Hamilton's principle/Equation index

This index lists every numbered equation in the volume, by page. It is generated from the page sources rather than maintained by hand, so it cannot drift out of step with them.

Equation numbers restart at (1) on each page. They are written inside the mathematics rather than with the usual tagging command, because MediaWiki's TeX checker does not accept it; see Hamilton's principle/Notation.

319 numbered equations across 35 pages.

Expressions longer than one table cell holds are not reproduced here: 90 of them are marked and left on their own pages. They are not truncated, because a cut expression is invalid mathematics and would fail the volume's own checks.

Hamilton's principle/History/d'Alembert and virtual work

no.	section	expression
(1)	Virtual work in statics	$\sum _i{𝐅}_i\cdot \delta {𝐫}_i=0$
(2)	D'Alembert's extension	$\sum _i({𝐅}_i-m_i{\ddot{𝐫}}_i)\cdot \delta {𝐫}_i=0$
(3)	Compare Hamilton's formula	$\Sigma .m(x^{″}\delta x+y^{″}\delta y+z^{″}\delta z)=\delta U$
(4)	From virtual work to a variational principle	long expression; see the page
(5)	From virtual work to a variational principle	${\int }_{t_0}^{t_1}\delta (T-V)dt=0$

Hamilton's principle/Mathematics/Functionals and the space of paths

no.	section	expression
(1)	The admissible set	$𝒜=\{y\in C^2[a,b]:y(a)=y_a,y(b)=y_b\}$
(2)	The space of variations	$𝒱=\{\eta \in C^2[a,b]:\eta (a)=\eta (b)=0\}$
(3)	The space of variations	$y_{\epsilon }=y+\epsilon \eta$
(4)	The functional	$J[y]={\int }_a^{b}L(x,y(x),y^{\prime }(x))dx$
(5)	Distance between paths	$\Vert \eta {\Vert }_1=\underset{[a,b]}{max}|\eta (x)|+\underset{[a,b]}{max}|{\eta }^{\prime }(x)|$
(6)	Distance between paths	$\Vert \eta {\Vert }_0=\underset{[a,b]}{max}|\eta (x)|$

Hamilton's principle/Mathematics/The first variation

no.	section	expression
(1)	The variation defined	long expression; see the page
(2)	The variation defined	long expression; see the page
(3)	Computing it	long expression; see the page
(4)	Computing it	$\delta J[y;\eta ]={\int }_a^b[L_y\eta +L_{y^{\prime }}{\eta }^{\prime }]dx$
(5)	Removing the derivative of the variation	long expression; see the page
(6)	Removing the derivative of the variation	$\delta J[y;\eta ]={\int }_a^b[L_y-\frac{d}{dx}{L}_{y^{\prime }}]\eta dx$
(7)	Removing the derivative of the variation	$E(x)=L_y-\frac{d}{dx}{L}_{y^{\prime }}$
(8)	The \delta notation	$\Sigma .m(x^{″}\delta x+y^{″}\delta y+z^{″}\delta z)=\delta U$

Hamilton's principle/Mathematics/The fundamental lemma

no.	section	expression
(1)	Statement	${\int }_a^{b}E(x)\eta (x)dx=0$
(2)	Proof	$E(x)>\frac{1}{2}E(x_0)>0\text{for}|x-x_0|<\delta$
(3)	Proof	long expression; see the page
(4)	Proof	long expression; see the page
(5)	Variants	${\int }_a^{b}F(x){\eta }^{\prime }(x)dx=0$

Hamilton's principle/Mathematics/The Euler-Lagrange equation

no.	section	expression
(1)	Statement	$J[y]={\int }_a^{b}L(x,y(x),y^{\prime }(x))dx$
(2)	Statement	long expression; see the page
(3)	Derivation	${\int }_a^b[L_y-\frac{d}{dx}{L}_{y^{\prime }}]\eta dx=0$
(4)	Expanded form	$\frac{d}{dx}{L}_{y^{\prime }}=L_{y^{\prime }x}+L_{y^{\prime }y}{y}^{\prime }+L_{y^{\prime }{y}^{\prime }}{y}^{″}$
(5)	Expanded form	$L_{y^{\prime }{y}^{\prime }}{y}^{″}+L_{y^{\prime }y}{y}^{\prime }+L_{y^{\prime }x}-L_y=0$
(6)	First integrals	$\frac{d}{dx}{L}_{y^{\prime }}=0⟹L_{y^{\prime }}=\text{const}$
(7)	First integrals	$y^{\prime }{L}_{y^{\prime }}-L=\text{const}$
(8)	Worked example: the shortest plane curve	$J[y]={\int }_a^b\sqrt{1+y{\text{'}}^2}dx$
(9)	Worked example: the shortest plane curve	$L_{y^{\prime }}=\frac{y^{\prime }}{\sqrt{1+y{\text{'}}^2}}=c$

Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions

no.	section	expression
(1)	The term that was discarded	long expression; see the page
(2)	The term that was discarded	long expression; see the page
(3)	The term that was discarded	long expression; see the page
(4)	Extracting two conditions from one requirement	$L_y-\frac{d}{dx}{L}_{y^{\prime }}=0$
(5)	Extracting two conditions from one requirement	$L_{y^{\prime }}(b,y(b),y^{\prime }(b))\eta (b)=0$
(6)	Extracting two conditions from one requirement	$L_{y^{\prime }}(b,y(b),y^{\prime }(b))=0$
(7)	Transversality: endpoints on a curve	$[L+(g^{\prime }-y^{\prime })L_{y^{\prime }}{]}_{x=b}=0$
(8)	Corner conditions	long expression; see the page

Hamilton's principle/Mathematics/The Beltrami identity

no.	section	expression
(1)	Statement	$L-y^{\prime }\frac{\partial L}{\partial y^{\prime }}=C$
(2)	Derivation	long expression; see the page
(3)	Derivation	long expression; see the page
(4)	Derivation	long expression; see the page
(5)	Relation to the Hamiltonian	$H=y^{\prime }p-L=-C$
(6)	Worked example: the catenary	$J[y]={\int }_a^{b}y\sqrt{1+y{\text{'}}^2}dx$
(7)	Worked example: the catenary	$y\sqrt{1+y{\text{'}}^2}-y^{\prime }\cdot \frac{y{y}^{\prime }}{\sqrt{1+y{\text{'}}^2}}=C$
(8)	Worked example: the catenary	$\frac{y}{\sqrt{1+y{\text{'}}^2}}=C$
(9)	Worked example: the catenary	$y=C\cosh \left(\frac{x-x_0}{C}\right)$

Hamilton's principle/Mathematics/Several dependent variables

no.	section	expression
(1)	Setup	$y_i(a)=y_{i,a},y_i(b)=y_{i,b},i=1,\dots ,n$
(2)	Setup	$J[y]={\int }_a^{b}L(x,y_i,{y_i}^{\prime })dx$
(3)	Derivation	long expression; see the page
(4)	Derivation	long expression; see the page
(5)	Derivation	$\frac{\partial L}{\partial y_k}-\frac{d}{dx}\frac{\partial L}{\partial {y_k}^{\prime }}=0$
(6)	Counting	long expression; see the page
(7)	First integrals	$p_k=\frac{\partial L}{\partial {y_k}^{\prime }}=\text{const}$
(8)	First integrals	$\sum _{i=1}^n{y_i}^{\prime }\frac{\partial L}{\partial {y_i}^{\prime }}-L=\text{const}$
(9)	Hamilton's own form	$\Sigma .m(x^{″}\delta x+y^{″}\delta y+z^{″}\delta z)=\delta U$
(10)	Hamilton's own form	long expression; see the page
(11)	Worked example: the free particle in the plane	$L=\frac{1}{2}m({\dot{y}}_1^2+{\dot{y}}_2^2)$

Hamilton's principle/Mathematics/Several independent variables

no.	section	expression
(1)	Setup	$u=u_0\text{on }\partial \Omega$
(2)	Setup	$J[u]={\int }_{\Omega }L(x,u,u_{,\alpha })d^{m}x$
(3)	Derivation	long expression; see the page
(4)	Derivation	long expression; see the page
(5)	Derivation	long expression; see the page
(6)	Derivation	long expression; see the page
(7)	Derivation	long expression; see the page
(8)	Several fields	long expression; see the page
(9)	The boundary term does not always vanish	long expression; see the page
(10)	Divergences and conservation	$\mathrm{D}\mathrm{i}\mathrm{v}A=\sum _{\alpha =1}^m\frac{\partial A_{\alpha }}{\partial x_{\alpha }}$
(11)	Divergences and conservation	$\sum {\psi }_i\delta u_i=\delta f+\mathrm{D}\mathrm{i}\mathrm{v}A$
(12)	Worked example: the vibrating string	$L=\frac{1}{2}\rho u_{,t}^2-\frac{1}{2}\tau u_{,x}^2$
(13)	Worked example: the vibrating string	long expression; see the page
(14)	Worked example: the vibrating string	$\rho u_{,tt}=\tau u_{,xx}$

