a holonomic constraint

A constraint that can be written purely as a relationship among positions (and possibly time), like 'this rod is rigid, so its two ends stay a fixed distance apart,' or 'this bead must stay on the wire.' It cleanly ties down where things are allowed to be.

It is expressible as an equation f(q_1, ..., q_n, t) = 0 involving only the coordinates and time, never the velocities. Each independent holonomic constraint removes one degree of freedom and lets you eliminate one coordinate, so you can work in a smaller independent set. A particle confined to a sphere of radius R, for instance, obeys x^2 + y^2 + z^2 - R^2 = 0.

Holonomic constraints are the friendly case: you can always choose generalized coordinates that satisfy them automatically, so the bare Euler-Lagrange machinery applies directly, or you keep the extra coordinate and use one Lagrange multiplier per constraint if you also want the force of constraint. If a constraint genuinely involves velocities and cannot be integrated back into a pure position relation, it is nonholonomic and this tidy reduction fails.