Integrable Hamiltonian system

Liouville integrability means having $n$ almost-everywhere independent first integrals in mutual Poisson involution.

For a Hamiltonian phase space of dimension $2n$, Liouville integrability means $n$ functionally independent conserved quantities with pairwise vanishing Poisson brackets. This is the Hamiltonian specialization of integrable systems.

A superintegrable Hamiltonian system with $n$ degrees of freedom has more than $n$ functionally independent integrals of motion, including an integrable set of $n$ in involution. At a regular nonzero Hamiltonian vector field, at most $2n-1$ independent integral differentials can annihilate the flow. A maximally superintegrable system attains that bound. The three-dimensional Kepler orbit has five independent integrals through energy, angular momentum and the Laplace-Runge-Lenz vector. A closed orbit on one isolated resonance is weaker than maximal superintegrability on an open phase-space region.

On the regular set of two uncoupled oscillators with irrational frequency ratio, every continuous globally defined integral of motion is constant on each fixed-action flat torus, because every orbit is dense there. A differentiable integral is consequently a function of the two actions, so its differential of a smooth map lies in their span. Local angle combinations can still give extra local integrals; they fail to be single-valued on the torus.

For a compact connected regular level of $n$ independent Poisson-commuting functions, the periods of their joint flow form a full-rank discrete subgroup $\Lambda \subset {\mathbb{R}}^n$. The level is ${\mathbb{R}}^n/\Lambda$. A local smooth basis of these periods determines action integrals and the normalized periodic fields used to construct action-angle variables.

A Stäckel potential allows additive separation of the Hamilton-Jacobi equation in an orthogonal coordinate system. In axisymmetric prolate spheroidal coordinates, the relative-potential form is $\psi =[G(\lambda )-G(\mu )]/(\lambda -\mu )$. It gives energy, axial angular momentum and an independent third separation integral. Here $G$ is a one-variable generating function, not the gravitational constant. Generic bounded orbits are regular invariant-torus motions; resonance and coordinate degeneracy require separate treatment.

For fixed $E,l,I$, separated motion obeys $p_{\tau }^2=[E\tau +G(\tau )-I+(\alpha -\beta )l^2/(2(\tau +\alpha ))]/[2(\tau +\alpha )(\tau +\beta )]$. Allowed intervals are where this is nonnegative, with turning points and coordinate-fold endpoints treated separately. A generic nonresonant bound orbit with $l\ne 0$ circulates azimuthally and densely samples a projected volume bounded by confocal spheroids and hyperboloids. It is not uniformly distributed in volume. Resonant or zero-angular-momentum orbits can project to lower-dimensional regions; integrability alone does not imply that every orbit fills a three-dimensional shell.

For $H=\frac{1}{2}\sum p_j^2/h_j^2-\psi$, $l=p_{\varphi }$ and a Stäckel potential, multiplying the separated Hamilton-Jacobi equation by $\lambda -\mu$ gives $2(\tau +\alpha )(\tau +\beta )p_{\tau }^2-(\alpha -\beta )l^2/[2(\tau +\alpha )]-G(\tau )-E\tau =-I$ for both $\tau =\lambda ,\mu$. Eliminating $E$ gives $I=\frac{1}{2}[\mu v_{\lambda }^2+\lambda v_{\mu }^2+(\lambda +\mu +\alpha )v_{\varphi }^2]-[\mu G(\lambda )-\lambda G(\mu )]/(\lambda -\mu )$. Physical components are $v_j=h_j{\dot{q}}_j$ and $p_j=h_j{v}_j$. Separation proves conservation locally; direct Poisson brackets extend it across regular coordinate patches.

First integrals are in involution when every pair has zero Poisson bracket. An $n$-degree-of-freedom Hamiltonian system is Liouville integrable when it has $n$ almost-everywhere independent first integrals in involution.

An invariant torus is a subset homeomorphic to a torus and preserved by a flow. In action-angle variables, an integrable Hamiltonian flow moves linearly in the angular coordinates of each regular invariant torus.

