Hamiltonian systems with nilpotent structures

• Main Author: Irving, Malcolm
• Published: University of Warwick 1983
• Subjects: 629.8; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.347658

Symplectic Geometry has proved a powerful method in extending the knowledge of the classical theory of Hamiltonian mechanics without external variables. In this thesis these methods are applied to a class of Hamiltonian systems with controls in order to answer fundamental questions arising from Systems Theory and Classical Mechanics. The theoretical aspects of this thesis deal with the extension of the Lie algebraic results of Engel and Lie on nilpotent and solvable Lie algebras, respectively, by the introduction of symplectic structures. It provides revealing results on the internal structure of symplectic vector spaces acted on by nilpotent or solvable Lie algebras. Then, using the methods of Kostant and Kirillov, these results are globalized to look at nilpotent transitive actions on simply connected symplectic manifolds and the consequent internal structures. This theory is then applied to realizations of finite Volterra series with the additional property that the realization is Hamiltonian. These realizations are known to have an underlying nilpotent structure. A canonical realization is found and then shown to be closely linked with the theory of interconnections. Finally, the concepts of complete integrability on free Hamiltonian systems is put into a feasible framework for Hamiltonian systems with controls which have an associated nilpotent Lie algebra. It is found that it is still possible to integrate these systems by quadratures but the structure is now much more complex.