Lectures on hyperhamiltonian dynamics and physical applications
Hyper-Hamiltonian dynamics was originally motivated by the desire to formulate an approach to dynamics that is capable of dealing with systems having spin. Such systems are described, for example, by the Dirac and Pauli equations. In [J. Phys. A, Math. Gen. 35, No. 17, 3925--3943 (2002; Zbl 1055.53033)] \textit{G. Gaeta} and \textit{P. Morando} introduced a generalization of standard Hamiltonian dynamics that moved from a symplectic to a hyper-symplectic setting, from Hamiltonian dynamics on Kähler manifolds to hyper-Hamiltonian dynamics on hyper-Kähler manifolds. A hyper-Kähler manifold is a real smooth orientable Riemannian manifold \((M,g)\) of dimension \(m=4n\) \((n=1,2,\dots)\) together with three almost-complex structures \(J_1\), \(J_2\), and \(J_3\) satisfying (i) covariant constancy under the Levy-Cività connection, \(\nabla J_\alpha= 0\), and (ii) quaternionic relations \(J_\alpha J_\beta= \varepsilon_{\alpha\beta\gamma}J_\gamma-\delta_{\alpha\beta}I\), where \(\varepsilon_{\alpha\beta\gamma}\) is the completely antisymmetric Levy-Civita tensor. This book is an introduction to the theory of hyper-Hamiltonian dynamics with a discussion of some of its physical applications. The first chapter provides a quick review of standard Hamiltonian dynamics and symplectic geometry, and then introduces hyper-Kähler structures and manifolds. The next chapter offers definitions of hyper-Kähler dynamics and introduces the hyper-Kähler form as an important tool. Canonical and hyper-Kähler maps are discussed in Chapters 3 and 4; the authors show that any hyper-Hamiltonian vector field is the generator of a one-parameter group of canonical transformations of the underlying hyper-Kähler structure. Then Chapter 5 takes up integrable hyper-Hamiltonian systems and Chapter 6 considers perturbations of these. Finally, Chapter 7 focuses on physical applications -- the Pauli and Dirac equations as well as Taub-NUT manifolds.
scientific article; zbMATH DE number 6718363