Homogeneous potentials, Lagrange's identity and Poisson geometry
A. V. Tsiganov
TL;DR
The paper extends Lagrange's identity for homogeneous potentials to reveal extra tensor invariants in Hamiltonian dynamics. It constructs invariant Poisson bivectors $\hat P$ and symplectic forms $\hat\omega$ from a basic Poisson structure, and further generalizes to a second family of inhomogeneous potentials via a shifted construction that yields another nondegenerate invariant bivector $\tilde P$ with $\tilde\omega=\tilde P^{-1}$. A tensor analogue of the Lagrange identity, $\hat P dH = \tfrac{k+2}{2} H P dH$ (and its inhomogeneous counterpart) encodes the invariant geometry, with local Darboux normal forms ensuring canonical representations. The results also extend to Lie-Poisson settings on $so^*(4)$ and $e^*(3)$, highlighting pervasive invariant structures beyond the basic $H$, $P$, and $\omega$, and connecting to Volterra-type invariant brackets in a geometric framework.
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.