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This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion
μ
and the nonlinearity
ε
satisfied

Dynamics research on Hamilton systems is an important subject in mechanics for a long time. Hamilton’s principles have also the big advantage of ensuring that one can build approximations with optimal “fit” among all the equations defining the problem at hand. The principles of Hamilton mechanics settled a series of problems effectively that could not be solved by other methods, which showed theoretically the importance of Hamilton mechanics. Whitham [

This Hamilton’s principle for incompressible and inviscid fluid is used to derive approximate wave models. The formulation of Madsen et al. [7,8] is most capable of treating highly non-linear waves to for dispersion, with accurate velocity profiles up to. Luke [

We consider an inviscid, irrotational flow of constant density subjected to a gravitational field g acting in the negative z-axis which is directed vertically downward. In its undisturbed state, the fluid, which is of infinite horizontal extent, is confined to a region

.

Here we have used Hamilton’s principle with Lagrange function

The relevant ingredients, needed in order to describe this flow, are:

is the velocity potential, ρ is the fluid density, g is the acceleration by the Earth’s gravity, x is the horizontal coordinate, x-axis represents undisturbed surface with constant depth H, z is the vertical coordinate, is the elevation of the free surface.

Free surface is the surface of a fluid that is subject to constant perpendicular normal stress and zero parallel shear stress, such as the boundary between two homogenous fluids, for example liquid water and the air in the Earth’s atmosphere. Unlike liquids, gases cannot form a free surface on their own. A liquid in a gravitational field will form a free surface if unconfined from above. Under mechanical equilibrium this free surface must be perpendicular to the forces acting on the liquid; if not there would be a force along the surface, and the liquid would flow in that direction. Thus, on the surface of the Earth, all free surfaces of liquids are horizontal unless disturbed (except near solids dipping into them, where surface tension distorts the surface locally). In a free liquid at rest, that is, one subject to internal attractive forces only and not affected by outside forces such as a gravitational field, its free surface will assume the shape with the least surface area for its volume—a perfect sphere.

Now are allowed to vary subject to the restrictions on the boundary of D.

According to the standard procedure of the calculus of variations, Hamilton’s principle gives

Now

since.

Integrating the z-integral by parts, it turns out that

In view of the fact that the first z-integral in each of the square brackets vanishes on the boundary of D, we obtain

We first choose; since is arbitrary, we deduce

Then, since can be given arbitrary independent values, we obtain

Evidently the Laplace equation, two free surface conditions, and the bottom boundary condition constitute the two-dimensional water wave equation. This system of equation has been used by Stoker [

We consider an inviscid irrotational flow of constant density subjected to a gravitational field g acting in the negative z-axis which is directed vertically downward. The fluid with a free surface is confined in a region. There exists a velocity potential such that the fluid velocity is given by the potential is lying between and. Then Hamilton’s principle with Lagrange function

and are allowed to vary subject to the restrictions on the boundary δD of D.

According to the standard procedure of the Calculus of variations, Hamilton’s principle becomes

Integrating the z-integral by parts along with r and integrals, it turns out that

In view of the fact that the first z-integral in each of the square bracket vanishes on the boundary, we obtain

We first choose; since is an arbitrary, we derive

Then since can be given arbitrary independent values, we deduce

Evidently, the Laplace equation, two flee-surface conditions and the bottom boundary condition constitute the non-axisymmetric water wave equations in cylindrical polar coordinates. This set of equations has also been used by several authors including Debnath [

In non-dimensional form, the shallow-water equations take the form

Here h represents the height of the free-surface, and the horizontal velocity field. The height h has been normalized by its mean value H, the velocity field by the characteristic speedConsidering and and are much smaller than one.

Hence we focus our attention here to irrotational flows. These are described by a scalar potential. For such flows,

where Lagrange function is

This system is Hamiltonian, with

The Hamiltonian form of the equations is

For the fully nonlinear shallow-water equations, waves and vorticity no longer decouple. However, it is still true that a flow which starts irrotational stays so forever. Hence we may restrict ourselves to introduce again the scalar potential, and this will take the form

where Lagrange function is

This system is also Hamiltonian, with

and canonical equations

In this case, the Hamiltonian is the sum of the potential and kinetic energy.

Here Hamilton’s Principle for irrotational water waves free of side conditions is used with Lagrange function

Then, we have variation of within the flow region

The variation of gives the dynamical boundary condition on the free surface:

Hongli et al. [

where is a constant.

Here we consider,

From Laplace equation using Equation (17), we obtain

Since z be an arbitrary value within the flow region, so each coefficient in power of must be zero, thus

On the other hand, using Equation (17) on the last free surface condition yields. Therefore, for all odds, , i.e.,

Supposing that, we have

Now, the expression of velocity potential is obtained:

By linear approximation, we also consider

Therefore, the velocity potential can be found to be:

using Equation (20)

Now

The case of, was considered by Benjamin [

Hou et al. [

Let

Expanding to th term, we have

Based on the dynamical boundary condition of the free surface, we have

From Equation (23), equating the coefficients of constant, and terms, we have

Substituting Equations (22) and (23) in Hamilton’s principle and neglecting the terms higher than order terms, we have the Lagrangian

Obviously, Lagrangian is a function of generalized coordinates and generalized velocity.

We also used the generalized momentum

Now Hamiltonian function

Hamilton’s canonical equation of motion

Firstly, we have generalized two dimensional water wave equation in Cartesian and in cylindrical polar coordinates. We have also discussed water wave equation with Lagrangian and Hamiltonian with canonical variables. Then the Lagrangian function within a certain flow region expanded up to. It is obvious that Lagrangian is a function of generalized coordinate and generalized velocity and Hamiltonian is the sum of kinetic energy and potential energy. Using generalized momentum Hamiltonian function is formulated and then Hamilton’s canonical equations of motion have been also developed.