Hamilton–Jacobi and integrability — Examples

Adopt the harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k>0$: one label of mass $m>0$ on $\mathbb{R}$ with spring constant $k>0$, force law $F(x)=-kx$, potential $U(x)=\frac{1}{2}k{x}^2$ and no residual force. For the CM12-CM14 presentation interfaces, declare the ambient one-label model on $Q_{\mathrm{a}\mathrm{m}\mathrm{b}}={\mathbb{R}}^3$ with mass operator $m{I}_3$, time window $\mathbb{R}$, potential $U_{\mathrm{a}\mathrm{m}\mathrm{b}}(y)=k{y}_1^2\mathrm{/}2$, $d=0$ and smooth force $F_{\mathrm{a}\mathrm{m}\mathrm{b}}(y,v)=(-k{y}_1,0,0)$ (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). The generalized-coordinate map $X:\mathbb{R}\to Q_{\mathrm{a}\mathrm{m}\mathrm{b}}$, $X(x)=(x,0,0)$ is a smooth immersion with length-valued coordinate $x$, $G=D{X}^{\mathsf{T}}m{I}_{3}DX=m$ and substituted potential $k{x}^2\mathrm{/}2$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). Thus its chart Hamiltonian $\tilde{H}$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions clause (b) is exactly the scalar Hamiltonian below; write it as $H$. A scalar harmonic motion reconstructs an ambient motion $X\circ x$ since the longitudinal equation is $m{x}^{\mathrm{\prime }\mathrm{\prime }}=-kx$ and both transverse equations are $0=0$. Write

$\omega :=\sqrt{\frac{k}{m}}>0,T:=\frac{2\pi }{\omega },$

so that $U(x)=\frac{1}{2}m{\omega }^2{x}^2$ and the branch Hamiltonian in the generalized-coordinate chart is

$H(x,p)=\frac{p^2}{2m}+\frac{m{\omega }^2}{2}{x}^2$

(Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions), with $[H]=\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). On the open set

$V:=\{(t,x,\alpha )\in {\mathbb{R}}^3:\sin (\omega t)\mathrm{\ne }0\}$

define

$S(t,x,\alpha ):=\frac{m\omega }{2}[(x^2+{\alpha }^2)\cot (\omega t)-2x\alpha \csc (\omega t)]\mathrm{.}$ The example checks the following claims of this page on this instance.

1. (complete integral) $S$ is of class $C^{\mathrm{\infty }}$ on $V$ and solves the model's Hamilton-Jacobi equation $S_t+H(x,S_x)=0$ on $V$; the mixed derivative is $S_{x\alpha }=-m\omega \csc (\omega t)$, so $S$ is a complete integral of the model's Hamilton-Jacobi equation at every point of $V$ (The Hamilton-Jacobi equation of a declared potential presentation: the class of action-dimension functions $S$ with $S_t+H(t,q,S_q)=0$ in one momentum chart, A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants).

2. (characteristics) for every $(t_0,x_0,{\alpha }_0)\in V$ and every real $\beta$, the curve $x(t)={\alpha }_0\cos (\omega t)-\frac{\beta }{m\omega }\sin (\omega t)$ satisfies $S_{\alpha }(t,x(t),{\alpha }_0)=\beta$, and with $p(t):=S_x(t,x(t),{\alpha }_0)$ the pair $(x,p)$ solves $x^{\mathrm{\prime }}=H_p$, $p^{\mathrm{\prime }}=-H_x$ on every interval on which $\sin (\omega t)\mathrm{\ne }0$; hence $x$ is a motion of the harmonic model with $p=m{x}^{\mathrm{\prime }}$ and $S_{\alpha }$ is constant along it (A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(\beta ,\alpha )$ labels).

3. (action-angle chart) the map $(\theta ,I)⟼(x,p)=(\sqrt{\frac{I}{\pi m\omega }}\sin (2\pi \theta ),\sqrt{\frac{Im\omega }{\pi }}\cos (2\pi \theta ))$

   for $I>0$ and $\theta \in \mathbb{R}\mathrm{/}\mathbb{Z}$ is a $C^{\mathrm{\infty }}$ parametrization of the punctured phase plane; local real lifts of $\theta$ give charts with $\{x,p\}=1$ for the presentation's bracket, i.e. a canonical chart of Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $\omega ={\sum }_{a=1}^{m}d{\theta }^a\wedge d{\theta }^{m+a}$ with $m$ the chart's half-dimension, and the model's own momentum charts with ${\omega }_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}=d\theta \wedge dI$ in the period-one convention of Action and angle coordinates, and in it $H=h(I)=\frac{I}{T},$ so that every motion satisfies $\dot{I}=0$, $\dot{\theta }=1\mathrm{/}T$, by the canonical Hamiltonian field of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian clause (a).

