"In a rigid body free to move in any manner there are six degrees of freedom, and the generalised coordinates most frequently chosen in this case are the three rectangular coordinates of some point in the body and three angular coordinates determining the orientation of the body about that point, generally the angles θ, φ, ψ of ordinary occurrence in rigid dynamical problems."
In analytical mechanics, generalized coordinates are a set of parameters used to represent the configuration of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference configuration. The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes thes
"When the body degenerates into a material straight line the number of degrees of freedom is reduced to five; and when this straight line is constrained to move parallel to some fixed plane the number of degrees of freedom is still further reduced to four."
"Full use of Lagranges own made the unification of the varied principles of statistics and dynamics possible—in statistics by the use of the principle of virtual velocities, in dynamics by the use of . This led... to generalized coordinates and to the equation of motion in their "Lagrangian" form... Newtons geometrical approach was now fully discarded; Lagranges book was a triumph of pure analysis."
"When the position of every point of a material system can be determined in terms of any independent variables n in number, the system is said to possess n degrees of freedom, and the n independent variables are called the generalised coordinates. The choice of the particular independent variables is perfectly arbitrary, and may be varied indefinitely, but the number of degrees of freedom cannot be either increased or diminished."