Constraint
A restriction on the motion of a system — a surface, wire, or fixed distance. In Lagrangian mechanics, absorbed into coordinate choice.
Definition
A constraint restricts where the degrees of freedom of a system can go. A bead threaded on a wire is constrained to the wire. A pendulum bob is constrained to a sphere of fixed radius. Rigid bodies are constrained to fixed inter-particle distances.
In Newtonian mechanics, each constraint produces an unknown constraint force (the normal reaction from the wire, the tension in the pendulum string) that must be carried through every free-body diagram and eliminated at the end. Lagrangian mechanics handles constraints by a coordinate change: you choose generalised coordinates along the constraint surface, write down T − V on that surface, crank the Euler-Lagrange equation, and the constraint force never appears. This is why solving a bead-on-wire problem by Lagrange takes three lines while solving it by Newton fills a page.