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In three-dimensional autonomous mechanics, a regular common level set of $N$ independent isolating integrals has dimension $6-N$ in phase space. If an orbit is dense in a $d$-dimensional invariant torus and the integrals characterize its closure, $N=6-d$. A generic three-frequency regular orbit has three integrals; a spherical nonclosed rosette has energy and three components of specific angular momentum, with two-dimensional closure and four integrals. A periodic nonstationary orbit has one-dimensional closure and can be isolated by five independent orbit labels on the invariant family. The maximum is five wherever the Hamiltonian vector field is nonzero, since all integral differentials annihilate that field. A single resonant orbit need not yield an additional smooth single-valued integral on an open neighborhood; global superintegrability is stronger than orbital closure.

One integral can occur in an autonomous system with a fully chaotic region and no additional continuous orbit constraints. The specific orbital energy is conserved in a time-independent gravitational potential. If a chaotic stellar orbit is dense in a connected region of its energy hypersurface, every continuous integral of motion is constant on that region. Thus it cannot supply a new independent constraint there, and energy is the only independent isolating integral. A generic nonaxisymmetric potential can contain such regions; not every orbit in such a potential is necessarily chaotic.

More than three integrals occur when additional symmetry or degeneracy restricts motion beyond a generic three-frequency invariant torus. In a time-independent central gravitational potential, energy and the three components of specific angular momentum are four generically independent isolating integrals. They fix the orbital plane as well as the radial-motion parameters. A nonclosed planar rosette typically has two-dimensional orbit closure, rather than a three-dimensional invariant torus.

The Kepler orbit is a stronger example. In addition to $E$ and $\mathbf{L}$, the conserved Laplace-Runge-Lenz vectorfixes the apsidal direction. Its constraints $\mathbf{A}\cdot \mathbf{L}=0$ and $A^2={\mu }^2+2E{L}^2$ leave five independent integrals among $E,\mathbf{L},\mathbf{A}$. It is a maximally superintegrable Hamiltonian system, with closed nondegenerate bound orbits.

More than three integrals need not form a set of Poisson-commuting functions; the three-integral limit applies to an independent commuting set, rather than to every conserved orbit constraint. At a nonstationary point in six-dimensional phase space, integral gradients annihilate the nonzero flow vector, so at most five are independent. This explains the isolating integrals and orbital closure dimension count. A resonance on one isolated orbit does not by itself prove an additional smooth global integral on a neighborhood.

An isolating integral is a single-valued integral of motion $I(\mathbf{x},\mathbf{v})$ whose fixed value confines an orbit to a lower-dimensional level set of phase space. Along the flow, $dI/dt=0$; for an autonomous Hamiltonian, this means $\{I,H\}=0$. Several isolating integrals are independent when their gradients are linearly independent on the regular region considered. Counting them requires a specified domain: merely local orbit labels in a flow box do not establish global single-valued integrals. The following cases concern regular globally meaningful orbit constraints.

An isolating integral restricts the accessible phase space to a lower-dimensional regular level set. For functionally independent integrals of motion $I_1,\dots ,I_N$, the differentials $d{I}_j$ must be independent, and $I_j=c_j$ impose $N$ independent restrictions. In six-dimensional phase space the common regular level set has dimension $6-N$. Nonisolating orbit labels or multivalued constants do not further reduce the closure explored by the motion. This is the isolating integrals and orbital closure dimension distinction.

A constant of motion is a quantity preserved along the evolution, possibly with explicit time dependence. For $C(\mathbf{x},\mathbf{v},t)$ in a gravitational potential $\Phi$, the preservation condition isFor free motion, $\mathbf{x}-\mathbf{v}t$ is a useful explicitly time-dependent example. Such initial-position labels should not be counted as stationary isolating integrals of a galactic orbit.

Zero stationary isolating integrals are possible when time dependence removes conservation of energy and no other symmetry or invariant survives. For instance, a generic time-dependent nonsymmetric gravitational potential need not have a stationary integral. This is not possible in an ordinary time-independent galactic gravitational potential: $E=\frac{1}{2}{v}^2+\Phi$ is already an isolating integral on a regular specific orbital energy surface. An explicitly time-dependent constant that encodes an initial condition does not contradict the stationary count.

More than three occur when symmetries or frequency degeneracies reduce the orbit's invariant closure below a generic three-dimensional invariant torus. For a nonradial orbit in a spherical gravitational potential, specific orbital energy and the three components of specific angular momentum are generically four independent isolating integrals. Their values fix the orbital plane, angular momentum magnitude and radial specific orbital energy, but a nonclosed rosette still fills a two-dimensional invariant torus. Thus more than three does not by itself imply closure. Orbital resonances can further reduce the number of independent orbital phases. The count concerns independent integrals, not arbitrarily many functions of the same invariants.

One isolating integral occurs when specific orbital energy is the only independent stationary invariant. An autonomous chaotic orbit that densely explores a connected five-dimensional specific orbital energy surface is the standard case. Every continuous integral of motion must then be constant on that surface, so it is locally a function of specific orbital energy rather than an additional independent integral. Irregular motion need not mean zero integrals; specific orbital energy still isolates the orbit from other specific orbital energy surfaces.

