Homogeneous potentials, Lagrange's identity and Poisson geometry

TL;DR

This paper explores how the Lagrange identity relates to Hamiltonian systems, revealing new tensor invariants in systems with homogeneous and inhomogeneous potentials, impacting stability analysis.

Contribution

It proves that Hamiltonian systems satisfying the Lagrange identity have additional tensor invariants, including a new class with inhomogeneous potentials.

Findings

Hamiltonian systems with Lagrange identity have extra tensor invariants.

A new class of systems with inhomogeneous potentials also exhibits similar invariants.

The work links Lagrange identity to stability and geometric properties of Hamiltonian systems.

Abstract

The Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through kinetic energy and homogeneous potential energy, from which follows the Jacobi well-known result on the instability of a system of gravitating bodies. In this work, it is proven that if a Hamiltonian system satisfies the Lagrange identity, then it possesses additional tensor invariants that are not expressed through the basic invariants existing for all Hamiltonian systems. A new class of Hamiltonian systems with inhomogeneous potentials is considered, which also possess similar additional tensor invariants.