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变分原理与非线性水波的Hamilton描述 总被引:4,自引:0,他引:4
本文用全变分方法导出水波动力学问题的基本方程组,再用平均势函数F(X,t)渐近表示速度势函数(X,y,t)和Lagrange函数L(X,y,t),导出具有Bousinesq形式的方程.研究了Hamilton正则方程的简单推导问题. 相似文献
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各向异性弹性力学问题Hamilton正则方程的一般形式 总被引:3,自引:0,他引:3
本文从修正后的Hellinger-Reissner变分原理出发,导出了由21个弹性常数组成的各向异性材料的混合方程,并证明它们即是Hamilton正则方程。由该统一形式还给出了角铺设材料和正交各向异性材料的Hanilton正则形式。 相似文献
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本文从修正后的Hellinger-Reissner变分原理出发,导出了由21个弹性常数组成的各向异性材料的混合方程,井证明它们即是Hamilton正则方程。由该统一形式还给出了角铺设材料和正交各向异性材料的Hanilton正则形式。 相似文献
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非惯性参考系中弹性壳的非线性振动分析 总被引:1,自引:0,他引:1
给出了弹性壳处于非惯性参系中的运动描述,基于Hamilton原理建立了中厚壳在非惯性参考系中的非线性运动控制方程,应用多尺度法及谐波平衡法具体地分析了圆柱壳的非线性振动问题 相似文献
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将非力学系统的微分方程化成Hamilton方程形式,引进无限小变换,研究微分方程或Hamilton作用量在无限小变换下的不变性,进而给出守恒量存在的条件以及守恒量的形式. 相似文献
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含初缺陷复合材料圆柱曲板的动力屈曲分析 总被引:1,自引:1,他引:1
基于修正的一阶剪切变形理论,利用Hamilton原理导出包含横向剪切变形和转动惯量的复合材料长圆柱曲板的非线性动力方程,通过将位移和载荷展开为Fourier级数,把非线性偏微分方程组转化为二阶常微分方程组,并可由四阶Runge-Kutta方法数值求解,通过算例,讨论了有关因素对迭层复合材料圆柱曲板动力屈曲的影响。 相似文献
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多刚体系统碰撞动力学方程及可解性判别准则 总被引:6,自引:0,他引:6
本文引入碰撞铰概念描述开环和闭环多刚体系统中各刚体间任意碰撞的情况,导出了适用于计算机编程求解的碰撞动力学方程,该方程适用于开环和闭环系统,文章对方程可能出现奇异的情况作了讨论,根据系统的碰撞结构导出了方程非奇异的必要和充分条件,文末以卫星帆板展开锁定过程作为算例。 相似文献
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A set of stable canonical equations of second order is derived, which describe the propagation of almost periodic waves in the horizontal plane, including weakly nonlinear interactions. The derivation is based on the Hamiltonian theory of surface waves, using an extension of the Ritz variational method. For waves of infinitesimal amplitude the well-known linear refraction-diffraction model (the mild-slope equation) is recovered. In deep water the nonlinear dispersion relation for Stokes waves is found. In shallow water the equations reduce to Airy's nonlinear shallow-water equations for very long waves. Periodic solutions with steady profile show the occurrence of a singularity at the crest, at a critical wave height. 相似文献
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小尺度波(扰动波)迭加在大尺度波(未受扰动波)上形成的波动一般之为“骑行波”。研究了有限可变深度的理想不可压缩流体中的骑行波的显式Hamliltn表示,考虑了自由面上流体与空气之间的表面张力。采用自由面高度和自由面上速度势构成的Hamilton正则变量表示骑行波的动能密度,并在未受扰动波的自由面上作一阶展开。运用复变函数论方法处理了二维流动。先用保角变换将物理平面上的流动区域变换到复势平面上的无限长带形区域,然后在复势平面上用Fourier变换解出Laplace方程,最后经Fourier逆变换求出了扰动波速度热所满足的积分方程。作为特例考虑了平坦底部的流动,导出了动能密度的显式表达式。这里给出的积分方程可以替代相当难解的Hamilton正则方程。通过求解积分方程可得出agrange密度的显式表达式。本文提出的方法约研究骑行波的Hamilton描述以及波的相互作用问题提供了新的途径,有助于了解海面的小尺度波的精细结构。 相似文献
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A note on Hamiltonian for long water waves in varying depth 总被引:1,自引:0,他引:1
The Hamiltonian for two-dimensional long waves over a slowly varying depth is derived. The vertical variation of the velocity field is obtained by using a perturbation method in terms of velocity potential. Employing the canonical theorem, the conventional Boussinesq equations are recovered. The Hamiltonian becomes negative when the wavelength becomes short. A modified Hamiltonian is constructed so that it remains positive and finite for short waves. The corresponding Boussinesq-type equations are then given. 相似文献
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The exact equations for surface waves over an uneven bottom can be formulated as a Hamiltonian system, with the total energy of the fluid as Hamiltonian. If the bottom variations are over a length scale L that is longer than the characteristic wavelength ℓ, approximating the kinetic energy for the case of “rather long and rather low” waves gives Boussinesq type of equations. If in the case of an even bottom one restricts further to uni-directional waves, the Korteweg-de Vries (KdV) is obtained. For slowly varying bottom this uni-directionalization will be studied in detail in this part I, in a very direct way which is simpler than other derivations found in the literature. The surface elevation is shown to be described by a forced KdV-type of equation. The modification of the obtained KdV-equation shares the property of the standard KdV-equation that it has a Hamiltonian structure, but now the structure map depends explicitly on the spatial variable through the bottom topography. The forcing is derived explicitly, and the order of the forcing, compared to the first order contributions of dispersion and nonlinearity in KdV, is shown to depend on the ratio between ℓ and L; for very mild bottom variations, the forcing is negligible. For localized topography the effect of this forcing is investigated. In part II the distortion of solitary waves will be studied. 相似文献
