Hamiltonian formalism of the inverse problem using Dirac geometry and its application on Linear systems

Hassan Najafi Alishah

Published 2020-04-07Version 1

We present the Hamiltonian formalism for the inverse problem having Dirac$\backslash$ big-isotropic structures as underlying geometry. We used the same idea at [1] to treat replicator equations. Here we state the procedure used there for general vector fields that can be written in a gradient form. For a linear system, we show that if representing matrix of the system has at least one pair of positive-negative non-zero eigenvalues or in the case of eigenvalue zero, at least one three dimensional Jordan block associated to it, then the linear system has a Hamiltonian description with respect to a non-trivial Dirac$\backslash$big-isotropic structure. More interestingly, we prove that every Hamiltonian linear system is Hamiltonian integrable. As a byproduct, we found a class of linear systems with eigenvalue zero that are Hamiltonian only with respect to a proper big-isotropic structure. Our approach provides, also, a clear picture for the alternative Hamiltonian descriptions of linear systems.