Variational formulations of steady rotational equatorial waves

Jifeng Chu; Joachim Escher

doi:10.1515/anona-2020-0146

Variational formulations of steady rotational equatorial waves

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0146 ◽

2020 ◽

Vol 10 (1) ◽

pp. 534-547

Author(s):

Jifeng Chu ◽

Joachim Escher

Keyword(s):

Linear Stability ◽

Stream Function ◽

Water Waves ◽

Variational Formulation ◽

Energy Functional ◽

Equatorial Waves ◽

Second Variation ◽

Governing Equations ◽

The Second Variation ◽

Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

The second variation of the stream function energy of water waves with vorticity

Journal of Differential Equations ◽

10.1016/j.jde.2012.07.005 ◽

2012 ◽

Vol 253 (9) ◽

pp. 2646-2656 ◽

Cited By ~ 1

Author(s):

Georg S. Weiss ◽

Guanghui Zhang

Keyword(s):

Stream Function ◽

Water Waves ◽

Second Variation ◽

The Second Variation

Correct Formulation of the Stability Problem for Timoshenko Beam

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.725-726.854 ◽

2015 ◽

Vol 725-726 ◽

pp. 854-862 ◽

Cited By ~ 1

Author(s):

Vladimir Lalin ◽

Daria Kushova

Keyword(s):

Nonlinear Problems ◽

Shear Stiffness ◽

Lagrangian Description ◽

Second Variation ◽

Rigid Support ◽

Stability Problems ◽

Lagrangian Functional ◽

The Stability ◽

The Second Variation ◽

Variational Formulations

This article is about the nonlinear problems of the theory of elastic Cosserat – Timoshenko’s rods in the material (Lagrangian) description with energy conjugated vectors of forces, moments and strains. The variational formulations of static problems was given. The differential equations of the plane stability problems were obtained from the second variation of the Lagrangian functional. The exact solutions of the stability problems for basic types of the end fixities of the rod were obtained for the Timoshenko’s rod (taking into account only bending and shear stiffness). It appears that classical well-known equilibrium stability functional and stability equations for the Timoshenko’s rod are incorrect. Also well-known Engesser formula (with bending and shear stiffness) is incorrect. The numerical solution of the stability problems for hinged Timoshenko’s rod with rigid support was obtained. Also, simplified formula for this problem was derived using asymptotic analysis.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

International Journal of Mathematics ◽

10.1142/s0129167x05003211 ◽

2005 ◽

Vol 16 (09) ◽

pp. 1017-1031 ◽

Cited By ~ 21

Author(s):

QUN HE ◽

YI-BING SHEN

Keyword(s):

Riemannian Manifold ◽

Harmonic Maps ◽

Harmonic Map ◽

Energy Functional ◽

Second Variation ◽

Finsler Manifolds ◽

Totally Geodesic ◽

The Stability ◽

The Second Variation ◽

Weitzenböck Formula

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

A fractal variational theory of the Broer-Kaup system in shallow water waves

Thermal Science ◽

10.2298/tsci180510087l ◽

2021 ◽

pp. 87-87

Author(s):

Wei-Wei Ling ◽

Pin-Xia Wu

Keyword(s):

Shallow Water ◽

Water Waves ◽

Water Wave ◽

Variational Formulation ◽

Inverse Method ◽

Variational Theory ◽

Shallow Water Waves ◽

Solution Structures ◽

Fractal Space ◽

Variational Formulations

The Broer-Kaup equation is one of many equations describing some phenomena of shallow water wave. There are many errors in scientific research because of the existence of the non-smooth boundaries. In this paper, we generalize the Broer-Kaup equation to the fractal space and establish fractal variational formulations through the semi-inverse method. The acquired fractal variational formulation reveals conservation laws in an energy form in the fractal space and suggests possible solution structures of the morphology of the solitary waves.

Formal stability of circular vortices

Journal of Fluid Mechanics ◽

10.1017/s0022112092002362 ◽

1992 ◽

Vol 242 ◽

pp. 249-278 ◽

Cited By ~ 7

Author(s):

R. C. Kloosterziel ◽

G. F. Carnevale

Keyword(s):

Angular Momentum ◽

Linear Combination ◽

Linear Stability ◽

Second Variation ◽

Piecewise Constant ◽

Constant Vorticity ◽

Formal Stability ◽

Stability Regime ◽

The Second Variation ◽

Equivalent Expressions

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

The second variation and neighboring optimal feedback guidance for multiple burn orbit transfers

10.2514/6.1995-3360 ◽

1995 ◽

Author(s):

C-HGoodson, Chuang, , Troy D ◽

Laura Ledsinger ◽

John Hanson

Keyword(s):

Second Variation ◽

Optimal Feedback ◽

The Second Variation

Integral criteria for closed surfaces with constant mean curvature

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.1818 ◽

2007 ◽

Vol 463 (2081) ◽

pp. 1199-1210

Author(s):

Luca Guzzardi ◽

Epifanio G Virga

Keyword(s):

Mean Curvature ◽

Constant Mean Curvature ◽

Three Dimensional ◽

Dimensional Euclidean Space ◽

Second Variation ◽

Closed Surfaces ◽

Area Functional ◽

Symmetric Surface ◽

The Second Variation ◽

Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

The Second Variation of Arclength, Index Forms, and Jacobi Fields

Fundamental Principles of Classical Mechanics ◽

10.1142/9789814551496_0030 ◽

2014 ◽

pp. 345-355

Keyword(s):

Second Variation ◽

Jacobi Fields ◽

The Second Variation

Autoparallel Deviation in the Geometry of Lyra

Canadian Mathematical Bulletin ◽

10.4153/cmb-1960-033-8 ◽

1960 ◽

Vol 3 (3) ◽

pp. 263-271 ◽

Cited By ~ 1

Author(s):

J. R. Vanstone

Keyword(s):

Riemannian Manifold ◽

Euclidean Space ◽

Calculus Of Variations ◽

Finsler Manifold ◽

Geometrical Approach ◽

Second Variation ◽

Geodesic Deviation ◽

The Difference ◽

The Second Variation

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

ESTIMATION OF WATER PARTICLE VELOCITIES OF SHALLOW WATER WAVES BY A MODIFIED TRANSFER FUNCTION METHOD

Coastal Engineering Proceedings ◽

10.9753/icce.v20.33 ◽

1986 ◽

Vol 1 (20) ◽

pp. 33 ◽

Cited By ~ 2

Author(s):

Hirofumi Koyama ◽

Koichiro Iwata

Keyword(s):

Transfer Function ◽

Shallow Water ◽

Approximate Method ◽

Stream Function ◽

Water Waves ◽

Function Method ◽

Finite Amplitude ◽

Water Particle ◽

Irregular Wave ◽

Particle Velocities

This paper Is intended to propose a simple, yet highly reliable approximate method which uses a modified transfer function in order to evaluate the water particle velocity of finite amplitude waves at shallow water depth in regular and irregular wave environments. Using Dean's stream function theory, the linear function is modified so as to include the nonlinear effect of finite amplitude wave. The approximate method proposed here employs the modified transfer function. Laboratory experiments have been carried out to examine the validity of the proposed method. The approximate method is shown to estimate well the experimental values, as accurately as Dean's stream function method, although its calculation procedure is much simpler than that of Dean's method.