Analysis of invariants in non-canonical Hamiltonian dynamics leading to hierarchies of coupled Volterra gyrostats

TL;DR

This work analyzes invariants in coupled Volterra gyrostats (GLOMs) arising from Galerkin projections of PDEs, focusing on how the number and form of quadratic invariants are constrained by model structure and non-canonical Hamiltonian constraints. It juxtaposes the standard quadratic-invariant approach with the Casimir-based constraints of non-canonical Hamiltonian dynamics, showing that general GLOMs typically have only the energy invariant, whereas Hamiltonian hierarchies yield additional, structurally consistent invariants across nested or fully coupled model hierarchies. The authors develop sparse and dense nested hierarchies, as well as fully coupled hierarchies, and provide incremental Jacobi-constraint conditions that govern the existence of Casimirs, along with gradient consistency under projection. These results offer a principled framework for constructing GLOMs with prescribed invariant structure, aiding faithful low-order representations of PDE dynamics and informing model reduction and attractor analysis.

Abstract

This work deals with the analysis of the existence and number of invariants in a hitherto unexplored class of dynamical systems that lie at the intersection of two major classes of dynamical systems. The first is the conservative core dynamics of low-order models that arise naturally when we project infinite dimensional PDEs into a finite dimensional subspace using the classical Galerkin method. These often have the form of coupled gyrostatic low order models (GLOMs). Second is the class of non-canonical Hamiltonian models obtained by systematically coupling the basic component systems, the Volterra gyrostat and its special cases such as the Euler gyrostat. While it is known that the Volterra gyrostat enjoys two invariants, it turns out that members of the GLOM exhibit varying numbers of invariants. The principal contribution of this study is to relate the structure of GLOMs to the number of invariants and describe the importance of the non-canonical Hamiltonian constraint in tackling this problem. Devising model hierarchies with consistent invariants is important because these constrain the evolution as well as asymptotic behavior of these dynamical systems.

Figures (4)

• Figure 1: Schematic describing the topic of this paper, hierarchically coupled systems of Volterra gyrostats in relation to low order models derived from PDEs as well as various classes of dynamical systems.
• Figure 2: Regression tree with leaves distinguishing conditions for $1-3$ invariants, for Model 1 where the nonlinear coefficients are nonzero.
• Figure 3: Regression tree with leaves distinguishing conditions for $2$ or $3$ invariants, for Model 2 where the nonlinear coefficients are all nonzero.
• Figure 4: Regression tree indicating conditions on number of invariants: a) for Model 2, where atleast one gyrostat has $3$ nonlinear terms. Many leaves for the general GLOM without the Hamiltonian constraint are not pure, so further partitioning is needed for complete characterization of number of invariants; b) for Model 2, where atleast one gyrostat has $3$ nonlinear terms, and additionally the model is of the non-canonical Hamiltonian form. Note the much simpler conditions for characterizing $2,3,4$ invariants among the Hamiltonian cases.