Lagrange's equations provide an alternative method to Newton's laws for deriving the equations of motion for mechanical systems. Lagrange's method uses generalized coordinates and the kinetic and potential energies of the system to derive scalar differential equations, avoiding the need to solve for constraint forces or accelerations directly. The number of degrees of freedom for a system, which determines the number of differential equations needed, depends only on the number of coordinates and constraints and is independent of the particular coordinate system used.