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We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation and using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators. 相似文献
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Results obtainable using the theory of progressive waves of small amplitude and certain fundamental solutions relating to waves of finite height are investigated. The theoretical findings are compared with existing experimental data. It is established that the best agreement between the theoretical and experimental profiles of a plane wave is achieved with constructions based on Kozhevnikov's [1] graphs and the equations of motion in the second approximation with respect to the wave height in the form proposed by Mich [2]. The limits within which it is expedient to use the theoretical formulas of the theory of small-amplitude waves and the theory of the second approximation with respect to the wave height are found and proved for the particle velocity, excess pressure, energy flux, and the energy of a single wave. 相似文献
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An approach is proposed to set up the dispersion equations for surface waves in a periodically stratified half-space contacting with a layer of a perfect compressible liquid. The approach is based on the formalism of periodic Hamiltonian systems. The dispersion equations derived are valid for an arbitrary law of variation in the properties with respect to the coordinate of periodicity. The effects of the liquid layer and the inhomogeneity of the elastic medium on the dispersion spectra of surface waves are studied 相似文献
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L. J. F. Broer 《Flow, Turbulence and Combustion》1975,31(5):377-395
In the first part of this paper the Hamiltonian theory of water waves is used to obtain some equations in local coordinates. These equations are approximations of the Boussinesq type. They are stable with respect to short wave perturbations, e.g. rounding off errors in digital computing. In the second part the relation of Boussinesq equations to Korteweg-de Vries and Benjamin-Bona-Mahony equations is investigated. 相似文献
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Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena
such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary-wave
or front solutions. In this paper, we present a new symplectic framework for analysing the spectral problem associated with
the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of
Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions,
with a third symplectic structure – with two-form denoted by Ω– associated with a coordinate frame moving at the speed of
the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization
about solitary waves and fronts.
We introduce the concept of the symplectic Evans matrix, a matrix consisting of restricted Ω-symplectic forms. By applying Hodge duality to the exterior algebra formulation of
the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic
Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral
parameter λ to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely
on the abstract geometry of the equations, and results for the large |λ| behaviour which use primarily the symplectic structure,
but also extend to the non-symplectic case.
The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems
at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation.
By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained.
The theory applies to a large class of solitary waves and fronts including waves which are bi-asymptotic to a nonconstant
manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model
from oceanography, a class of nonlinear Schr?dinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric
dynamics.
Accepted August 7, 2000 ?Published online January 22, 2001 相似文献