4.3. Integrals of motion¶
An integral of motion is any quantity \(I(\vec{x},\vec{v})\) that only depends on the current phase-space position \((\vec{x},\vec{v})\) (and not on the time) that is conserved along the orbit: \begin{equation}\label{eq-spherorb-intmotion-definition} I(\vec{x}_1,\vec{v}_1) = I(\vec{x}_2,\vec{v}_2)\,, \end{equation} for all pairs \((\vec{x}_1,\vec{v}_1)\) and \((\vec{x}_2,\vec{v}_2)\) along the same orbit. Integrals of motion play a large role in the study of galactic dynamics.
We have already encountered a few quantities that have this property. In Section 4.1 above, we demonstrated that the specific angular momentum vector \(\vec{L}\) is conserved along each orbit in a spherical potential and, because \(\vec{L} = \vec{x}\times\vec{v}\), it satisfies Equation (4.44). Thus, each orbit in a spherical mass distribution has at least three integrals of motion: the three components of the angular momentum. In Chapter 3.1, we showed that the specific energy \(E\) is conserved for any static potential (whether spherical or not; Equation 3.12). Because for a static potential, \(E = |\vec{v}|^2/2 + \Phi(\vec{x})\), it also satisfies Equation (4.44) and is therefore an integral of motion. Orbits in static spherical potentials therefore have at least four integrals of motion.
Integrals of motion like the specific energy and angular momentum are of high importance in the study of galactic orbits and of galactic equilibria, because they are isolating integrals. This means that these integrals restrict the orbit to a sub-space (typical of lower dimensionality) of the full phase-space volume that they would otherwise have access to. For example, in a static, spherical mass distribution, an orbit could in principle reach all of the six-dimensional phase space, but as we demonstrated in Section 4.1, the conservation of the direction of the angular momentum limits the orbit to a two-dimensional plane in configuration space, or a four-dimensional sub-space of the full phase space. Within this four-dimensional sub-space, the conservation of the magnitude of the angular momentum (the remaining component after accounting for two for the direction) restricts the orbit further to the three-dimensional sub-space where the angular velocity is determined by the radius through \(\dot{\psi} = L/r^2\) (Equation 4.7). Finally, energy conservation relates the radial velocity \(\dot{r}\) to the radius as well through Equation (4.8). Most spherical potentials do not have any further, independent isolating integrals of motion and orbits can explore the full two-dimensional area in \((r,\psi)\) at \(r_p \leq r \leq r_a\). Orbits can also have non-isolating integrals, but these are not generally useful because they do not meaningfully restrict the complexity of orbits.
When one thinks of integrals of motion, one typically thinks of such quantities as “angular momentum” or “energy”. But as soon as an orbit has one integral of motion, the orbit in fact has infinitely many of them. This is because any function \(f(I)\) of one or more integrals of motion is itself an integral of motion, because Equation 4.44 holds for any function of \(I\). Such a function can be explicit or implicit. For example, we have introduced the peri- and apocenter radius in a static, spherical potential in Section 4.1 and because these are implicitly defined as the solution of Equation (4.11), which otherwise only depends on \(E\) and \(L\), they are also conserved and they can be computed based on the current phase-space position alone. So the peri- and apocenter radii are also integrals of motion. Similarly, one could use the radius of a circular orbit with specific angular momentum \(L\) (the radius \(r_L\) such that \(L = r_L\,v_c(r_L)\); again an implicit definition, in terms of \(L\) alone this time). Another useful alternative set of integrals of motion are the actions that we discussed in Chapter 3.4.3, because they are in many ways the natural coordinates to use when studying orbits and galactic equilibria. All of these alternatives are in fact useful when using integrals of motion to build models of galaxies. However, keep in mind that such integrals of motion that are fully degenerate with other integrals of motion do not lead to additional restrictions on the phase-space volume covered by an orbit (even though they can be isolating integrals as well). We will refer to a set of integrals of motion where no individual integral can be computed solely based on the others as a set of independent integrals of motion.