Hamilton's principle/Mathematics/Holonomic constraints

no.	section	expression
(1)	Definition	$g(x,y_1,\dots ,y_n)=0$
(2)	Why the independence argument fails	$g(x,y+\epsilon \eta )=0$
(3)	Why the independence argument fails	$\sum _{i=1}^n\frac{\partial g}{\partial y_i}{\eta }_i=0$
(4)	Why the independence argument fails	long expression; see the page
(5)	Constraint forces	$E_i=-\sum _{\beta =1}^r{\lambda }_{\beta }\frac{\partial g_{\beta }}{\partial y_i}$
(6)	Worked example: the simple pendulum	long expression; see the page
(7)	Worked example: the simple pendulum	$L=\frac{1}{2}m{\ell }^2{\dot{\theta }}^2+m{g}_0\ell \cos \theta$
(8)	Worked example: the simple pendulum	$m{\ell }^2\ddot{\theta }+m{g}_0\ell \sin \theta =0$
(9)	Non-holonomic constraints	$\sum _{i=1}^n{a}_i(x,y){y_i}^{\prime }+a_0(x,y)=0$

Hamilton's principle/Mathematics/Lagrange multipliers

no.	section	expression
(1)	Statement	$J[y]={\int }_a^{b}L(x,y_i,{y_i}^{\prime })dx$
(2)	Statement	long expression; see the page
(3)	Derivation for one constraint	${\int }_a^b\sum _{i=1}^n{E}_i{\eta }_{i}dx=0$
(4)	Derivation for one constraint	$\sum _{i=1}^n\frac{\partial g}{\partial y_i}{\eta }_i=0$
(5)	Derivation for one constraint	long expression; see the page
(6)	Derivation for one constraint	$\lambda (x)=-E_n{\left(\frac{\partial g}{\partial y_n}\right)}^{-1}$
(7)	Derivation for one constraint	$E_n+\lambda \frac{\partial g}{\partial y_n}=0$
(8)	Derivation for one constraint	long expression; see the page
(9)	Derivation for one constraint	$E_i+\lambda \frac{\partial g}{\partial y_i}=0,i=1,\dots ,n-1$
(10)	Why the augmented functional works	$\frac{\partial J^{\ast }}{\partial {\lambda }_{\beta }}:g_{\beta }(x,y)=0$
(11)	Worked example: the pendulum, with tension	$L=\frac{1}{2}m({\dot{y}}_1^2+{\dot{y}}_2^2)+m{g}_0{y}_2,g=y_1^2+y_2^2-{\ell }^2$
(12)	Worked example: the pendulum, with tension	$m{\ddot{y}}_1=-2\lambda y_1,m{\ddot{y}}_2-m{g}_0=-2\lambda y_2$
(13)	Worked example: the pendulum, with tension	$m{\ell }^2\ddot{\theta }+m{g}_0\ell \sin \theta =0$
(14)	Worked example: the pendulum, with tension	$2\lambda \ell =m{g}_0\cos \theta -m\ell {\dot{\theta }}^2$

Hamilton's principle/Mathematics/Isoperimetric problems

no.	section	expression
(1)	Statement	$J[y]={\int }_a^{b}L(x,y,y^{\prime })dx$
(2)	Statement	$K[y]={\int }_a^{b}M(x,y,y^{\prime })dx=\ell$
(3)	Statement	$L^{\ast }=L+\mu M$
(4)	Derivation	$y_{{\epsilon }_1{\epsilon }_2}=y+{\epsilon }_1{\eta }_1+{\epsilon }_2{\eta }_2$
(5)	Derivation	long expression; see the page
(6)	Derivation	$\mathrm{\nabla }\Phi (0,0)+\mu \mathrm{\nabla }\Psi (0,0)=0$
(7)	Derivation	$\delta J[y;{\eta }_j]+\mu \delta K[y;{\eta }_j]=0,j=1,2$
(8)	Derivation	long expression; see the page
(9)	Worked example: the catenary, done properly	$J[y]={\int }_a^{b}y\sqrt{1+y{\text{'}}^2}dx$
(10)	Worked example: the catenary, done properly	$K[y]={\int }_a^b\sqrt{1+y{\text{'}}^2}dx=\ell$
(11)	Worked example: the catenary, done properly	$L^{\ast }=(y+\mu )\sqrt{1+y{\text{'}}^2}$
(12)	Worked example: the catenary, done properly	$\frac{y+\mu }{\sqrt{1+y{\text{'}}^2}}=C$
(13)	Worked example: the catenary, done properly	$y=-\mu +C\cosh \left(\frac{x-x_0}{C}\right)$
(14)	Several constraints	$L^{\ast }=L+\sum _{\gamma =1}^s{\mu }_{\gamma }{M}_{\gamma }$
(15)	Mixed problems	$L^{\ast }=L+\sum _{\beta }{\lambda }_{\beta }(x)g_{\beta }+\sum _{\gamma }{\mu }_{\gamma }{M}_{\gamma }$

