a nonholonomic constraint
A constraint that restricts how a system may move at each instant without restricting where it can ultimately end up. The classic picture is a ball or a coin rolling without slipping: its velocity is limited at every moment, yet by rolling around cleverly you can bring it to any position and orientation you like.
It is expressed as a relation among the velocities that cannot be integrated into a pure coordinate relation, typically a non-integrable differential form, a sum of a_i dq_i + a_t dt = 0 that is not an exact differential. Rolling without slipping is the archetype: it links translational and rotational velocities but forbids no configuration. It thus reduces the accessible velocities (the kinematic freedom) but not the dimension of configuration space.
The ordinary 'eliminate a coordinate' trick fails here, because you cannot solve the constraint to remove a variable, so you must keep all coordinates and enforce the constraint with Lagrange multipliers. And an honest warning: applying Hamilton's principle blindly to a nonholonomic system (so-called vakonomic mechanics) gives equations that differ from the physically correct ones. The right route is the Lagrange-d'Alembert principle, built on d'Alembert's principle rather than on naive variation of the action.
A coin rolling upright without slipping on a table obeys a nonholonomic constraint, yet it can be maneuvered to any position and heading on the surface.
Velocities constrained, full configuration space still reachable.
Nonholonomic means a non-integrable equality constraint on velocities, not merely an inequality; and naive action-variation gives the wrong equations here.