This book provides the mathematical foundations of the theory of hyperhamiltonian dynamics, together with a discussion of physical applications. In addition, some open problems are discussed. Hyperhamiltonian mechanics represents a generalization of Hamiltonian mechanics, in which the role of the symplectic structure is taken by a hyperkähler one (thus there are three Kähler/symplectic forms satisfying quaternionic relations). This has proved to be of use in the description of physical systems with spin, including those which do not admit a Hamiltonian formulation. The book is the first monograph on the subject, which has previously been treated only in research papers.
Giuseppe Gaeta is currently Professor of Mathematical Physics at the Università degli Studi di Milano, Milan, Italy. Professor Gaeta studied Physics in Rome and subsequently held research or teaching positions in CRM, Montreal (Canada); IHES, Bures-sur-Yvette (France); École Polytechnique, Palaiseau (France); Rijksuniversiteit, Utrecht (the Netherlands); Universidad Carlos III and Universidad Complutense, Madrid (Spain); and Loughborough University (UK). He is the author of about 140 scientific research papers and two research monographs (one in collaboration with G. Cicogna). His main scientific interests are the roles of Symmetry in physical theories and in nonlinear dynamics.
Miguel A. Rodríguez is full professor in the Department of Theoretical Physics of Universidad Complutense of Madrid, Spain. His teaching is mainly related to courses on Mathematics applied to Physics. He has been visiting professor at Université de Montréal (Canada), University of California at Los Angeles (USA), and Università di Roma Tre (Italy). His research field includes several areas of Mathematical Physics; in particular, Integrable Systems, Group Theory, and Difference Equations. In recent years he has collaborated with research groups in Spain, Canada, and Italy on spin chains and their statistical and thermodynamic properties, invariant schemes for difference equations, and hyperhamiltonian dynamics.
Introduction.- 1 Background material.- 2 Hyperhamiltonian dynamics.- 3 Quaternionic transformations for Hyperkahler structures in Euclidean spaces.- 4 Integrable hyperhamiltonian systems.- 5 Physical applications.- References.- Index.