On the structure of the energy conserving low-order models and their relation to Volterra gyrostat

Abstract

Low-order models (LOM) described by a system of n th-order (nonlinear) ordinary differential equations (ODE) of the type x ˙ i = x T A ( i ) x + B i x + c i , i = 1 , 2 , … , n (where x is a column vector, A ( i ) is a n × n matrix, B i is a row vector, c i is a scalar and T denotes the transpose) routinely arise when we apply the Galerkin type projection techniques to the quasi-geostrophic potential vorticity equation (with forcing, dissipation and topography), Rayleigh–Bernard convection and Burgers’ equation, to mention a few. To our knowledge there is no systematic method for testing if a given LOM conserves energy. Our goal in this paper is twofold. First, we derive a set of sufficient conditions on the structural parameters ( A ( i ) , B i and c i for i = 1 , 2 , … , n ) for conserving energy. It is well known in Mathematical Physics that the Volterra gyrostat and many of its special cases including the Euler gyroscope represent a prototype of energy conserving dynamical systems. It turns out that a special case of our sufficient condition is closely related to the Volterra gyrostats. Exploiting this relation, we then derive an algorithm for rewriting the LOM (corresponding to the special case of our sufficient conditions) as a system of coupled gyrostats which brings out the inherent relation between the energy conserving LOM and the system of coupled gyrostats.