Hamilton's principle/Mathematics/The second variation

no.	section	expression
(1)	Definition	${\delta }^{2}J[y;\eta ]={\Phi }^{″}(0)$
(2)	Definition	long expression; see the page
(3)	Computing it	long expression; see the page
(4)	Computing it	long expression; see the page
(5)	Computing it	${\delta }^{2}J[y;\eta ]={\int }_a^b[P\eta {\text{'}}^2+Q{\eta }^2]dx$
(6)	Computing it	$P=L_{y^{\prime }{y}^{\prime }},Q=L_{yy}-\frac{d}{dx}{L}_{y{y}^{\prime }}$
(7)	Necessary conditions, and the trap	${\delta }^{2}J[y;\eta ]\ge 0\text{for all }\eta \in 𝒱$
(8)	The accessory problem	$\frac{d}{dx}\left(P{\eta }^{\prime }\right)-Q\eta =0$
(9)	A first sight of the failure	${\delta }^{2}J[0;\eta ]={\int }_0^b(\eta {\text{'}}^2-k^2{\eta }^2)dx$
(10)	A first sight of the failure	${\delta }^{2}J=\frac{b}{2}(\frac{{\pi }^2}{b^2}-k^2)$

Hamilton's principle/Mathematics/The Legendre condition

no.	section	expression
(1)	Statement	$P(x)=L_{y^{\prime }{y}^{\prime }}(x,y(x),y^{\prime }(x))\ge 0$
(2)	Derivation	long expression; see the page
(3)	Derivation	$\int {\eta }_h^{2}dx=O(h^{-1}),\int {\eta }_h{\text{'}}^{2}dx=O(h^{-3})$
(4)	Derivation	${\delta }^{2}J[y;{\eta }_h]={\int }_I[P{\eta }_h{\text{'}}^2+Q{\eta }_h^2]dx$
(5)	Derivation	${\delta }^{2}J[y;{\eta }_h]<0$
(6)	What it does not buy	${\delta }^{2}J=\frac{b}{2}(\frac{{\pi }^2}{b^2}-k^2)$

Hamilton's principle/Mathematics/The Jacobi condition and conjugate points

no.	section	expression
(1)	The accessory equation	$\frac{d}{dx}(P(x){\eta }^{\prime })-Q(x)\eta =0$
(2)	Why zeros of the accessory equation matter	${\eta }^{\ast }(x)=\{\begin{array}{cc}u(x),& a\le x\le c\\ 0,& c<x\le b\end{array}$
(3)	Why zeros of the accessory equation matter	long expression; see the page
(4)	Why zeros of the accessory equation matter	${\delta }^{2}J[y;{\eta }^{\ast }]=0$
(5)	The analytic example, revisited	${\eta }^{″}+k^2\eta =0$
(6)	The geometric counterexample: geodesics on a sphere	$u^{″}+Ku=0⟹u^{″}+u=0$

Hamilton's principle/Mathematics/The Weierstrass excess function

no.	section	expression
(1)	The excess function	$E(x,y,p,q)=L(x,y,q)-L(x,y,p)-(q-p)L_{y^{\prime }}(x,y,p)$
(2)	The excess function	$E=\frac{1}{2}(q-p{)}^2{L}_{y^{\prime }{y}^{\prime }}(x,y,\overline{p})$
(3)	The conditions assembled	$E(x,y(x),y^{\prime }(x),q)\ge 0$
(4)	Worked example: where the strong claim fails	$J[y]={\int }_0^1(y{\text{'}}^2-y{\text{'}}^4)dx$
(5)	Worked example: where the strong claim fails	$L_{y^{\prime }{y}^{\prime }}=2-12y{\text{'}}^2$
(6)	Worked example: where the strong claim fails	$E(x,y,0,q)=(q^2-q^4)-0-q\cdot 0=q^2(1-q^2)$
(7)	Worked example: where the strong claim fails	$J[y_n]={\int }_0^1(4-16)dx=-12$

Hamilton's principle/Mathematics/The Legendre transform

no.	section	expression
(1)	The construction	$p=f^{\prime }(v)$
(2)	The construction	$g(p)=pv(p)-f(v(p))$
(3)	The geometric reading	$h(v)=pv-f(v)$
(4)	The geometric reading	$g(p)=\underset{v}{max}[pv-f(v)]$
(5)	It is an involution	long expression; see the page
(6)	It is an involution	$\frac{d^{2}g}{d{p}^2}=\frac{dv}{dp}=\frac{1}{f^{″}(v)}$
(7)	Several variables	$g(p)=\sum _{i=1}^n{p}_i{v}_i-f(v)$
(8)	Passive variables	$\frac{\partial g}{\partial u}=-\frac{\partial f}{\partial u}$
(9)	Passive variables	long expression; see the page
(10)	The transform in mechanics	$p=\frac{\partial L}{\partial y^{\prime }}$
(11)	The transform in mechanics	$H(x,y,p)=p{y}^{\prime }-L(x,y,y^{\prime })$
(12)	The transform in mechanics	$\frac{\partial H}{\partial y}=-\frac{\partial L}{\partial y}$

Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations

no.	section	expression
(1)	Setup	$p_i=\frac{\partial L}{\partial {\dot{q}}_i}$
(2)	Setup	$H(t,q,p)=\sum _{i=1}^n{p}_i{\dot{q}}_i-L$
(3)	Derivation	long expression; see the page
(4)	Derivation	long expression; see the page
(5)	Derivation	long expression; see the page
(6)	Derivation	long expression; see the page
(7)	Derivation	long expression; see the page
(8)	Conservation	long expression; see the page
(9)	When is H the energy?	$H=2T-(T-V)=T+V$
(10)	Worked example: the harmonic oscillator	$L=\frac{1}{2}m{\dot{q}}^2-\frac{1}{2}m{\omega }^2{q}^2$
(11)	Worked example: the harmonic oscillator	long expression; see the page
(12)	Worked example: the harmonic oscillator	$\dot{q}=\frac{p}{m},\dot{p}=-m{\omega }^{2}q$

Hamilton's principle/Mathematics/The Hamilton-Jacobi equation

no.	section	expression
(1)	The idea	long expression; see the page
(2)	Statement	long expression; see the page
(3)	Statement	$\frac{\partial S}{\partial {\alpha }_i}={\beta }_i=\text{const}$
(4)	Separation of variables	$S(t,q)=W(q)-Et$
(5)	Separation of variables	$H(q,\frac{\partial W}{\partial q})=E$
(6)	The optical analogy	$|\mathrm{\nabla }W{|}^2=2m(E-V(q))$
(7)	The optical analogy	$n(q)⟷\sqrt{2m(E-V(q))}$
(8)	Hamilton's pair	long expression; see the page

Hamilton's principle/Mathematics/Action as a function of its endpoints

no.	section	expression
(1)	The characteristic function	$V=\int \Sigma .m(x^{\prime }dx+y^{\prime }dy+z^{\prime }dz)={\int }_0^{t}2Tdt$
(2)	The law of varying action	long expression; see the page
(3)	The law of varying action	long expression; see the page
(4)	The law of varying action	long expression; see the page
(5)	The law of varying action	$\frac{\delta V}{\delta H}=t$
(6)	Relation to the modern S	$dS=\sum _i{p}_{i}d{q}_i-\sum _i{p}_{0i}d{q}_{0i}-Hdt+H_{0}d{t}_0$

Hamilton's principle/Mathematics/Noether's first theorem

no.	section	expression
(1)	Setup and notation	$I=\int \cdots \int f(x,u,\frac{\partial u}{\partial x},\dots )dx$
(2)	Setup and notation	$\delta I=\int \cdots \int \sum {\psi }_i\delta u_{i}dx$
(3)	Setup and notation	$\sum {\psi }_i\delta u_i=\delta f+\mathrm{D}\mathrm{i}\mathrm{v}A$
(4)	The two kinds of variation	$y_i=x_i+\Delta x_i,v_i(y)=u_i+\Delta u_i$
(5)	The two kinds of variation	long expression; see the page
(6)	Deriving the divergence relations	$\delta f+\mathrm{D}\mathrm{i}\mathrm{v}(f\cdot \Delta x)=0$
(7)	Deriving the divergence relations	$\sum {\psi }_i\delta u_i=\mathrm{D}\mathrm{i}\mathrm{v}B,B=A-f\cdot \Delta x$
(8)	Deriving the divergence relations	long expression; see the page
(9)	Passing to conservation laws	$\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(1)}=0,\dots \mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\rho )}=0$

Hamilton's principle/Mathematics/Noether's second theorem

no.	section	expression
(1)	Derivation	long expression; see the page
(2)	Derivation	long expression; see the page
(3)	Derivation	long expression; see the page
(4)	Derivation	long expression; see the page
(5)	Noether's own examples	$\sum {\psi }_i\frac{d{u}_i}{dx}=0$
(6)	Proper and improper conservation laws	$\sum {\psi }_i\frac{\partial u_i}{\partial x_{\lambda }}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}$

Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton

no.	section	expression
(1)	The two quantities	$S={\int }_{t_0}^{t_1}Ldt={\int }_{t_0}^{t_1}(T-V)dt$
(2)	The two quantities	$W=\int \sum _i{p}_{i}d{q}_i=\int 2Tdt$
(3)	The two quantities	$W=S+\int Hdt=S+E(t_1-t_0)$
(4)	Deriving Maupertuis from Hamilton	$\delta W=\delta S+E\delta (t_1-t_0)$
(5)	Deriving Maupertuis from Hamilton	$\delta W=0$
(6)	Jacobi's form	$W=\int \sqrt{2(E-V(q))}ds,d{s}^2=\sum _{i,k}{g}_{ik}d{q}_{i}d{q}_k$
(7)	Jacobi's form	${\tilde{g}}_{ik}=2(E-V(q))g_{ik}$

