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By Bruce J. West (auth.)

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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.

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