The principle of virtual work characterizes equilibrium by the vanishing of the work of applied forces under every infinitesimal displacement compatible with bilateral constraints at a fixed time. For ideal constraints, whose reaction forces do no virtual work, it takes the form
 |
where the dot product pairs each applied force 
with a permissible displacement 
(Goldstein 1980).
Writing the positions in independent coordinates 
, define the generalized forces by 
. Then
 |
so equilibrium requires 
for each independent coordinate. For conservative forces
with scalar potential 
, the relation 
makes equilibrium a stationarity condition on
.
It is not necessarily a minimum; stability requires additional conditions. This connects
the principle with the calculus of variations
and the use of a Lagrangian function in mechanics.
