  Amorphous media at finite temperatures, be them liquids, colloids or glasses,
are made of interacting particles that move chaotically due to thermal energy,
colliding and scattering continuously off each other. When the average
configuration in these systems relaxes only at long times, one can introduce
{\em effective interactions} that keep the {\em mean positions} in mechanical
equilibrium. We introduce a new framework to determine these effective
force-laws that define an effective Hessian that can be employed to discuss
stability properties and density of states of the amorphous system. We
exemplify the approach with a thermal glass of hard spheres; these feel zero
forces when not in contact and infinite forces when they touch. The present
approach recaptures the effective interactions which for sufficiently dense
spheres at temperature $T$ depends on the gap $h$ between spheres as $T/h$ [C.
Brito and M. Wyart, Europhys. Lett. 76 149 (2006)]. In systems at lower
densities or with longer microscopic interaction (say like Lennard-Jones), the
emergent force laws will include ternary, quaternary and generally higher order
many-body terms, even if the microscopic interactions are strictly binary.
