Matched Pair Analysis of Euler-Poincar\'{e} Flow on Hamiltonian Vector Fields

TL;DR

This paper develops a matched pair decomposition for Hamiltonian vector fields, revealing two interacting subdynamics: one for isentropic fluid flows and another for higher-order tensor flows, advancing geometric fluid dynamics understanding.

Contribution

It introduces a novel matched pair decomposition of symmetric contravariant tensors and Hamiltonian vector fields, elucidating their subdynamics and interactions.

Findings

Decomposition of Hamiltonian vector fields into two subdynamics.

Identification of Euler--Poincaré equations for isentropic fluids.

Extension to higher-order contravariant tensors (n≥2).

Abstract

In this paper we provide a matched pair decomposition of the space of symmetric contravariant tensors $\mathfrak{T}\mathcal{Q}$. From this procedure two complementary Lie subalgebras of $\mathfrak{T}\mathcal{Q}$ under \textit{mutual} interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to these realizations, Euler-Poincar\'{e} flows on such spaces are decomposed into two subdynamics: one of which is the Euler--Poincar\'{e} formulation of isentropic fluid flows, and the other one corresponds with Euler--Poincar\'{e} equations on higher order contravariant tensors ($n\geq 2$)