Heat and Gravitation. II. Stability

TL;DR

This paper introduces a Hamiltonian-based approach to stability criteria in hydrodynamics, presenting a new virial principle within the Eulerian framework that simplifies the derivation of stellar stability conditions.

Contribution

It proposes a Hamiltonian functional for stability analysis and formulates a new virial principle in Eulerian hydrodynamics, simplifying stellar stability criteria derivation.

Findings

Derived a stability criterion for polytropic stellar configurations.

Formulated a virial principle within Eulerian hydrodynamics.

Established boundary conditions based on mass conservation.

Abstract

In this second article of a series we propose to base criteria of stability on the hamiltonian functional that is provided by the variational principle, to replace the reliance that has often been placed on {\it ad hoc} definitions of the "energy". We introduce a new virial principle that is formulated entirely within the Eulerian description of hydrodynamics, which allows a simpler derivation of a well known stability criterion for polytropic stellar configurations. Boundary conditions are based entirely on mass conservation.