Starting from Newtonian kinetic equations of a particle system, the energy of higher order-velocity of the system is introduced; higher-order Lagrange equations, higher-order Nielsen equations and higher-order Appell equations of a holonomic mechanical system are derived, from which we prove that the three kinds of higher-order differential equations of motion of the holonomic system are equivalent to each other. The result indicates that the higher-order differential equations of motion of the holonomic system reveal the relationship between the chan ges of the system's motion state and the rate of change of force at every order, which cannot be obtained by using Newtonian kinetic equations and the tradition al analytical mechanical equations. Therefore, the higher-order differential equ ations of motion of the holonomic system are a complement to the second-order di fferential equations of motion, including Newtonian kinetic equations and the tr aditional Lagrange equations, Nielsen equations and Appell equations.