Generalized coordinates
![Open straight pathOpen curved path F(x, y) = 0Closed curved path C(x, y) = 0One generalized coordinate, one degree of freedom, on paths in 2d. Only one number is needed to uniquely specify positions on the curve, the examples shown are the arc length s or angle θ. Both of the Cartesian coordinates (x, y) are unnecessary since either x or y is related to the other by the equations of the curves. They can also be parameterized by s or θ.](/Images/godic/202501/15/Generalized_coordinates_open_straight_path_2d_1df.svg1300.png")
![](/Images/godic/202501/15/Generalized_coordinates_open_curved_path_2d_1df.svg1300.png")
![](/Images/godic/202501/15/Generalized_coordinates_closed_path_2d_1df.svg1300.png")
![Open curved path F(x, y) = 0. Multiple intersections of radius with path.Closed curved path C(x, y) = 0. Self-intersection of path.The arc length s along the curve is a legitimate generalized coordinate since the position is uniquely determined, but the angle θ is not since there are multiple positions for a single value of θ.](/Images/godic/202501/15/Non_generalized_coordinates_open_curved_path_2d_1df.svg1300.png")
In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration. The generalized velocities are the time derivatives of the generalized coordinates of the system.