完整力学系统的高阶运动微分方程

Higher-order differential equations of motion of a holonomic mechanical system

从质点系的牛顿动力学方程出发,引入系统的高阶速度能量,导出完整力学系统的高阶Lagrange方程、高阶Nielsen方程以及高阶Appell方程,并证明了完整系统三种形式的高阶运动微分方程是等价的.结果表明,完整系统高阶运动微分方程揭示了系统运动状态的改变与力的各阶变化率之间的联系,这是牛顿动力学方程以及传统分析力学方程不能直接反映的.因此,完整系统高阶运动微分方程是对牛顿动力学方程及传统Lagrange方程、Nielsen方程、Appell方程等二阶运动微分方程的进一步补充.

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 changes 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 traditional analytical mechanical equations. Therefore, the higher-order differential equations of motion of the holonomic system are a complement to the second-order differential equations of motion, including Newtonian kinetic equations and the traditional Lagrange equations, Nielsen equations and Appell equations.