Hamilton's principle/Applications/The brachistochrone

no.	section	expression
(1)	Setting up the functional	$\frac{1}{2}m{v}^2=m{g}_{0}y⟹v=\sqrt{2g_{0}y}$
(2)	Setting up the functional	$J[y]={\int }_0^{x_1}\frac{\sqrt{1+y{\text{'}}^2}}{\sqrt{2g_{0}y}}dx$
(3)	Solving	$L=\sqrt{\frac{1+y{\text{'}}^2}{y}}$
(4)	Solving	$L-y^{\prime }\frac{\partial L}{\partial y^{\prime }}=C_1$
(5)	Solving	$\frac{\partial L}{\partial y^{\prime }}=\frac{y^{\prime }}{\sqrt{y}\sqrt{1+y{\text{'}}^2}}$
(6)	Solving	long expression; see the page
(7)	Solving	$y(1+y{\text{'}}^2)=k^2,k^2=1/C_1^2$
(8)	Integrating	$y^{\prime }=\sqrt{\frac{k^2-y}{y}}$
(9)	Integrating	$dx=\sqrt{\frac{y}{k^2-y}}dy$
(10)	Integrating	long expression; see the page
(11)	Integrating	long expression; see the page
(12)	Integrating	$x=a(\theta -\sin \theta ),y=a(1-\cos \theta )$
(13)	Fixing the constants	$\frac{x_1}{y_1}=\frac{{\theta }_1-\sin {\theta }_1}{1-\cos {\theta }_1}$
(14)	The descent time	$t=\sqrt{\frac{a}{g_0}}{\theta }_1$

Hamilton's principle/Applications/The catenary

no.	section	expression
(1)	The functional and the constraint	$J[y]=\rho g_0{\int }_{x_0}^{x_1}y\sqrt{1+y{\text{'}}^2}dx$
(2)	The functional and the constraint	$K[y]={\int }_{x_0}^{x_1}\sqrt{1+y{\text{'}}^2}dx=\ell$
(3)	The functional and the constraint	$L^{\ast }=(y+\mu )\sqrt{1+y{\text{'}}^2}$
(4)	The shape	$\frac{y+\mu }{\sqrt{1+y{\text{'}}^2}}=C$
(5)	The shape	$\frac{dy}{\sqrt{(y+\mu {)}^2/C^2-1}}=dx$
(6)	The shape	$y=-\mu +C\cosh \left(\frac{x-{x_0}^{\prime }}{C}\right)$
(7)	Fixing the three constants	long expression; see the page
(8)	Fixing the three constants	$\ell =2C\sinh \left(\frac{d}{C}\right)$
(9)	Reading the multiplier	$T_x=\rho g_{0}C$
(10)	Reading the multiplier	$T=\rho g_0(y+\mu )$
(11)	The related surface	$A=2\pi \int y\sqrt{1+y{\text{'}}^2}dx$

Hamilton's principle/Applications/Geodesics

no.	section	expression
(1)	The functional	$d{s}^2=\sum _{i,k}{g}_{ik}(q)d{q}_{i}d{q}_k$
(2)	The functional	long expression; see the page
(3)	The functional	$E[q]=\frac{1}{2}\int \sum _{i,k}{g}_{ik}{\dot{q}}_i{\dot{q}}_{k}d\lambda$
(4)	The geodesic equation	long expression; see the page
(5)	The geodesic equation	long expression; see the page
(6)	The geodesic equation	${\ddot{q}}_m+{\Gamma }_{ik}^m{\dot{q}}_i{\dot{q}}_k=0$
(7)	The geodesic equation	long expression; see the page
(8)	The sphere	$p_{\phi }={\sin }^2\theta \dot{\phi }=h=\text{const}$
(9)	The sphere	${\dot{\theta }}^2+{\sin }^2\theta {\dot{\phi }}^2=1$
(10)	The sphere	${\dot{\theta }}^2=1-\frac{h^2}{{\sin }^2\theta }$
(11)	The sphere	long expression; see the page
(12)	The sphere	$d\phi =\frac{-hdu}{\sqrt{1-h^2-h^2{u}^2}}$
(13)	The sphere	$\cot \theta =\alpha \cos (\phi -{\phi }_0)$

