Non-canonical Hamiltonian structure and Poisson bracket for two-dimensional hydrodynamics with free surface
Non-canonical Hamiltonian structure and Poisson bracket for two-dimensional hydrodynamics with free surface
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DOI：
10.1017/jfm.2019.219
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发表时间：
2018-09
影响因子：
3.7
通讯作者：
A. Dyachenko;P. Lushnikov;Vladimir E Zakharov
中科院分区：
文献类型：
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作者：
A. Dyachenko;P. Lushnikov;Vladimir E Zakharov
本文考虑二维几何中无限深理想不可压缩流体有自由面势流的欧拉方程。重力和表面张力都被考虑在内。一个时间相关的保角映射的辅助复变量$W$下复杂的半平面映射到流体的面积，与真实的线$W$映射到自由流体的表面。我们重新制定的确切欧拉动力学通过一个非正则的非局部哈密顿结构的一对哈密顿变量。这两个变量是保形映射的虚部和流体的速度势，两者都在流体的自由表面处进行计算。相应的泊松括号是非退化的，即它没有任何卡西米尔不变量。保角映射的任意两个泛函关于泊松括号可交换。新的哈密顿结构是Zakharov（J. Appl. Mech. Tech.物理、第9（2）卷，1968年，第100页。190-194），其仅对自然表面参数化为单值的解有效，即，水平坐标的每个值仅对应于自由表面上的单个点。相比之下，新的非正则哈密顿方程对任意非线性解（包括多值自然表面参数化）都是有效的，并且等价于欧拉方程。我们还考虑了广义流体力学的附加物理项的哈密顿超越欧拉方程。在这种情况下，我们确定强大的减少，使人们能够找到一般类的特定解决方案。
We consider the Euler equations for the potential flow of an ideal incompressible fluid of infinite depth with a free surface in two-dimensional geometry. Both gravity and surface tension forces are taken into account. A time-dependent conformal mapping is used which maps the lower complex half-plane of the auxiliary complex variable $w$ into the fluid’s area, with the real line of $w$ mapped into the free fluid’s surface. We reformulate the exact Eulerian dynamics through a non-canonical non-local Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid’s velocity potential, both evaluated at the fluid’s free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. The new Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Zakharov (J. Appl. Mech. Tech. Phys., vol. 9(2), 1968, pp. 190–194) which is valid only for solutions for which the natural surface parametrization is single-valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, the new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to the Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations. In that case we identify powerful reductions that allow one to find general classes of particular solutions.