# Download Deep Water Gravity Waves: On the Simpler Aspects of by Bruce J. West (auth.) PDF

By Bruce J. West (auth.)

**Read or Download Deep Water Gravity Waves: On the Simpler Aspects of Nonlinear Fluctuating (Weak Interaction Theory) PDF**

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**Extra resources for Deep Water Gravity Waves: On the Simpler Aspects of Nonlinear Fluctuating (Weak Interaction Theory)**

**Sample text**

Returning to the system of mode rate equations generated by the Hamiltonian (1) we note that classical dynamics prescribes how to solve such a set of equations. Hamilton-Jacobi theory and canonical perturbation theory were developed to integrate such equations of motion. However, i t is not uncommon for Hamilton-Jacobi theory to yield divergent perturbation expansions when applied to nonlinear mechanical systems~4'35'37 I t is useful for us to understand the fundamental limitations of applying classical techniques to the analysis of such nonlinear physical phenomena.

The Hamiltonian representation (34) c l e a r l y specifies the wave dynamics as an ordering of the n o n ~ i n ~ i ~ t ~ a ~ t i o n ~ in powers of the surface slope. Each term in (34) is independent of the v e r t i c a l coordinate z and the e x p l i c i t expressions f o r H2, H3 and H4 are discussed in the next lecture. Also t h i s perturbation theory is contrasted with the more f a m i l i a r perturbation series expansions of the v e l o c i t y potential such as given in Stoker 2 or by Hasselmann I I .

This orbit is in the two-dimensional phase space for the system and does not correspond to the orbital motion of a f l u i d parcel in a physical water wave. For a system with two degrees of freedom the motion in the four-dimensional phase space is on the surface of a doughnut (torus), the action variables are the major and minor radii and the rotations of these radii the corresponding angle variables. The action-angle descrip- tion in general displays the dynamics of a mechanical system with N degrees of freedom as a N-dimensional torus embedded in a 2N dimensional phase space.