| The Lagrange mechanical system can provide a unified and comprehensive description of the system’s dynamics using a Lagrange function,which brings great convenience to the study of the system’s properties.Therefore,for a given differential equation(system),finding its variational structure,also known as the Lagrange structure,is a very important task.For some ordinary differential equation,by solving their Jacobi last multipliers and combining the Lie symmetry of the equations,we can obtain the Lagrange functions of these equations,thus transforming them into the equations of motion of some variational problems.Applying Noether theorem to Lagrange function will get the corresponding first integral of the equation,namely conserved quantity,and then reduce the order of the equation.In this article,we have derived multiple Jacobi last multipliers and multiple Lagrange functions for several types of differential equations,thus proving that common differential equations have many non equivalent Lagrange structures.In Chapter 2,we introduced two second-order differential equations with two and three Lie point symmetries analyzed by Ibragimov,and obtained several Lagrange functions of the equation.These results can be compared with the existing results of Ibragimov.At the same time,we discussed a special class of second-order ordinary differential equation.For this class of equations with eight Lie point symmetries,we obtained more Jacobi last multipliers and Lagrange functions.By discussing three types of equations with different numbers of Lie point symmetries,it is found that the equation has multiple Lagrange functions and first integration.In chapter 3,we discuss a class of biological systems and two special first order ordinary differential equation.Due to the existence of Jacobi last multiplier,many second-order differential equations have Lagrange descriptions.If a set of planar differential equations is given,but it cannot be reduced to a second-order differential equation,Jacobi’s last multiplier provides a solution for solving the Lagrange function of the system.Using Noether theorem,we obtain the conserved quantity of the equation.For different Lagrange quantities,they can all characterize the evolutionary laws of the biological system.The combination of Lagrange quantities and Euler-Lagrange equations yields dynamic motion equations,which also ensures the effectiveness of the variational principle.In this paper,we use the method of Jacobi last multiplier to solve the inverse variational problems of some ordinary differential equation(systems),that is,to find multiple variational structures of these equations.The concept of multiplier introduced by Jacobi,the combination of Lie’s idea of invariance under continuous transformation groups and Noether theorem of conserved quantity provide us with an efficient tool to solve inverse variational problems of differential equations.Compared to the formal or operator methods proposed by Helmholtz and Volterra,the Jacobi last multiplier method is more user-friendly,reflecting the intersection and fusion of abstract mathematical theories and specific physical models. |
