Basis function expansions for galactic dynamics: Spherical versus
  cylindrical coordinates
  Aims. The orbital structure of galaxies is strongly influenced by the
accuracy of the force calculation during orbit integration. We explore the
accuracy of force calculations for two expansion methods and determine which
one is preferable for orbit integration.
  Methods. We specifically compare two methods, one was introduced by Hernquist
\& Ostriker (HO), which uses a spherical coordinate system and was built
specifically for the Hernquist model, and the other by Vasiliev \& Athanassoula
(CylSP) has a cylindrical coordinate system. Our comparisons include the Dehnen
profile, its triaxial extension (of which the Hernquist profile is a special
case) and a multicomponent system including a bar and disk density
distributions for both analytical models and $N$-body realizations.We
specifically compare two methods, one was introduced by Hernquist \& Ostriker
(HO), which uses a spherical coordinate system and was built specifically for
the Hernquist model, and the other by Vasiliev \& Athanassoula (CylSP) has a
cylindrical coordinate system. Our comparisons include the Dehnen profile, its
triaxial extension (of which the Hernquist profile is a special case) and a
multicomponent system including a bar and disk density distributions for both
analytical models and $N$-body realizations.
  Results. For the generalized Dehnen density, the CylSP method is more
accurate than the HO method for nearly all inner power-law indices and shapes
at all radii. For $N$-body realizations of the Dehnen model, or snapshots of an
$N$-body simulation, the CylSP method is more accurate than the HO method in
the central region for the oblate, prolate, and triaxial Hernquist profiles if
the particle number is more than $5\times 10^5$. For snapshots of the Hernquist
models with spherical shape, the HO method is preferred. (Abridged)
