Dynamical invariants and diffusion of merger substructures in
  time-dependent gravitational potentials
  This paper explores a mathematical technique for deriving dynamical
invariants (i.e. constants of motion) in time-dependent gravitational
potentials. The method relies on the construction of a canonical transformation
that removes the explicit time-dependence from the Hamiltonian of the system.
By referring the phase-space locations of particles to a coordinate frame in
which the potential remains `static' the dynamical effects introduced by the
time evolution vanish. It follows that dynamical invariants correspond to the
integrals of motion for the static potential expressed in the transformed
coordinates. The main difficulty of the method reduces to solving the
differential equations that define the canonical transformation, which are
typically coupled with the equations of motion. We discuss a few examples where
both sets of equations can be exactly de-coupled, and cases that require
approximations. The construction of dynamical invariants has far-reaching
applications. These quantities allow us, for example, to describe the evolution
of (statistical) microcanonical ensembles in time-dependent gravitational
potentials without relying on ergodicity or probability assumptions. As an
illustration, we follow the evolution of dynamical fossils in galaxies that
build up mass hierarchically. It is shown that the growth of the host potential
tends to efface tidal substructures in the integral-of-motion space through an
orbital diffusion process. The inexorable cycle of deposition, and progressive
dissolution, of tidal clumps naturally leads to the formation of a `smooth'
stellar halo.
