中文摘要：

就一般非完整约束系统，从约束方程满足的变分恒等式出发，利用增广位形流形上的向量场定义三类非自由变分，即非完整变分：vakonomic变分、HSlder变分、Suslov变分，并讨论它们之间的关系以及它们成为自由变分的充要条件．利用非完整变分以及相应的积分变分原理建立两类动力学方程：vakonomic方程和Routh方程或Chaplygin方程．通过vakonomic方程分别与Routh方程和Chaplygin方程比较，得到它们具有共同解的两类充分必要条件．这些条件并不是约束的可积性条件．

英文摘要：

For general nonholonomically constrained systems, variation identity is used to define three kinds of unfree variations, i.e., nonholonomic variations ： the vakonomic, the Htilder and the Suslov by means of vector fields on the extended configuration manifold. The relations among the three kinds variations are discussed and a necessary and sufficient condition for the variations to become free ones is obtained. The nonholonomic variations and the corresponding integral variational principles are utilized to derive the two kinds of dynamical equations： vakonomic equations and Routh＇s equations or Chaplygin＇s equations. By comparing vakonomic equations with Routh＇s equations and Chaplygin＇s equations respectively, two necessary and sufficient conditions for the two kinds of equations to have common solutions are obtained, which are not integrable conditions of the constraints.