i}(x) +\varepsilon\phi^{i}\) and \(\bar y^{i}(x+\varepsilon\xi) =\bar y^{i}(x)+\varepsilon\xi\,y'^{i}(x)+O(\varepsilon^{2})\), so the vertical (evolutionary) variation is

Renaming the dummy variable \(\bar x\) back to \(x\), the left-hand side of Equation (16.75) is \(\int_{x_{1}+\varepsilon\xi_{1}}^{x_{2}+\varepsilon\xi_{2}} f\left(x,\bar y(x),\bar y'(x)\right)\dd x\), and moving the limits back costs one boundary term to first order:

By Proposition 16.17 applied to the vertical variation \(\bar\phi\), and integrating the \(\bar\phi'\) term by parts,

with \(E_{i}\) the Euler–Lagrange expression of Equation (16.40). Substituting both displays into Equation (16.75), dividing by \(\varepsilon\) and letting \(\varepsilon\to0\),

This holds for every subinterval \([x_{1},x_{2}]\subseteq[a,b]\) and every curve, so the integrand, which is continuous, vanishes identically. On a solution of Equation (16.30) the first term is zero, leaving \(\dd I/\dd x=0\) with \(I\) as in Equation (16.76), since \(\bar\phi^{i}=\phi^{i}-y'^{i}\xi\).

Derives Corollary 16.87. With \(\bar x=x+\varepsilon\) and \(\bar y=y\) the integrand is unchanged and the integration measure is unchanged, so Equation (16.75) holds with \(\Lambda=0\) precisely because \(f\) carries no explicit \(x\). Then \(I=\left(\pp f/\pp y'^{i}\right)\left(-y'^{i}\right)+f=-h\).

Derives Corollary 16.88. In both cases \(\xi=0\) and \(\Lambda=0\), so Equation (16.76) reduces to \(I=\left(\pp f/\pp y'^{i}\right)\phi^{i}=p_{i}\phi^{i}\). Case (1) gives \(I=p_{k}\), which is also Proposition 16.28. Case (2) gives \(I=\vect{p}\cdot\left(\vect{n}\times\vect{y}\right) =\vect{n}\cdot\left(\vect{y}\times\vect{p}\right)\) by the cyclic invariance of the scalar triple product.

Derives Theorem 16.90. The argument of Theorem 16.84 carries over word for word with the divergence theorem (Theorem 7.101) in place of the one-dimensional integration by parts. The vertical variation is \(\bar\Phi^{A}=\Phi^{A}-\xi^{\nu}\pp_{\nu}\varphi^{A}\), obtained exactly as Equation (16.77). Moving the domain \(\bar\Omega\) back to \(\Omega\) contributes, to first order, the flux of \(\Lag\,\xi^{\mu}\) through \(\pp\Omega\), that is \(\varepsilon\int_{\Omega}\pp_{\mu}\left(\Lag\xi^{\mu}\right)\dd^{D}x\), because the Jacobian of Equation (16.80) is \(1+\varepsilon\,\pp_{\mu}\xi^{\mu}+O(\varepsilon^{2})\). The vertical part contributes, by Equation (16.33) before the boundary term is discarded,

Invariance up to \(\pp_{\mu}\Lambda^{\mu}\) therefore gives, for every subregion and hence pointwise,

On shell the bracket in the first term vanishes by Equation (16.32), leaving \(\pp_{\mu}j^{\mu}=0\) with \(j^{\mu}\) as in Equation (16.81).

Derives Corollary 16.91. Substituting \(\xi^{\mu}=a^{\mu}\) and \(\Phi^{A}=0\) into Equation (16.81) gives \(j^{\mu}=-a^{\nu}\left[\pi^{\mu}_{A}\,\pp_{\nu}\varphi^{A} -\delta^{\mu}_{\nu}\Lag\right]=-a^{\nu}T^{\mu}{}_{\nu}\), and \(\pp_{\mu}j^{\mu}=0\) for every constant \(a^{\nu}\) forces \(\pp_{\mu}T^{\mu}{}_{\nu}=0\) componentwise.

Derives Theorem 16.93. Because the \(\epsilon^{r}\) have compact support, all boundary terms in Equation (16.33) vanish and invariance of the action reads

for every choice of the \(\epsilon^{r}\), and for arbitrary field configurations, since the invariance is a property of the action and not of its extremals. Integrating the second term by parts, again with no boundary contribution,

The bracket is continuous and the \(\epsilon^{r}\) are arbitrary, so Lemma 16.18, in its \(D\)-dimensional form used in Theorem 16.36, gives Equation (16.85), one identity for each \(r\).

