Abstract: In this paper, by using Mathematica software to disclose the relationship between partial derivative and high-order derivative of different variables in the function r=r(q(t),t), the symmetry of Yang Hui Triangle of the function r=r(q(t),t) is discovered. Combining the symmetry of Yang Hui triangle with the Newtonian second law, the high-order differential equations of motion are deduced. Finally the high-order differential equations of systems under ideal constraints are discussed.

Keywords: 
• symmetry of Yang Hui Triangle /
• Newtonian second law /
• high-order differential equations of motion /
• high-order force /
• energy of high-order velocity /
• the ideal constraint