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GENERALIZED COORDINATES

Coordinates, such as cartesian, spherical, or polar, are independent parameters used to specify a particle’s position and orientation in space. The locations and orientations of n particles can be described uniquely by 6n coordinates.
It is often advantageous to define these coordinates in another set of independent quantities of the form
(1) x1 =  x1(q1, q2, ..., qN)     y1 =  y1(q1, q2, ..., qN)     z1 =  z1(q1, q2, ..., qN)     φ1 =  φ1(q1, q2, ..., qN)     θ1 =  θ1(q1, q2, ..., qN)     ψ1 =  ψ1(q1, q2, ..., qN)     ...   zn =  zn(q1, q2, ..., qN)     ...   ψn =  ψn(q1, q2, ..., qN)    
where N is the number of unconstrained degrees of freedom. N depends on the number of constraints should always be less than or equal to 6n (see the explanation of degrees of freedom and constraints in Deriving Equations of Motion). The variables qi are known as generalized coordinates, and Eq. 1 defines the coordinate transformation from generalized coordinates to cartesian coordinates with orientations defined by the Euler angles θ, φ, and ψ.
Any set of parameters that can uniquely describe the position and orientation of the bodies in a system is a set of generalized coordinates. To ensure that the set of generalized coordinates uniquely describes the position and orientation of the system, there must be N generalized coordinates for a system with N unconstrained degrees of freedom.