Degrees of Freedom, Number of

in mechanics, the number of virtual displacements of a mechanical system that are independent of each other. The number of degrees of freedom depends on the number of material particles forming the system and the number and nature of the constraints applied on the system. It is equal to 3 for a free particle, 6 for a solid body, and 1 for a body having a fixed axis of rotation. For any holonomic system, that is, one with geometric constraints, it equals the number s of independent coordinates determining the position of the system and is given by the equality s = 3n – k, where n is the number of particles of the system and k is the number of geometric constraints. In a nonholonomic system, the number of degrees of freedom is less than the number of coordinates determining the position of the system; the difference is equal to the number of kinematic constraints that do not reduce to geometric (nonintegrable) constraints. The number of equations of motion and the equilibrium conditions of a mechanical system depend on the number of degrees of freedom.