  The gauge/string dualities have drawn attention to a class of variational
problems on a boundary at infinity, which are not well defined unless a certain
boundary term is added to the classical action. In the context of supergravity
in asymptotically AdS spaces these problems are systematically addressed by the
method of holographic renormalization. We argue that this class of a priori ill
defined variational problems extends far beyond the realm of holographic
dualities. As we show, exactly the same issues arise in gravity in non
asymptotically AdS spaces, in point particles with certain unbounded from below
potentials, and even fundamental strings in flat or AdS backgrounds. We show
that the variational problem in all such cases can be made well defined by the
following procedure, which is intrinsic to the system in question and does not
rely on the existence of a holographically dual theory: (i) The first step is
the construction of the space of the most general asymptotic solutions of the
classical equations of motion that inherits a well defined symplectic form from
that on phase space. The requirement of a well defined symplectic form is
essential and often leads to a necessary repackaging of the degrees of freedom.
(ii) Once the space of asymptotic solutions has been constructed in terms of
the correct degrees of freedom, then there exists a boundary term that is
obtained as a certain solution of the Hamilton-Jacobi equation which
simultaneously makes the variational problem well defined and preserves the
symplectic form. This procedure is identical to holographic renormalization in
the case of asymptotically AdS gravity, but it is applicable to any Hamiltonian
system.
