In `n` space dimensions, many orbits can formally be
decomposed into `n` independent periodic motions. Such
*regular* orbits may be represented as winding paths on an
`n`-dimensional torus. On the other hand, *irregular* or
stochastic orbits defy any such representation; such orbits may wander
anywhere permitted by conservation of energy.

**Constants of motion** are functions of phase-space coordinates
and time which are constant along orbits:

(1) C(r(t), v(t), t) = constant ,where

**Integrals of motion** are functions of phase-space coordinates
alone which are constant along orbits:

(2) I(r(t), v(t)) = constant .An integral of motion can't depend on time; thus all integrals are constants of motion, but not all constants are integrals. Integrals are important because they constrain the shapes of orbits. In a phase-space of

Consider the motion of a star in a spherically-symmetric potential,
`Phi = Phi(|r|)`. The orbit of the star remains in a plane
perpendicular to the angular momentum vector, and it's natural to
adopt a polar coordinate system; call the coordinates `R = |r|`
and `phi`. The system has `n = 2` degrees of freedom,
so the phase space has `4` dimensions.

The **equations of motion** can be derived by starting with the
lagrangian,

1 2 2 2 (3) L(R,phi,Rdot,phidot) = - (Rdot + R phidot ) - Phi(R) , 2where

dL (4) ------- = Rdot = v_R , d(Rdot) dL 2 (5) --------- = R phidot = R v_phi = J ; d(phidot)here

1 2 2 2 (6) H(R,phi,v_R,J) = - (v_R + J / R ) + Phi(R) . 2Then the equations of motion are

dR dH (7) -- = ------ = v_r , dt d(v_r) d(phi) dH J (8) ------ = -- = --- , dt dJ R^2 d(v_r) dH d(Phi) J^2 (9) ------ = - -- = - ------ + --- , dt dR dR R^3 dJ dH (10) -- = - ------ = 0 . dt d(phi)Here

The system has two **integrals of motion**. One, of course, is
the total energy `E`, numerically equal to the value of
`H`. The other is the angular momentum `J`. These
quantities are given by

2 2 (11) E = - (v_R + v_phi ) + Phi(R) , (12) J = R v_phi .Each of these integrals defines a hypersurface in phase space, and the orbit must remain in the intersection of these hypersurfaces. This can be visualized by ignoring the

For an orbit of a given `J`, the system may be reduced to
one degree of freedom by defining the *effective potential*,

J^2 (13) Psi(R) = Phi(R) + ----- ; 2 R^2the corresponding equations of motion are then just

dR (14) -- = v_R , dt d(v_R) d(Psi) (15) ------ = - ------ . dt dRBecause

In addition to its periodic radial motion described by Eqs. 14 &
15, a star also executes a periodic azimuthal motion as it orbits the
center of the potential. If the radial and azimuthal periods are
incommensurate, as is usually the case, the resulting orbit never
returns to its starting point in phase space; in coordinate space such
an orbit is a rosette (BT87, Fig. 3-1). The Keplerian potential is a
very special case in which the radial and azimuthal periods of all
bound orbits are *equal*. The only other potential in which all
orbits are closed is the harmonic potential generated by a uniform
sphere; here the radial period is half the azimuthal one and all bound
orbits are ellipses centered on the bottom of the potential well.
Thus in the Keplerian case all stars advance in azimuth by `Delta
phi = 2 pi` between successive pericenters, while in the harmonic
case they advance by `Delta phi = pi`. Galaxies typically have
mass distributions intermediate between these extreme cases, so most
orbits in spherical galaxies are rosettes advancing by `pi <
Delta phi < 2 pi` between pericenters.

This combination of radial and azimuthal motions can be represented
as a path on a 2-torus; that is, on a rectangle with its opposing
edges glued together. Associate the azimuthal direction with the long
edge of the rectangle, and the closed loop formed by the intersection
of `E = const.` and `J = const.` with the short edge of
the rectangle; then glue opposing edges together to render both
coordinates periodic and produce what is known as an *invariant
torus*. Stellar orbits wind solenoid-like on the surface of the
torus, typically making between one and two turns through the hole of
the torus for each turn made around its circumference.

Points on the surface of the torus may be parameterized by a pair
of angles `(theta_1,theta_2)`; moreover, by stretching the
torus appropriately, the motion of a star can be described by a pair
of linear relations:

theta_1(t) = theta_1(0) + omega_1 t , (16) theta_2(t) = theta_2(0) + omega_2 t ,where the

1 2 2 (17) H(theta_1,theta_2,omega_1,omega_2) = - (omega_1 + omega_2 ) . 2

In describing axisymmetric galaxy models it's natural to use
cylindrical coordinates `(R,phi,z)`, where `R` and
`phi` are polar coordinates in the equatorial plane, and
`z` is the coordinate perpendicular to that plane. In these
coordinates, the potential has the form `Phi = Phi(R,z)`. The
equations of motion are identical to Eqs. 7-10, with additional
expressions for `z` and `v_z`:

dz (18) -- = v_z , dt d(v_z) d(Phi) (19) ------ = - ------ . dt dz

Once again, there are two classical integrals of motion:

2 2 2 (20) E = - (v_R + v_phi + v_z ) + Phi(R,z) , (21) J_z = R v_phi .

