研究内容：1）运用变分法构造非完整系统非水平的H-J方程，建立非完整系统的H-J定理，积分非完整系统哈密顿方程。2）运用化零保辛类正则变换和奇异哈密顿系统的狄拉克约束理论，分别构造B-H-J方程和D-H-J方程并求解其完全积分，利用广义H-J定理，积分伯克霍夫方程。3）构造非完整系统纤维化的约束流形的拉格朗日子流形，运用联络理论研究其与纤维空间的横向关系，推广和证明非完整系统的H-J定理；构造一般辛流形的拉格朗日子流形及其上的B-H-J向量场的局部恰当实现。4）利用生成函数法研究非完整系统的H-J方程的几何数值积分问题，构造数值积分子和B-H-J方程的变分自伴随理论。研究意义：1）推广H-J定理和积分非完整系统哈密顿方程和伯克霍夫方程的H-J方法。2) 以拉格朗日子流形理论和联络理论构建非完整系统H-J理论的几何基础。3）完善生成函数法研究非完整系统的B-H-J方程的几何数值积分问题。

Contents: 1) Calculus of varaitions is used to construct the non-horizontal Hamilton-Jacobi equation for nonholonomic systems and to establish the Hamilton-Jacobi theorem for integrating the non-homogenous Hamilton's equations of the systems. By means of some symplectic-preserving canonical transformations whose transformed Hamiltonian is identicall null and theory of Dirac constraint, the Birkhoff-Hamilton-Jacobi equation and Dirac-Hamilton-Jacobi equation are respectively derived, whose complete integrals can be found out. Generalized Hamilton-Jacobi theorem is utilized to integrate Birkhoff's equations. Lagrangian submanifolds of fibred constraint manifolds are made up, whose transversality to fibres is discussed with the help of connection theory. The Hamilton-Jacobi theorem for nonholonomic systems is generalized and proved geometrically. Lagrangian submanifolds of a general symplectic manifold are constructed, on which the Birkhoff-Hamilton-Jacobi vector fields are represented exactly and locally. 4) The geometric numerical integration of Hamilton-Jacobi equation for nonholonomic systems is investigated making use of generating functions, togother with the geometric numerical integrator and variational self-adjoint theory of the Birkhoff-Hamilton-Jacobi equation..Meaning: 1) Hamilton-Jacobi theorem will be generalized, to be applied to integrate non-canonical Hamilton's equations of nonholonomic systems and Birkhoff's equations. 2) Lagrangian submanifolds and connection theory are taken as the geometric foundation of Hamilton-Jacobi theory for nonholonomic systems. 3) The generating function method will be improved to be applicable to the geometric numerical integration of Birkhoff-Hamilton-Jacobi equation for nonholonomic systems.