Closed orbits, conserved quantities, and symmetries in the Kepler problem
In some cases, unexpectedly deep connections allow far-reaching conclusions to be drawn from commonplace observations. This post provides an example of such an occurrence.
A readily observable fact about the orbits of bound celestial objects is that they regularly repeat. Closed orbits are not only prevalent in the solar system, but can also be directly observed, for instance, in visual binary stars, as shown in the figure below. This simple observation places strong constraints on the nature of the gravitational force.
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| Observed orbit projected in the plane of the sky of the secondary component of the Sirius system relative to the primary (open and filled circles represent data obtained with different techniques, as detailed in the legend; the black filled circle at the origin marks the position of the primary). The best-fitting orbit is shown by the black line (data and orbital solution from Bond et al. 2017, ApJ 840, 70). |
A somewhat obscure theorem of classical mechanics [1] states that the only central forces that result in closed orbits for all bound particles are Hooke's law and the inverse-square law (i.e., the force must be either directly proportional to the distance from the center of attraction, or inversely proportional to the square of that distance). Furthermore, since a linear restoring force would increase without limit as the distance increases, it is physically implausible in the context of celestial mechanics. The existence of closed orbits for a wide variety of initial conditions thus implies that the gravitational force scales as the inverse square of the distance.
This conclusion hints at the deep connection between the topology of the orbits and the underlying dynamics. The rest of the post explores this connection in more detail.
A hierarchy of central forces
- Stability of closed orbits against small perturbations requires $n > -3$;
- Slightly perturbed closed orbits remain closed if $n=-3 + \beta^2$, where $\beta$ is a rational number;
- Closed orbits are preserved even for arbitrarily large perturbations if $\beta^2=1$ or $\beta^2=4$.
Closed orbits and integrals of motion
A central force field conserves energy and angular momentum, hence the motion is confined to a fixed plane, and can always be reduced to quadratures. Using these two conservation laws, we can write, in terms of polar coordinates $(r, \theta)$:
$t = \int_{r_0}^r [G(r)]^{-1/2}dr$,
$t = \frac{1}{h}\int_{\theta_0}^\theta r^2 d\theta$,
where
$G(r) = \frac{2E}{m} + \frac{2}{m}\int f(r) dr - \frac{h^2}{r^2}$,
and $m$, $E$, and $h$ are the mass, energy, and angular momentum per unit mass of the particle. Along with the initial conditions $(r_0, \theta_0)$, these relations define implicitly the trajectory equation, either in the parametric form $[r = r(t), \, \theta = \theta(t)]$, or, after elimination of the time variable, in the form $r = r(\theta)$.
For a given force law and appropriate initial conditions, bounded orbits exist if the equation $\dot r^2 \equiv G(r) = 0$ admits two roots $r_1$, $r_2$ such that the motion is restricted to $r_1 \leq r \leq r_2$ at all times. Bounded orbits are, of course, not necessarily closed. Non-closed bounded orbits eventually pass arbitrarily near any point in the region between the two circles of radius $r_1$ and $r_2$, and the function $r(\theta)$ in the trajectory equation is a multi-valued function. Conversely, if all bound orbits are closed, a well-defined, single-valued functional relation between $r$ and $\theta$ must exist. This function corresponds to a
conserved quantity that is an algebraic function of ($\bf r$, $\bf v$), i.e., an integral of motion.
Thus the closure of bound orbits implies the existence of an integral of motion in addition to energy and angular momentum. For the inverse-square law, i.e., the Kepler problem, we have:
$\dfrac{d{\bf v}}{dt} = -\dfrac{\mu}{r^3} {\bf r}$; $\dfrac{d{\bf h}}{dt} = 0$,
where $\mu$ is the gravitational parameter of the central body, and ${\bf h} = {\bf r} \times {\bf v}$ is the angular momentum per unit mass. These two relations can be used to derive the identity:
$\dfrac{d}{dt} ({\bf v} \times {\bf h}) = \mu \dfrac{d}{dt}\left( \dfrac{\bf r}{r} \right)$,
which implies the constancy of the vector:
${\bf e} \equiv \dfrac{{\bf v} \times {\bf h}}{\mu} - \dfrac{\bf r}{r}$.
The vector $\bf e$ is usually called eccentricity vector, or Laplace-Runge-Lenz vector.
Closed orbits in superintegrable dynamical systems
The connection between closed orbits and extra conserved quantities in the Kepler problem points to the exceptional integrability properties of this dynamical system.
An integrable system with $d$ degrees of freedom has $d$ independent integrals of motion [2], which constrain the motion of the system to a $d$-dimensional manifold within the $2d$-dimensional phase space. If more than $d$ conserved quantities exist, the system is called superintegrable, and the dimensionality of the accessible region in the phase space is further reduced.
