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

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.