The torus is a particular invariant manifold; its toroidal topology is an extra condition beyond invariance.

A quasiperiodic orbit on a torus has angular coordinates $\theta (t)=\theta (0)+\omega t$ modulo $2\pi$. It is periodic exactly when the nonzero frequencies are commensurate; rationally independent frequencies produce a dense orbit.

For $x(t)=A\cos {\omega }_{1}t+B\cos {\omega }_{2}t$ with irrational frequency ratio and nonzero amplitudes, $|x|\le |A|+|B|$. The phase trajectory is dense on the two-torus, so this bound is a supremum. The required exact phase alignment may never occur at finite time. Thus an exam's maximum-amplitude estimate can be a limiting bound rather than an attained displacement; finite observation intervals have smaller maxima.

For two uncoupled harmonic oscillators with nonzero amplitudes and irrational frequency ratio ${\omega }_1/{\omega }_2$, the orbit is bounded but not periodic. It winds quasiperiodically on a two-dimensional invariant torus; a common period would force the frequency ratio to be rational.

The direct product of integrable Hamiltonian systems is integrable. Pull each family's commuting integrals of motion back to the product phase space. Integrals from different factors have zero Poisson bracket because they use disjoint canonical coordinates, and their differentials are independent wherever the differentials within each factor are independent.

Forthe functions $I_j=(q_j^2+p_j^2)/2$ are independent almost everywhere, Poisson commute, and sum to $H$. A regular common level $I_j=c_j>0$ is the product of circles of radii $\sqrt{2c_j}$, hence an invariant $n$-torus. Allowing some $c_j=0$ gives lower-dimensional singular tori.

A compact connected regular common level of $n$ commuting integrals is an $n$-torus with local action-angle coordinates.

For a one-degree-of-freedom closed orbit, action-angle coordinates make the Hamiltonian a function $H(I)$ and give $\dot{I}=0$, $\dot{\varphi }=dH/dI$.

Linearizing the Collisionless Boltzmann equation around $f_0(\mathbf{J})$ in an integrable background Hamiltonian $H_0(\mathbf{J})$ gives $({\partial }_t+i\mathbf{k}\cdot \mathbf{\Omega })\delta f_{\mathbf{k}}=i\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0{\Phi }_{\mathbf{k}}$, with $\mathbf{\Omega }={\partial }_{\mathbf{J}}{H}_0$. An integrating factor gives the displayed zero-initial-perturbation solution. A nonzero initial perturbation adds the freely evolving term $e^{-i\mathbf{k}\cdot \mathbf{\Omega }t}\delta f_{\mathbf{k}}(0)$.

A periodic canonical coordinate paired with an action variable. In an integrable background Hamiltonian $H_0(\mathbf{J})$, the angles advance uniformly at ${\dot{\theta }}_i=\partial H_0/\partial J_i$, while the action variables remain constant. The periodicity permits an integer-indexed Fourier series on each regular orbital torus.

For a harmonic oscillator with Hamiltonian $H=p^2/(2m)+m{\omega }^2{q}^2/2$, where $m,\omega >0$, defineThese action-angle variables are defined for $I>0$, with a local angle $\theta$ modulo $2\pi$. A proof of canonicity is ${\theta }_q=p/(2I)$ and ${\theta }_p=-q/(2I)$, which give $\{\theta ,I\}=1$ using the Poisson bracket. Substitution gives $H=\omega I$. Hamilton's equations then prove $I(t)=I(0)$ and $\theta (t)=\theta (0)+\omega t$. The action integral around the corresponding phase space ellipse is $\oint pdq=2\pi I$, so $I$ is the usual normalized action variable. The angle is undefined at the zero-energy origin.