Just as for spherical potentials, it's possible to define an effective potential

J_z^2 (22) Psi(R,z) = Phi(R,z) + ----- . 2 R^2Instead of governing motion along a line as in the spherical case, the effective potential now governs the star's motion in the

On the meridional plane, which has coordinates `(R,z)`, the
effective potential `Psi` has a minimum at `R > 0, z =
0` and a steep angular momentum barrier as `R -> 0`
(BT87, Fig. 3-2). If only the energy `E` constrains the motion
of a star on this plane, one might expect a star to travel everywhere
within some closed contour of constant `Psi`. But in many
cases this is *not* observed; instead, stars launched from rest
at different points along a contour of constant `Psi` follow
distinct trajectories. This implies the existence of a *third
integral* besides the classic integrals given by Eqs. 20-21. No
general expression for a third integral exists, but it can be
evaluated by following a given orbit until its velocity on the
meridional plane vanishes, and noting where on the zero-velocity
contour the star lies. The result is a proper integral of motion,
rather than just a constant, because this procedure makes no reference
to the current time `t`.

The existence of a third integral implies that a star's orbit is a
combination of *three* periodic motions: radial, azimuthal, and
vertical. Thus the orbit can be represented as a path on an invariant
3-torus, with action-angle variables `(theta_i,omega_i)`, where
`i = 1, 2, 3`.

Some axisymmetric potentials also have orbits which wander
everywhere energetically permitted on the meridional plane. For such
orbits, the description in terms of motion on a invariant 3-torus
breaks down. These are examples of *irregular* or *stochastic
orbits*; in an axisymmetric potential, they respect only the two
classical integrals, `E` and `J_z`.

Non-axisymmetric potentials, with `Phi = Phi(x,y)` or
`Phi(x,y,z)` in Cartesian coordinates `(x,y,z)`, admit
an even richer variety of orbits. The only classical integral of
motion in such a potential is the energy,

1 2 (23) E = - |v| + Phi(r) . 2Some potentials nonetheless permit other integrals of motion, and in such potentials regular orbits may be mapped onto invariant tori. But not all regular orbits can be continuously deformed into one another; consequently, orbits can be grouped in to topologically distinct

In a *separable potential* all orbits are regular and the
mapping to the invariant tori can be constructed analytically; all
integrals of motion are known. Separable potentials are rather
special, mathematically speaking, and it's highly unlikely that real
galaxies have such potentials. However, numerical experiments show
that non-axisymmetric galaxy models with finite cores or shallow cusps
usually generate potentials with many key features of separable
potentials.

The orbits in a separable potential may be classified into distinct
*families*, each associated with a set of closed and stable
orbits. In two dimensions, for example, there are two types of
closed, stable orbits; one type (i) oscillates back and forth along
the major axis, and the other type (ii) loops around the center.
Because these orbits are stable, other orbits which start nearby will
remain nearby at later times. The families associated with types (i)
and (ii) are known as *box* and *loop* orbits, respectively
(BT87, Ch. 3.3.1).

In three dimensions, a separable potential permits four distinct orbit families:

*box*orbits,*short-axis tube*orbits,*inner long-axis tube*orbits, and*outer long-axis tube*orbits.

In scale-free models all properties have either a power-law or a
logarithmic dependence on radius. In particular, scale-free models
with density profiles proportional to `r^-2` have logarithmic
potentials and flat rotation curves. While real galaxies are not
entirely scale-free, such steep power-law density distributions are
reasonable approximations to the central regions of some elliptical
galaxies and to the halos of galaxies in general.

If the density falls as `r^-2` or faster, then box orbits
are replaced by minor orbital families called *boxlets* (Gerhard
& Binney 1985, Miralda-Escude & Schwarzschild 1989). Each
boxlet family is associated with a closed and stable orbit arising
from a resonance between the motions in the `x` and `y`
directions. Moreover, some irregular orbits also exist; these have no
integrals of motion apart from the energy `E`. In principle,
such an orbit can wander everywhere on the phase-space hypersurface of
constant `E`, but in actuality such orbits show a complicated
and often fractal-like structure.

- Binney, J. & Tremaine, S. 1987,
*Galactic Dynamics*(BT87). - Gerhard, O.E. & Binney, J.J. 1985,
*M.N.R.A.S.***216**, 467. - Miralda-Escude, J. & Schwarzschild, M. 1989,
*Ap.J.***339**, 752.

Joshua E. Barnes (barnes@galileo.ifa.hawaii.edu)

Last modified: February 8, 1997