Hamiltonians of Spherically Symmetric, Scale-Free Galaxies in
  Action-Angle Coordinates
  We present a simple formula for the Hamiltonian in terms of the actions for
spherically symmetric, scale-free potentials. The Hamiltonian is a power-law or
logarithmic function of a linear combination of the actions. Our expression
reduces to the well-known results for the familiar cases of the harmonic
oscillator and the Kepler potential. For other power-laws, as well as for the
singular isothermal sphere, it is exact for the radial and circular orbits, and
very accurate for general orbits. Numerical tests show that the errors are
always small, with mean errors across a grid of actions always less than 1 %
and maximum errors less than 2.5 %. Simple first-order corrections can reduce
mean errors to less than 0.6 % and maximum errors to less than 1 %. We use our
new result to show that :[1] the misalignment angle between debris in a stream
and a progenitor is always very nearly zero in spherical scale-free potentials,
demonstrating that streams can be sometimes well approximated by orbits, [2]
the effects of an adiabatic change in the stellar density profile in the inner
regions of a galaxy weaken any existing 1/r density cusp, which is reduced to
$1/r^{1/3}$. More generally, we derive the full range of adiabatic cusp
transformations and show how to relate the starting cusp index to the final
cusp index. It follows that adiabatic transformations can never erase a dark
matter cusp.