Derives Corollary 16.94. Equation (16.85) is a linear relation among the \(E_{A}\) and their first derivatives holding identically, so the map \(\varphi\longmapsto\left(E_{1},\dots,E_{n}\right)\) has image in a set of codimension at least \(m\); equivalently, once \(n-m\) of the equations and suitable initial data are imposed, the remaining \(m\) follow by integration. That a gauge transform of a solution is a solution is immediate from invariance of the action.

Derives Theorem 16.97. The three Euler–Lagrange equations Equation (16.30) with \(x=t\) and \(y^{i}=x^{i}\) read

which is Equation (16.87) componentwise. Conversely a solution of Equation (16.87) satisfies the Euler–Lagrange system, hence makes the first variation vanish by Proposition 16.17 and the integration by parts of Theorem 16.22.

Derives Lemma 16.100. The integrand \(f=n\left(\vect{x}\right)\abs{\vect{x}'}\) is homogeneous of degree one in \(\vect{x}'\), so the functional does not depend on the parametrisation and the three Euler–Lagrange equations Equation (16.30) are not independent; the arclength gauge \(\abs{\vect{x}'}=1\) fixes the redundancy. With

setting \(\abs{\vect{x}'}=1\) after the differentiation turns Equation (16.30) into \(\dd\left(n\,T^{i}\right)/\dd s=\pp n/\pp x^{i}\), which is Equation (16.90).

Derives Theorem 16.101. Along a motion of energy \(E\) the speed is \(v=\abs{\dot{\vect{x}}}=\sqrt{2\left(E-V\right)/m}\), so the momentum has magnitude \(\abs{\vect{p}}=mv=\sqrt{2m\left(E-V\right)}\) and points along \(\vect{T}\). Hence \(p_{i}\,\dd q^{i}=\abs{\vect{p}}\,\dd s\) and Equation (16.88) coincides with Equation (16.91); this is Equation (16.89) with \(n=\sqrt{2m\left(E-V\right)}\), in \(\mathrm{kg}\,\mathrm{m}/\mathrm{s}\).

By Lemma 16.100 the extremals satisfy \(\dd\left(n\vect{T}\right)/\dd s=\nabla n\). Now \(n=\abs{\vect{p}}=mv\), so \(n\vect{T}=\vect{p}\), and

Since \(\dd/\dd s=v^{-1}\dd/\dd t\) along the motion, the ray equation reads \(v^{-1}\dd\vect{p}/\dd t=-v^{-1}\nabla V\), that is Equation (16.87). Every step is reversible, so the extremals of Equation (16.91) traversed with the stated time law are precisely the Newtonian orbits of energy \(E\).

Derives Theorem 16.104. Within each homogeneous medium \(\nabla n=0\), so Equation (16.90) gives \(\dd\vect{T}/\dd s=\vect{0}\): the ray is straight. The path is therefore determined by the single point at which it crosses the interface, and coplanarity follows because moving that point off the plane containing the endpoints and the normal lengthens both straight segments.

Place the interface at \(z=0\), the source at perpendicular distance \(a\) on one side and the target at perpendicular distance \(b\) on the other, their feet separated by \(d\), and let the crossing point be at distance \(u\) from the first foot. The optical length is

a smooth function of the single real variable \(u\), whose stationary points satisfy, by Lemma 7.33,

The two fractions are exactly \(\sin\vartheta_{1}\) and \(\sin\vartheta_{2}\), so the condition is Equation (16.93). Since \(\mathcal{W}''>0\) everywhere, the stationary point is the unique minimum.

Derives Proposition 16.105. Parametrise the ray in the vertical plane as \(x\longmapsto z(x)\), so that \(\mathcal{W}=\int f\,\dd x\) with \(f\left(z,z'\right)=n(z)\sqrt{1+z'^{2}}\). The integrand carries no explicit dependence on \(x\), so the Beltrami identity Equation (16.25) applies:

The unit tangent of the ray is \(\left(1,z'\right)/\sqrt{1+z'^{2}}\), so the angle \(\vartheta\) it makes with the vertical satisfies \(\sin\vartheta=\left(1+z'^{2}\right)^{-1/2}\), and the conserved quantity is \(n(z)\sin\vartheta\), which is Equation (16.94).