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.