A method for Hamiltonian truncation: A four-wave example

TL;DR

This paper introduces a novel Hamiltonian truncation method for 2+1 Hamiltonian mean field theories with noncanonical brackets, using beatification to facilitate finite-dimensional approximations, demonstrated through vortex dynamics examples.

Contribution

It presents a new beatification-based approach for Hamiltonian truncation applicable to noncanonical systems, with explicit examples in vortex dynamics.

Findings

Beatification enables Hamiltonian truncation of noncanonical systems.

Comparison shows beatified truncation preserves Hamiltonian structure.

Method successfully applied to four-wave vortex dynamics examples.

Abstract

A method for extracting finite-dimensional Hamiltonian systems from a class of 2+1 Hamiltonian mean field theories is presented. These theories possess noncanonical Poisson brackets, which normally resist Hamiltonian truncation, but a process of beatification by coordinate transformation near a reference state is described in order to perturbatively overcome this difficulty. Two examples of four-wave truncation of Euler's equation for scalar vortex dynamics are given and compared: one a direct non-Hamiltonian truncation of the equations of motion, the other obtained by beatifying the Poisson bracket and then truncating.