Write $\mu =G{M}_{\odot }$, and interpret the specified radial force field as acceleration, or force per unit test-particle mass. Since it is a central force, the specific angular momentum $\mathbf{h}=\mathbf{r}\times \mathbf{v}$ is conserved: $\dot{\mathbf{h}}=\mathbf{r}\times \mathbf{F}=0$. For $h=|\mathbf{h}|\ne 0$, motion is confined to the fixed plane perpendicular to $\mathbf{h}$. Choose the orientation of the polar angle so that $h=r^2\dot{\theta }>0$.

To derive the Binet equation here, let $u=1/r$ and differentiate with respect to $\theta$. Then $\dot{r}=-h{u}^{\prime }$ and $\ddot{r}-r{\dot{\theta }}^2=-h^2{u}^2(u^{\prime \prime }+u)$. Equating this to the radial acceleration givesPut $K=h^2+C$ and $\alpha =\sqrt{K/h^2}$. For $K>0$, the solution isThe radial potential energy per unit mass is $V(r)=-\mu /r+C/(2r^2)$, so the conserved specific orbital energy isSubstitute the displayed $u$ and $\dot{r}=-h{u}^{\prime }$ to obtain $\mathcal{E}={\mu }^2(e^2-1)/(2K)$. Therefore defineFor regular bound motion, $K>0$ and $-{\mu }^2/(2K)\le \mathcal{E}<0$, giving $a>0$ and $0\le e<1$. Since $K/\mu =a(1-e^2)$, choose the angular origin at a pericentre, so ${\theta }_0=0$, and obtainThe turning radii are $r_{\mathrm{p}}=a(1-e)$ and $r_{\mathrm{a}}=a(1+e)$. Thus $a$ is the arithmetic mean of the turning radii; when the orbit precesses it should not be interpreted as the semimajor axis of a fixed ellipse. If $K\le 0$, the centrifugal barrier needed for a regular two-turning-point bound orbit is absent, and this bound-orbit parametrization does not apply.

For $\alpha =1$, one has $C=0$ and the orbit is a closed Kepler orbit: an ellipse of eccentricity $e$ with the Sun at a focus. In addition to $\mathcal{E}$ and the three components of $\mathbf{h}$, the Laplace-Runge-Lenz vectoris conserved. Indeed,so $\dot{\mathbf{A}}=0$. It points toward the fixed pericentre. The constraints $\mathbf{A}\cdot \mathbf{h}=0$ and $A^2={\mu }^2+2\mathcal{E}{h}^2$ leave five generically independent isolating integrals. One may takechoosing a component $A_j$ whose differential is independent of the other four on the local region under consideration. The extra integral fixes the apsidal direction in the already fixed orbital plane.

For $\alpha \ne 1$, successive pericentres are separated by $2\pi /\alpha$ in polar angle. The generic bound orbit is a rosette in that same plane. Its perihelion advance per radial cycle isFor $\alpha <1$ this is prograde; for $\alpha >1$ it is retrograde. If $\alpha$ is rational, the orbit closes after a finite number of radial cycles; if it is irrational and $0<e<1$, its successive radial phases densely fill the accessible annulus. A generic independent set of single-valued isolating integrals isThe apsidal direction is no longer fixed. For irrational frequency ratio, any continuous integral of motion must be constant on the whole two-dimensional orbital torus, because one orbit is dense there; there is no further independent continuous integral specifying a fixed apsidal direction. Exceptional closed resonant orbits do not provide a fifth global integral on an open region of phase space. Circular and radial orbits are degenerate cases of these generic counts.

The plotted precession is enlarged for visibility. Both examples have $e=0.5$ and the same turning radii; the second has $\alpha =\sqrt{0.9}$.

For the precession rate, it is useful to derive the radial period rather than assume an unperturbed angular period. Set $r=a(1-e\cos \chi )$ in the radial energy equation. On a complete radial cycle,Hence $n=2\pi /P_r=\sqrt{\mu /a^3}$. Also $K=\mu a(1-e^2)$ and $h^2=K-C$, so ${\alpha }^{-1}=\sqrt{1-C/K}$. For $|C|\ll K$, inverse-cube perturbation of Kepler apsidal motion givesThere is a dimensional issue with the printed parameter: $C$ has units $L^4/T^2$ and $\mu$ has units $L^3/T^2$, so $\eta =aC/\mu$ has units $L^2$. Retaining that definition literally, the requested expression isA dimensionless perturbation coefficient would instead be $\tilde{\eta }=C/(\mu a)=\eta /a^2$. In that convention the rate is $-n\tilde{\eta }/[2(1-e^2)]$. These two conventions must not be conflated.

Use the stated residual precession and period. The advance in one radial cycle isWith $e=0.206$, the small-perturbation estimate becomesThe negative sign is required: an additional attractive inverse-cube acceleration advances perihelion, whereas the positive-$C$ term would make it regress. If a length-unit estimate of the printed $\eta$ is desired, $P_r=0.24\mathrm{yr}$ and the solar value $\mu \simeq 1.3271\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$ give $a\simeq 5.78\times 10^{10}\mathrm{m}\simeq 0.386\mathrm{au}$ through the radial-period law. ThusThe dimensionless value is the estimate obtainable directly from the supplied period, eccentricity and advance; the dimensional value additionally uses the solar gravitational parameter to set the length scale.