标题： 自引力多方流动的能量原理：I. 一般理论 显示英文标题

标题： Energy Principles for Self-Gravitating Barotropic Flows: I. General Theory

摘要： 以下的最小能量原理可能是动态摄动方法的一个有力替代方案，当后者难以应用时。 自引力绝热流的流体单元，其涡线延伸至流体边界，被标记为使得任何试验构型的变化自动保持质量和环流。 速度场由质量守恒的克布施表示法给出。 通过三个独立的拉格朗日函数，在固定线性和角动量的流动的所有小变化下，总能量是驻定的，前提是满足稳定运动的欧拉方程。 因此，如果能量最小，则稳态流动是稳定的。 由于在这里最小化能量时，局部和全局运动常数被固定，用这种方法得到的稳定性极限预计接近动态摄动方法给出的极限。 此外，这些稳定性极限是对任意的，不一定是小的扰动而言的。 还给出了能量原理的一个较弱形式，这可能更容易应用。 具有相同三个拉格朗日变量的拉格朗日泛函对于完全时间相关的欧拉方程也是驻定的。 由此可知，如果拉格朗日中没有关于时间导数的一阶项（陀螺项），则最小能量原理给出的稳定性条件既是必要的也是充分的。 对于围绕稳态流动的小偏差，陀螺项被明确给出。 关键词：能量变分原理；自引力系统；流体稳定性。 显示英文摘要

摘要： The following principle of minimum energy may be a powerful substitute to the dynamical perturbation method, when the latter is hard to apply. Fluid elements of self-gravitating barotropic flows, whose vortex lines extend to the boundary of the fluid, are labelled in such a way that any change of trial configurations automatically preserves mass and circulation. The velocity field is given by a mass conserving Clebsch representation. With three independent Lagrangian functions, the total energy is stationary for all small variations about a flow with fixed linear and angular momenta provided Euler's equations for steady motion are satisfied. Thus, steady flows are stable if their energy is minimum. Since energy is here minimized subject to having local and global contants of the motion fixed, stability limits obtained that way are expected to be close to limits given by dynamical perturbation methods. Moreover, the stability limits are with respect to arbitrary, not necessary small, perturbations. A weaker form of the energy principle is also given which may be easier to apply. The Lagrangian functional, with the same three Lagrange variables is stationary for the fully time dependent Euler equations. It follows that the principle of minimum energy gives stability conditions that are both necessary and sufficient if terms linear in time derivatives (gyroscopic terms) are absent from the Lagrangian. The gyroscopic term for small deviations around steady flows is given explicitly. Key words: Energy variational principle; Self-gravitating systems; Stability of fluids.