Lagrange子流形理论在约束系统动力学中的应用

Contents: 1) In the framework of Birkhoffian systems the Birkhoff-Jacobi theory will be established to solve the dynamical problems of nonconservative systems after the condition of fibred general symplectic manifolds is obtained and its horizontal Lagrangian submanifolds is constructed. 2) When the fibred presymplectic structure and its horizotal Lagrangian submanifolds are constructed the Dirac-Hamilton-Jacobi theory can be established to solve the dynamical problems of singular systems. And the Dirac-Hamilton-Jacobi theory can also be extended to the generalized Birkhoff-Jacobi theory which can be utilized to solve the dynamical problems of nonholonomic constrained systems in the framework of generalized Bikrhoffian systems. 3) If we let the dynamical variables of Birkhoffian systems as the local coordinates on the base manifolds of the vector bundles, the Lagrangian submanifolds which is transversal to the fibres or horizontal to the base manifolds of the tangent bundles and the cotangent bundles will be established. In this way the Lagrange-Birkhoff-Jacobi theory and the Hamilton-Birkhoff-Jacobi theory can be obtained which will be utilized to nonconservative dynamical systems and singular dynamical systems. 4) The existing conditions of Lagrangian submanifolds which is transversal to the fibres or horizontal to the base manifolds of fibred bundles for nonholonomic constrained manifolds can be derived by utilizing of the horizontal distribution theory of Ehresmann connections. At the same time the generalized Hamilton-Jacobi theorem of nonholonomic constrained systems is obtained. And the differential equations of motion for nonholonomic constrained systems can be solved by using of the generalized Hamilton-Jacobi theorem..Meaning: 1) The relations of constrained manifolds and its submanifolds to the dynamical problems of constrained systems will be more and more clearly to be understand by meaning of the research of Lagrangian submanifolds theory. 2) New approaches which can be utilized to solve the problems of singularity or nonholonomicity, dynamical modeling and the integration of differential equations for nonholonomic constrained systems or singular constrained systems can be derived by using of Lagrangian submanifolds theory.

内容：1）得到一般辛流形纤维化条件并构造其拉格朗日子流形，在伯克霍夫系统框架内建立解决非保守系统动力学问题的B-J理论。2）构造预辛流形的纤维化结构及其水平拉格朗日子流形，建立解决奇异系统动力学问题的D-H-J理论，并将其推广为在广义伯克霍夫系统框架内解决非完整系统动力学问题的广义B-J理论。3）以伯克霍夫系统动力学变量为一般辛向量丛底流形的局域坐标，构建其切丛和余切丛上水平拉格朗日子流形，建立应用于非保守和奇异动力学系统的L-B-J理论和H-B-J理论。4）利用埃雷斯曼联络的水平分布，得到非完整系统纤维丛上水平拉格朗日子流形存在条件，建立非完整约束系统的广义H-J 定理，实现非完整系统的求解。意义：1）拉格朗日子流形理论研究使约束流形及其子流形与约束系统动力学问题的关系更清晰。2）拉格朗日子流形理论为研究奇异约束系统和非完整系统的奇异性和非完整性、动力学建模、积分等动力学问题提供新途径。

研究背景：哈密顿-雅克比理论的几何本质是建立在辛流形的横向或者水平的Lagrange子流形上的动力学理论体系，依据哈密顿-雅克比理论的本质几何特征：纤维化约束流形横向或者水平的Lagrange子流形理论，从约束动力学系统的辛结构、纤维丛结构的角度，将哈密顿-雅克比理论拓展到非保守约束和非完整约束动力学系统。实现了在非保守、非完整系统的哈密顿-雅克比理论框架内对非保守系统、非完整系统的可积性、动力学建模、运动方程的积分等动力学问题的研究，并为非保守、非完整约束系统的应用研究开拓新的途径。.研究内容及重要结果：（1）关于Lagrange子流形理论在约束动力学系统的应用研究中，利用辛同胚映射和Lagrange子流形之间的对应关系，解释了正则变换和勒让德变换的几何意义。（2）在非完整系统Riemann-Cartan空间上的准动量映射问题研究中，给出了非完整约束系统在Riemann-Cartan空间中的准牛顿定律、准动量定理和Lagrange方程。（3）关于与哈密顿-雅克比方法类似的场积分方法的改进，并将其应用于线性非完整约束系统的积分问题研究。（4）关于非定常完整约束系统的线性映射方法研究，基于线性映射理论，提出了约束系统降维的方法。（5）在非完整约束系统的准正则化及其哈密顿-雅克比理论的研究中，通过非完整约束系统的准正则化理论，研究了线性齐次非完整约束系统动力学方程积分的哈密顿-雅克比方法。（6）在非保守力学系统的哈密顿-雅克比理论及其应用研究中，推广偏微分方程的解曲面和其特征曲线的正交性关系，构建了非保守约束系统的广义哈密顿-雅克比理论。.研究意义：开展Lagrange子流形理论在约束系统动力学中的应用研究，澄清了约束流形及其Lagrange子流形的几何结构与Lagrange动力学、Hamilton动力学和哈密顿-雅克比动力学之间的关系，并为推广哈密顿-雅克比理论的应用研究奠定了几何基础。通过构造非完整约束系统纤维化子流形横向的Lagrange子流形结构，实现了对保守系统哈密顿-雅克比定理的拓展，研究了非保守约束系统和非完整约运动微分方程的积分问题，为非保守和非完整约束系统的动力学应用研究奠定了理论基础。