For a bound noncollision Kepler orbit with gravitational parameter $\mu$, the canonical actions per unit mass are $\mathcal{L}=\sqrt{\mu a}$, $\mathcal{G}=\mathcal{L}\sqrt{1-e^2}$ and $\mathcal{H}=\mathcal{G}\cos i$. Their angles are mean anomaly $\ell$, argument of periapsis $\omega$ and longitude of ascending node $\Omega$. The Hamiltonian is $-{\mu }^2/(2{\mathcal{L}}^2)$, so only $\ell$ advances in the unperturbed problem. In the prograde spherical-action chart, $J_r=\mathcal{L}-\mathcal{G}$, $J_{\theta }=\mathcal{G}-\mathcal{H}$, $J_{\varphi }=\mathcal{H}$; the type-two canonical transformation is generated by $F_2=(\mathcal{L}-\mathcal{G})w_r+(\mathcal{G}-\mathcal{H})w_{\theta }+\mathcal{H}{w}_{\varphi }$. Retrograde spherical actions have $J_{\theta }=\mathcal{G}-|\mathcal{H}|$, so the absolute value cannot be dropped globally. Circular and coplanar limits are singular angle charts, not singular orbits.

Near a regular Lagrangian torus, choose a one-form $\lambda$ with $d\lambda =-\omega$ and a basis of fibre cycles ${\gamma }_j$. The action $I_j=(2\pi {)}^{-1}{\int }_{{\gamma }_j}\lambda$ has differential of a smooth map $d{I}_j=(2\pi {)}^{-1}{\sum }_i{t}_i^{(j)}d{F}_i$, where $t^{(j)}$ is the corresponding joint-flow period. Nonsingularity of the period matrix makes these actions local coordinates transverse to the tori.

With angle period $2\pi$, the action is the enclosed phase-space area divided by $2\pi$:

This is the action component of action-angle variables, rather than the entire coordinate construction.

For $H=p_r^2/2+{\ell }^2/(2r^2)-k/r$, a noncollision bound orbit has $\ell \ne 0$ and $-k^2/(2{\ell }^2)\le E<0$. Its radial action is $\sqrt{-2E}/\pi$ times ${\int }_{\alpha }^{\beta }\sqrt{(r-\alpha )(\beta -r)}dr/r$, where $\alpha ,\beta =[k\mp \sqrt{k^2+2E{\ell }^2}]/(-2E)$. Evaluating the integral gives the displayed action and $H=-k^2/[2(I_r+|\ell |{)}^2]$. The angular and radial frequencies have equal magnitudes, explaining closed orbits rather than generic quasiperiodicity. The circular limit has $I_r=0$; zero angular momentum gives collision in the unregularized phase space.

When a Hamiltonian parameter varies slowly compared with the orbital period and no separatrix is crossed, the action changes only at higher adiabatic order.

The action is an adiabatic invariant under these assumptions; other slowly varying systems can have different adiabatic invariants.

For a one-dimensional particle of energy $E$ between walls separated by $L$, the unnormalized action isAdiabatic conservation gives $E(t)L(t{)}^2=\text{constant}$.

If an action has parameter dependence $I\propto \sqrt{m/|E|}$, adiabatic conservation implies $|E|\propto m$.

When a gravitational potential changes slowly relative to an orbital period, the orbital actions remain constant provided no separatrix is crossed. In a spherical potential, angular momentum is conserved exactly and the radial action is adiabatically conserved.

Adiabatic invariance of an orbital action preserves action values during slow regular changes, but frequencies $\partial H/\partial J_i$ can vary. Closure requires rational frequency ratios, which need not persist. A slowly added harmonic component in a Kepler orbit potential produces generic apsidal precession.

For radial motion between turning radii $r_{-}$ and $r_{+}$ in a central potential, the radial action per unit mass isIt vanishes on a circular orbit and is canonically conjugate to the radial angle.

For a bound Kepler orbit of semimajor axis $a$ around mass $M$,Adiabatic central-mass change therefore preserves $Ma$ and orbital eccentricity.

For $H=p^2/(2m)-1/|q|=-|E|$, collision continuation between the two turning points gives $I=\sqrt{2m/|E|}$.