In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.

Lie symmetry of Appell equation and conserved quantity deduced directly by Lie symmetry for a Chetaev's type constrained mechanical system are investigated. The relations between Lagrange function and A function are analyzed. A general approach of studying conserved quantity deduced by Lie symmetry of Appell equation for a Chetaev's type constrained mechanical system is discussed. The definition and the criterion of Lie symmetry of Appell equations under the infinitesimal transformations of groups are given. The structural equation of Lie symmetry and the expression of conserved quantity deduced directly by Lie symmetry are obtained. An example is given to illustrate the application of the results.

The characteristics of stationary and non-stationary skew-gradient systems are studied. The skew-gradient representations of holonomic systems, Birkhoffian systems, generalized Birkhoffian systems, and generalized Hamiltonian systems are given. The characteristics of skew-gradient systems are used to study integration and stability of the solution of constrained mechanical systems. Examples are given to illustrate applications of the result.

Mei symmetry and Mei conserved quantity of Nielsen equations for a nonholonomic system of unilateral non-Chetaev's type in the event space are studied. The differential equations of motion for the system are established. The definition and the criteria of Mei symmetry, loose Mei symmetry and strict Mei symmetry for the system are respectively given. The existence condition and the expression of Mei conserved quantity deduced directly from Mei symmetry are obtained. An example is given to illustrate the application of the results.

The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.