A constraint is a restriction on the allowed motion of a mechanical system that appears mathematically as an equation relating the coordinates, so that those coordinates can no longer vary independently; the number of independent variables that survive the constraints is the system's degrees of freedom, counted as f = n − k for n coordinates and k independent constraint equations. The Lagrangian formulation takes this geometry of allowed motion — rather than the inventory of applied forces — as its starting point, describing dynamics through the scalar function L = T − V built from kinetic and potential energy, so that a constraint satisfied identically by the chosen coordinate requires no constraint force in the equations of motion. This belongs to analytical (Lagrangian) mechanics within classical mechanics, and supplies the prerequisite structure for generalized coordinates, the Euler–Lagrange equation, generalized momentum, and the Hamiltonian reformulation.
A constraint is a restriction on the allowed motion of a mechanical system that appears mathematically as an equation relating the coordinates, so that those coordinates can no longer vary independen…