| In this thesis, based on the theorem of traditional analytical mechanics, such as Lagrangian equations , Hamilton's canonical equations and Hamilton's principle ,the researches of the high-order differential equations of motion are performed. These results can directly describe the motion laws of mechanical systems which have concern with the time rate of change of force and the high-order time rate of change of force. And they are complements to the second-order differential equations of motion, including Newtonian kinetic equations and the traditional analytical mechanics. My central work and some conclusions of the thesis are as follows:(1) using the principle of jerk and Newton second law, the three-order Lagrange equation is derived, it's the most important differential equation. Then based on the three-order Lagrange equation, Hamilton's function of acceleration H *and generalized momentum P *are defined and pseudo-Hamilton's canonical equations corresponding with three-order Lagrange equation are obtained in this thesis. The equations are similar to Hamilton's canonical equations of analytical mechanics in the form.(2) based on the theorem of the high-order velocity energy, the three-order Hamilton's principle, the four-order Hamilton's principle the high-order Hamilton's principle for general holonomic systems is deduced by use of integration and variation principle. Then, three-order Lagrangian equations and four-order Lagrangian equations have been obtained from high-order Hamilton's principle.(3) The Hamilton's principle on high-order Lagrangian function (L n) is given by use of integration and variation principle. From the Hamilton's principle on high-order Lagrangian function, High-order Lagrangian equations can be obtained.(4) If the condition of variationδt =0 is satisfied, the higher-order motive differential equations which are corresponding to higher-order Lagrangian function (L n) are deduced. The results can enrich the theory of analytical mechanics...
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