  In thermal glasses at temperatures sufficiently lower than the glass
transition, the constituent particles are trapped in their cages for
sufficiently long time such that their {\em time-averaged positions} can be
determined before diffusion and structural relaxation takes place. The
effective forces are those that hold these average positions in place. In
numerical simulations the effective forces $\B F_{ij}$ between any pair of
particles can be measured as a time average of the {\em bare} forces $\B
f_{ij}(\B r_{ij}(t))$. In general even if the bare forces come from two-body
interactions, thermal dynamics dress the effective forces to contain many-body
interactions. Here we develop the effective theory for systems with generic
interactions, where the effective forces are derivable from an effective
potential and in turn they give rise to an effective Hessian whose eigenvalues
are all positive when the system is stable. In this Letter we offer analytic
expressions for the effective theory, and demonstrate the usefulness and the
predictive power of the approach.
