  We study the Hamilton-Jacobi formulation of effective mechanical actions
associated with holographic renormalization group flows when the field theory
is put on the sphere and mass terms are turned on. Although the system is
supersymmetric and it is described by a superpotential, Hamilton's
characteristic function is not readily given by the superpotential when the
boundary of AdS is curved. We propose a method to construct the solution as a
series expansion in scalar field degrees of freedom. The coefficients are
functions of the warp factor to be determined by a differential equation one
obtains when the ansatz is substituted into the Hamilton-Jacobi equation. We
also show how the solution can be derived from the BPS equations without having
to solve differential equations. The characteristic function readily provides
information on holographic counterterms which cancel divergences of the
on-shell action near the boundary of AdS.
