分析廣義歐拉方程, 用一些方法看方程式平衡點的穩定性.

The generalized Euler equations are changed to the generalized vorticity equations, conservation of energy, Casimir functions and a momentum function are checked.
The energy Casimir method is applied to find a
stability criterion for some generalized Euler flows. The momentum Casimir method provides another stability criterion of some other flows. Find the stability of nonincreasing equilibria of our generalized vorticity equation.

[1]Arnold V., Mathematical Method of Classical
Mechanics. 2nd ed., GTM Vol. 60. Springer-Verlag, 1989.
[2]Holm, D., J. Marsden, T. Ratiu, and A. Weinstein,
Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1-116, 1985.
[3]Marchioro C. and Pulvirenti M., Mathematical Theory
of Incompressible Nonviscous fluids. Springer-Verlag, New York, 1994.
[4]Marchioro C. and Pulvirenti M., Some considerations on the nonlinear stability of stationary planar Euler flows.
Comm. Math. Phys. 100, 343-354, 1985.
[5]McLachlan, R. I. and Marsland S., N-particle dynamics of the Euler equations for planar diffeomorphisms. Dynam. Systems, submitted. http://arxiv.org/abs/nlin.CD/0512030