4. (action normalization) the action coordinate is the period integral $I=\oint pdx={\int }_0^{T}p(t)x^{\mathrm{\prime }}(t)dt$ along one period of the oscillator motion, in the period-one normalization discussed in the remark of Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H=h(I)$ and linear motion on the tori.

5. (radian convention) with ${\theta }^{\mathrm{r}\mathrm{a}\mathrm{d}}:=2\pi \theta$ and $I^{\mathrm{r}\mathrm{a}\mathrm{d}}:=I\mathrm{/}(2\pi )$ the chart becomes $x=\sqrt{2I^{\mathrm{r}\mathrm{a}\mathrm{d}}\mathrm{/}(m\omega )}\sin {\theta }^{\mathrm{r}\mathrm{a}\mathrm{d}}$, $p=\sqrt{2I^{\mathrm{r}\mathrm{a}\mathrm{d}}m\omega }\cos {\theta }^{\mathrm{r}\mathrm{a}\mathrm{d}}$ with $H=\omega I^{\mathrm{r}\mathrm{a}\mathrm{d}}$ and ${\dot{\theta }}^{\mathrm{r}\mathrm{a}\mathrm{d}}=\omega$, which is the radian chart of the sources (Period-one to radian action-angle rescaling: ${\theta }^{\mathrm{r}\mathrm{a}\mathrm{d}}=2\pi \theta$ and $I^{\mathrm{r}\mathrm{a}\mathrm{d}}=I\mathrm{/}(2\pi )$ preserve $\omega$).

All data $m,k$ are declared model parameters; the example asserts no measured value and no claim about a real oscillator.

• Where the construction breaks. The family $S$ is defined only for $\sin (\omega t)\mathrm{\ne }0$, and the mixed derivative $S_{x\alpha }=-m\omega \csc (\omega t)$ is unbounded as $\omega t\to \pi \mathbb{Z}$; the complete integral is a local object between the conjugate times of the oscillator and is not continued across them. The companion example computes the focusing of the characteristic family at those times.

• No empirical content. The oscillator data are declared model parameters, and the displayed identities are statements about the model; no measured frequency, amplitude or period is asserted.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the conic, ellipse and period suppliers.

Adopt the two-label gravitational model of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (i) inside the central-force model of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions: inertial masses $m_1,m_2>0$, $M=m_1+m_2$, reduced mass $\mu =m_1{m}_2\mathrm{/}M$, gravitational couplings $m_1^g,m_2^g>0$, $\kappa =G_{\mathrm{N}}{m}_1^g{m}_2^g>0$ and the pair potential of Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection. Let $x:I\to Q$ be a motion whose relative coordinate $r$ has constant angular momentum $j=J{e}_3$ with $J>0$ and constant energy $E<0$. Let $\bar{r}:\mathbb{R}\to {\mathbb{R}}^3\setminus \{0\}$ be the unique global central-force extension of $r$ supplied by Completeness and periodicity of the negative-energy Kepler orbit, with $T=2\pi \mu ab\mathrm{/}J$ and $T^2=4{\pi }^2\mu a^3\mathrm{/}\kappa$ clause 1 and Kepler's third law for the gravitational two-label model: the relative period is $T=2\pi \mu ab\mathrm{/}J=2\pi \sqrt{\mu \mathrm{/}\kappa }{a}^{3\mathrm{/}2}$, so $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ under the mass identification, and both labels are periodic relative to the centre of mass clause 1. In all subsequent period and quadrature statements, write $r:=\bar{r}$, $\rho :=\mathrm{\mid }\bar{r}\mathrm{\mid }$ and let $\theta :\mathbb{R}\to \mathbb{R}$ be its increasing polar-angle lift. Its restriction to the initial interval is the original relative motion; no global physical time window is assumed. Let $0\le e<1$, ${\theta }_0$,