The angular momentum is in fact a bit of a special set of integrals of motion even among other isolating integrals of motion. The reason for this is that the spherical symmetry of the mass distribution associated with the angular momentum allows the Hamilton-Jacobi equation that we discussed in Chapter 3.4.3 to be simplified (and in fact solved) through the technique of separation of variables (see Section 4.4 below). What this means in practice is that the motion becomes explicitly the combination of three oscillations: a first trivial one of the orbit’s position with respect to the plane perpendicular to the angular-momentum vector (trivial, because the oscillation amplitude is zero, as the orbit does not leave this plane), a second one in \(\psi\) with dynamics \(\dot{\psi} = L/r^2\) that couples to the third oscillation in \(r\) determined by the effective potential \(\Phi_\mathrm{eff}(r;L)\). Thus, the conservation of the angular momentum does not only restrict the orbit to lie within a three-dimensional subspace of six-dimensional phase-space, it is also directly responsible for the fact that the orbit can be described as a simple \(r\) oscillation in the effective potential, and an \(r\)-dependent \(\phi\) oscillation. When one actually solves the Hamilton-Jacobi relation, one finds that two components of the angular momentum vector are in fact actions (see Chapter 3.4.3; the third component of the angular momentum is not an action, but instead related to one of the angles in the action-angle formalism, which is constant for spherical mass distributions rather than increasing linearly as usual). In Chapter 9.1, we will see that the conservation of the \(z\)-component of the angular momentum in axisymmetric potentials has a similar simplifying effect. Without this special property, the study of galactic orbits would be significantly harder!
We discussed Noether’s theorem in Chapter 3.3. By Noether’s theorem, a continuous symmetry property results in an integral of motion. However, the opposite direction of this implication does not hold in any useful manner: just because a system has an integral of motion, does not mean that it necessarily has a corresponding continuous symmetry property that is in any way useful (given any integral of motion, it is nevertheless possible to define a convoluted symmetry using Hamiltonian mechanics). For example, when we discuss orbits in static, axisymmetric disks, we will see that such orbits appear to have an exact integral of motion in addition to the specific energy and \(z\)-component of the angular momentum, but this integral cannot generally be calculated analytically and does not correspond to a simple symmetry property of the system.
Regular orbits in general mass distributions typically have three independent integrals of the motion, but not more. Orbits in static spherical potentials are therefore special because they have at least four independent integrals of motion. Physically, the additional integral of motion causes one of the oscillations that regular orbits normally have, the oscillation of the orbit with respect to the average orbital plane, to vanish. Orbits in the point-mass and homogeneous-sphere mass distributions that we discussed above, actually have an additional, fifth integral of the motion. This fifth integral of motion is responsible for the fact that the orbits in these mass distributions close and form ellipses, thus requiring only a single variable that is a function of \((\vec{x},\vec{v})\) to describe where a body is along the orbit (for Keplerian orbits, this is, e.g., the true or eccentric anomaly). For Keplerian orbits, this fifth integral of the motion is typically taken to be the Laplace–Runge–Lenz vector. But for both the point-mass and homogeneous-sphere potential, it can be taken to be the unit vector in the orbital plane (thus, only requiring a single number to specify) that points to the pericenter (at \(x\geq 0\) for the homogeneous sphere). For general spherical potentials, this vector is not conserved, because the azimuth of the pericenter changes from radial period to radial period, but for the point-mass and homogeneous-sphere potentials, the pericenter remains fixed in place for all time.
We will often use spherical mass distributions, and even the point-mass potential, as approximations of galactic mass distributions, because their simple properties make them so much easier to deal with. However, the fact that orbits in spherical potentials have more independent integrals of motion than orbits in more realistic galactic mass distributions is something to remain vigilant about when doing this. For example, in problems where any motion perpendicular to the average orbital plane is important, the fact that this motion is zero for spherical potentials renders them inapplicable to studying these problems. Similarly, Keplerian orbits are sometimes a nice approximation to make, because they are known analytically and are quite simple. But when one, for example, studies how a population of stars fills the entire volume of phase space allowed by its integrals of motion, the fact that Keplerian orbits only fill a one-dimensional sub-space, rather than the three-dimensional one of typical regular orbits, can trick one into making incorrect inferences and predictions about this process.
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