A system with $d$ degrees of freedom can have at most $2d-1$ integrals of motion, in which case the dimensionality of the trajectory is reduced to one: for such a system, all bound trajectories are closed. This is the case of the Kepler problem, which has three degrees of freedom and five independent integrals of motion (two relations exist among the seven quantities E, $\bf h$, and $\bf e$, so only five of them are independent).
Thus the closure of orbits in the Kepler problem is a manifestation of its maximal superintegrability.
Closed orbits and hidden symmetries
A further implication of the closure of orbits is the existence of hidden symmetries of the system.
This connection is best explored employing the formalism of the Hamilton-Jacobi equation, which describes the dynamics of a system through a single partial differential equation. If a system with $d$ degrees of freedom is integrable, its Hamilton-Jacobi equation can be solved analytically by separation of variables, with the separation constants related to the $d$ integrals of motion. Since the solution of the Hamilton-Jacobi equation
in one coordinate system can yield at most $d$ constants of motion,
superintegrable systems must be separable in more than one coordinate
system.
The Hamilton-Jacobi equation for the Kepler problem is separable, and therefore analytically solvable, in both spherical and parabolic coordinates. This property corresponds to the availability of multiple
choices of the three constants of motion required for the variable separation. While the spherical coordinates emphasize the spherical symmetry associated with the conservation of the magnitude of the total angular
momentum vector, the parabolic coordinates emphasize the conservation of
the Laplace-Runge-Lenz vector.
1. Starting from the Hamiltonian in spherical coordinates $(r, \vartheta, \varphi)$:
${\cal H} = \dfrac{1}{2}\left[p_r^2 + \dfrac{1}{r^2}p_\vartheta^2 + \dfrac{1}{r^2\sin^2\vartheta} p_\varphi^2 \right] - \dfrac{\mu}{r}$,
we can write the Hamilton-Jacobi equation for the Kepler problem as:
$\dfrac{1}{2}\left[\left(\dfrac{\partial S}{\partial r}\right)^2 + \dfrac{1}{r^2} \left(\dfrac{\partial S}{\partial \vartheta}\right)^2 + \dfrac{1}{r^2\sin^2\vartheta} \left(\dfrac{\partial S}{\partial \varphi}\right)^2 \right] - \dfrac{\mu}{r} + \dfrac{\partial S}{\partial t} = 0$,
which is separable, leading to the formal solution:
$S = - E\, t + h_z\, \varphi + \int \sqrt{h^2 - \frac{h_z^2}{\sin^2 \vartheta}} d\vartheta + \int \sqrt{2(E - \frac{\mu}{r}) - \frac{h^2}{r^2}} dr$,
where the separation constants $E$, $h_z$, $h$ are the energy, the component of the angular momentum along the polar axis, and the magnitude of the total angular momentum, respectively.
2. In (cylindrical) parabolic coordinates $(\xi, \eta, \phi)$, which can be defined in terms of standard cylindrical coordinates $(s, \phi, z)$ through the relations $z = \frac{1}{2}(\xi - \eta)$, $s = \sqrt{\xi \eta}$, the Hamiltonian is:
${\cal H} = \dfrac{2(\xi \, p_\xi^2 + \eta \, p_\eta^2)}{\xi + \eta} + \dfrac{p_\phi^2}{2\xi \eta} - \dfrac{2\mu}{\xi + \eta}$,
and the Hamilton-Jacobi equation becomes:
$\dfrac{2}{\xi + \eta}\left[ \xi \left(\dfrac{\partial S}{\partial \xi}\right)^2 + \eta \left(\dfrac{\partial S}{\partial \eta}\right)^2 \right] + \dfrac{1}{2\xi \eta} \left(\dfrac{\partial S}{\partial \phi}\right)^2 - \dfrac{2\mu}{\xi + \eta} + \dfrac{\partial S}{\partial t} = 0$,
which is again separable, leading to the solution:
$S = -E\, t + h_z\, \phi + \int \sqrt{\frac{E}{2} + \frac{c_z-\mu}{2\xi} - \frac{h_z^2}{4\xi^2}}d\xi + \int \sqrt{\frac{E}{2} - \frac{c_z+\mu}{2\eta} - \frac{h_z^2}{4\eta^2}}d\eta$,
where the constant $c_z$, related to the Laplace-Runge-Lenz vector, appears instead of the total angular momentum.
These complementary viewpoints hint at the existence of hidden symmetries of higher order than the obvious 3D rotational symmetry in the Kepler problem. By mapping the system to a 4D sphere, the conserved quantities are revealed as components of a single, higher-dimensional conserved quantity. This unified perspective arises from the invariance of the system under 4D rotations, reflecting the full symmetry group of the Kepler problem.
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