Hamilton's principle/Applications/Fermat's principle

no.	section	expression
(1)	Statement	$T=\int \frac{ds}{v}=\frac{1}{c}\int n(x,y,z)ds$
(2)	Snell's law	$cT(x)=n_1\sqrt{a^2+x^2}+n_2\sqrt{b^2+(d-x{)}^2}$
(3)	Snell's law	$c\frac{dT}{dx}=\frac{n_{1}x}{\sqrt{a^2+x^2}}-\frac{n_2(d-x)}{\sqrt{b^2+(d-x{)}^2}}$
(4)	Snell's law	$n_1\sin {\theta }_1=n_2\sin {\theta }_2$
(5)	The continuous case	$J[y]=\int n(x,y)\sqrt{1+y{\text{'}}^2}dx$
(6)	The continuous case	$\frac{n(y)}{\sqrt{1+y{\text{'}}^2}}=\text{const}$
(7)	Rays are geodesics	$d{\tilde{s}}^2=n^2(q)(d{x}^2+d{y}^2+d{z}^2)$
(8)	Rays are geodesics	$n(q)⟷\sqrt{2(E-V(q))}$

Hamilton's principle/Applications/The optical-mechanical analogy

no.	section	expression
(1)	The correspondence	$\int n(q)ds$
(2)	The correspondence	$\int \sqrt{2m(E-V(q))}ds$
(3)	The correspondence	$n(q)⟷\sqrt{2m(E-V(q))}$
(4)	Both are Hamilton-Jacobi	$|\mathrm{\nabla }\Psi {|}^2=n^2(q),|\mathrm{\nabla }W{|}^2=2m(E-V(q))$
(5)	Filling the hole	$\psi =A(q)\exp \left(\frac{iW(q)}{\hslash }\right)$
(6)	Filling the hole	$|\mathrm{\nabla }W{|}^2-{\hslash }^2\frac{{\mathrm{\nabla }}^{2}A}{A}={\hslash }^2{k}^2$
(7)	Filling the hole	$-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\psi +V\psi =E\psi$

Hamilton's principle/Applications/Central force and the Kepler problem

no.	section	expression
(1)	Reduction	$U=m_1{m}_{2}f(r)$
(2)	Reduction	$r^2{\theta }^{\prime }=r_0^2{{\theta }_0}^{\prime }=h$
(3)	The inverse-square case	$f(r)=\frac{1}{r}$
(4)	The inverse-square case	$m_1+m_2=\mu ,\frac{h^2}{\mu }=p,\frac{-m_1{m}_2}{2H_0}=a$
(5)	The inverse-square case	long expression; see the page
(6)	Kepler's equation	$e=\sqrt{1-\frac{p}{a}},\upsilon ={\cos }^{-1}\left(\frac{a-r}{ae}\right)$
(7)	Kepler's equation	long expression; see the page
(8)	Kepler's equation	long expression; see the page
(9)	Lambert's theorem	$V_0=\frac{m_1{m}_{2}w}{m_1+m_2},w=w(\sigma ,\tau ,a)$
(10)	Lambert's theorem	$t=\frac{2a^2}{\mu }\frac{\delta w}{\delta a}$
(11)	Lambert's theorem	$w=2\sqrt{\mu }(\sqrt{\sigma +\tau }\mp \sqrt{\sigma -\tau })$
(12)	The modern route, for comparison	$L=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2)+\frac{k}{r}$
(13)	The modern route, for comparison	$E=\frac{1}{2}m{\dot{r}}^2+\frac{{\ell }^2}{2m{r}^2}-\frac{k}{r}$

Hamilton's principle/Applications/The rigid body

no.	section	expression
(1)	The configuration space	$T=\frac{1}{2}M|\dot{R}{|}^2+T_{\text{rot}}$
(2)	Rotational kinetic energy	$T_{\text{rot}}=\frac{1}{2}(I_1{\omega }_1^2+I_2{\omega }_2^2+I_3{\omega }_3^2)$
(3)	Euler angles, and their defect	long expression; see the page
(4)	The torque-free case	$2T=I_1{\omega }_1^2+I_2{\omega }_2^2+I_3{\omega }_3^2=\text{const}$
(5)	The torque-free case	long expression; see the page
(6)	The torque-free case	$I_1{\dot{\omega }}_1=(I_2-I_3){\omega }_2{\omega }_3$
(7)	The intermediate axis theorem	${\lambda }^2=\frac{(I_2-I_1)(I_3-I_1)}{I_2{I}_3}{\omega }_1^2$
(8)	The heavy symmetric top	long expression; see the page
(9)	The heavy symmetric top	$\frac{1}{2}{I}_1{\dot{\vartheta }}^2+V_{\text{eff}}(\vartheta )=E^{\prime }$

