Jets, Lifts and Dynamics

TL;DR

This paper develops a geometric framework using cotangent lifts of vector fields to unify and analyze Eulerian equations in continuum mechanics, plasma dynamics, and particle flow, revealing new structural insights.

Contribution

It introduces a novel geometric approach based on cotangent lifts and their decomposition, connecting various continuum and kinetic equations in a unified framework.

Findings

Unified geometric framework for Eulerian equations

Connections between lifts of divergence-free and Hamiltonian fields

Derivation of kinetic equations from Lie-Poisson reductions

Abstract

We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie-Poisson reductions and with the techniques of present framework.