$p=\frac{J^2}{\mu \kappa },a=\frac{p}{1-e^2},b=a\sqrt{1-e^2},T=2\pi \sqrt{\frac{\mu }{\kappa }}{a}^{3\mathrm{/}2}$ be its conic data (The Kepler orbit equation is a conic: $u=c(1+e\cos (\phi -{\theta }_0))$ with $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, Kepler's third law for the gravitational two-label model: the relative period is $T=2\pi \mu ab\mathrm{/}J=2\pi \sqrt{\mu \mathrm{/}\kappa }{a}^{3\mathrm{/}2}$, so $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ under the mass identification, and both labels are periodic relative to the centre of mass). Write

$I_0:=(-\frac{\mu {\kappa }^2}{2J^2},0),r_{\ast }:=\frac{J^2}{\mu \kappa }=p,U^{\mathrm{e}\mathrm{f}\mathrm{f}}(\rho ):=-\frac{\kappa }{\rho }+\frac{J^2}{2\mu {\rho }^2}(\rho >0)$ for the effective potential of Reduction of the central problem to the effective one-dimensional potential: $\mu {\rho }^{\mathrm{\prime }\mathrm{\prime }}=-(U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }}(\rho )$, the energy identity, and the converse reconstruction of the planar trajectory from the radial data (its effective potential $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}$). The example verifies:

1. (two simple turning points) $U^{\mathrm{e}\mathrm{f}\mathrm{f}}$ is of class $C^{\mathrm{\infty }}$ on $(0,\mathrm{\infty })$, has the unique critical point $r_{\ast }$ with $U^{\mathrm{e}\mathrm{f}\mathrm{f}}(r_{\ast })=-\mu {\kappa }^2\mathrm{/}(2J^2)$, is strictly decreasing on $(0,r_{\ast })$ and strictly increasing on $(r_{\ast },\mathrm{\infty })$; hence for every $E\in I_0$ there are exactly two solutions $0<r_{-}(E)<r_{\ast }<r_{+}(E)$ of $U^{\mathrm{e}\mathrm{f}\mathrm{f}}(\rho )=E$, with $(U^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }}(r_{-}(E))<0<(U^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }}(r_{+}(E))$ and $U^{\mathrm{e}\mathrm{f}\mathrm{f}}<E$ between them, and the datum $(J_0,U^{\mathrm{e}\mathrm{f}\mathrm{f}},\mu )$ with $J_0=(0,\mathrm{\infty })$ satisfies the hypotheses of The action variable of a two-simple-turning-point family is $C^1$ with derivative $T(E)\mathrm{/}(2\pi )$, the period quadrature over $2\pi$ on the energy window $I_0$.

2. (action derivative) consequently the action variable $J_{\mathrm{a}}$ of The one-dimensional action variable of a family of two-turning-point energy levels is of class $C^1$ on $I_0$ with $J_{\mathrm{a}}^{\mathrm{\prime }}(E)=\frac{T_{\mathrm{r}\mathrm{a}\mathrm{d}}(E)}{2\pi },T_{\mathrm{r}\mathrm{a}\mathrm{d}}(E)=\sqrt{2\mu }{\int }_{r_{-}(E)}^{r_{+}(E)}\frac{\mathrm{d}\rho }{\sqrt{E-U^{\mathrm{e}\mathrm{f}\mathrm{f}}(\rho )}}\in (0,\mathrm{\infty })\mathrm{.}$

3. (quadrature equals the orbital period) for a noncircular orbit $0<e<1$ (equivalently $E\in I_0$), the radial coordinate of this global extension solves the one-dimensional problem of clause 1 of Reduction of the central problem to the effective one-dimensional potential: $\mu {\rho }^{\mathrm{\prime }\mathrm{\prime }}=-(U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }}(\rho )$, the energy identity, and the converse reconstruction of the planar trajectory from the radial data with the same energy, and its least positive period is the orbital period $T$ of the conic data; the time quadrature of that problem is $T_{\mathrm{r}\mathrm{a}\mathrm{d}}(E)=T$, so that $J_{\mathrm{a}}^{\mathrm{\prime }}(E)=\frac{T}{2\pi }=\sqrt{\frac{\mu }{\kappa }}{a}^{3\mathrm{/}2},T^2=\frac{4{\pi }^2\mu a^3}{\kappa }\mathrm{.}$

4. (sharpness of the hypotheses) for $e=0$ the two turning points coalesce at the critical point $r_{\ast }=a$ and the simple-turning-point hypothesis of the action-variable theorem fails; the value $J=0$ is outside this example (no two-sided bound radial motion), so both hypotheses of the construction are genuinely used.

The couplings $\kappa$ and the masses are declared model parameters; no measured value of $G_{\mathrm{N}}$ and no empirical premise enters.