Hamilton's principle/Applications/Continuum systems: the vibrating string

no.	section	expression
(1)	The discrete chain	$\sqrt{h^2+(u_{j+1}-u_j{)}^2}-h\approx \frac{(u_{j+1}-u_j{)}^2}{2h}$
(2)	The discrete chain	long expression; see the page
(3)	The discrete chain	$m{\ddot{u}}_j=\frac{\tau }{h}(u_{j+1}-2u_j+u_{j-1})$
(4)	The discrete chain	long expression; see the page
(5)	The discrete chain	long expression; see the page
(6)	The field derivation	$\rho u_{,tt}-\tau u_{,xx}=0$
(7)	The field derivation	$c=\sqrt{\tau /\rho }$
(8)	Conservation laws	long expression; see the page
(9)	The free end	$\tau u_{,x}(\ell ,t)=0$
(10)	Separation and modes	${\omega }^2=\frac{4\tau }{\rho h^2}{\sin }^2\left(\frac{kh}{2}\right)$

Hamilton's principle/Applications/Classical field theory

no.	section	expression
(1)	The construction	$S[\varphi ]={\int }_{\Omega }ℒ({\varphi }_i,{\partial }_{\alpha }{\varphi }_i,x)d^{4}x$
(2)	The construction	long expression; see the page
(3)	Building a Lagrangian from symmetry	$ℒ=\frac{1}{2}{\partial }_{\alpha }\varphi {\partial }^{\alpha }\varphi -V(\varphi )$
(4)	Building a Lagrangian from symmetry	${\partial }_{\alpha }{\partial }^{\alpha }\varphi +V^{\prime }(\varphi )=0$
(5)	The stress-energy tensor	long expression; see the page
(6)	Fields with internal symmetry	$\varphi \to e^{i\alpha }\varphi$
(7)	Fields with internal symmetry	$j^{\alpha }=i({\varphi }^{\ast }{\partial }^{\alpha }\varphi -\varphi {\partial }^{\alpha }{\varphi }^{\ast })$

Hamilton's principle/Applications/The electromagnetic Lagrangian

no.	section	expression
(1)	Making the phase local	$\varphi (x)\to e^{i\alpha (x)}\varphi (x)$
(2)	Making the phase local	long expression; see the page
(3)	Making the phase local	$A_{\beta }\to A_{\beta }-{\partial }_{\beta }\alpha$
(4)	Making the phase local	$D_{\beta }\varphi =({\partial }_{\beta }+i{A}_{\beta })\varphi$
(5)	The field Lagrangian	$F_{\alpha \beta }={\partial }_{\alpha }{A}_{\beta }-{\partial }_{\beta }{A}_{\alpha }$
(6)	The field Lagrangian	$ℒ=-\frac{1}{4}{F}_{\alpha \beta }{F}^{\alpha \beta }-j^{\alpha }{A}_{\alpha }$
(7)	The field Lagrangian	${\partial }_{\alpha }{F}^{\alpha \beta }=j^{\beta }$
(8)	The field Lagrangian	long expression; see the page
(9)	The action must be gauge invariant, and that constrains the source	long expression; see the page
(10)	The action must be gauge invariant, and that constrains the source	${\partial }_{\alpha }{j}^{\alpha }=0$
(11)	Degeneracy	${\pi }^{\beta }=\frac{\partial ℒ}{\partial ({\partial }_0{A}_{\beta })}=-F^{0\beta }$
(12)	The two facts are one fact	${\partial }_{\beta }({\partial }_{\alpha }{F}^{\alpha \beta }-j^{\beta })=0$
(13)	The charged particle	long expression; see the page

Hamilton's principle/Applications/The Einstein-Hilbert action

no.	section	expression
(1)	The action	$S_{\text{EH}}=\frac{1}{2\kappa }\int R\sqrt{-g}{d}^{4}x$
(2)	Varying	long expression; see the page
(3)	Varying	$g^{\alpha \beta }\delta R_{\alpha \beta }={\mathrm{\nabla }}_{\gamma }{v}^{\gamma }$
(4)	Varying	$R_{\alpha \beta }-\frac{1}{2}R{g}_{\alpha \beta }=0$
(5)	The surface term	$S=S_{\text{EH}}+\frac{1}{\kappa }{\oint }_{\partial \Omega }K\sqrt{|h|}{d}^{3}x$
(6)	Noether's second theorem, realised	${\mathrm{\nabla }}_{\alpha }(R^{\alpha \beta }-\frac{1}{2}R{g}^{\alpha \beta })=0$
(7)	Geodesics recovered	$S=-mc\int ds$

Hamilton's principle/Applications/The Feynman path integral

no.	section	expression
(1)	The postulate	$K(b,a)=\sum _{\text{all paths}}\exp (\frac{i}{\hslash }S[q])$
(2)	Making the sum meaningful	long expression; see the page
(3)	Recovering Schrödinger's equation	long expression; see the page
(4)	Recovering Schrödinger's equation	long expression; see the page
(5)	The semiclassical limit and conjugate points	long expression; see the page