Giuseppe Marmo

POISSON STRUCTURES ON THE COTANGENT BUNDLE OF A LIE GROUP OR A PRINCIPLE BUNDLE AND THEIR REDUCTIONS

On a cotangent bundle T*G of a Lie group G one can describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of G on itself), and also in terms of the left Maurer–Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic structures on T*G and to Poisson structures by replacing the canonical momenta of the right or left actions of G on itself by arbitrary ones, followed by reduction (to G cross a Weyl‐chamber, e.g.). This method also works on principal bundles.

Recursion operators: Meaning and existence for completely integrable systems

Comments are made on the meaning and existence of recursion operators for completely integrable systems. It is shown that any nonresonant integrable system admits infinitely many Hamiltonian descriptions and recursion operators.

POISSON STRUCTURES ON DOUBLE LIE GROUPS

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gaus-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.

CONSTRUCTION OF COMPLETELY INTEGRABLE SYSTEMS BY POISSON MAPPINGS

Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail.

Alternative linear structures for classical and quantum systems

The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as adapted to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different nonequivalent ways, evading, so to speak, the von Neumann uniqueness theorem.

Completely Integrable Systems — A Generalization

We present a slight generalization of the notion of completely integrable systems such that they can be integrated by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.

The Language of Geometry and Dynamical Systems: The Linearity Paradigm

We can infer from the examples given in Chap. 1 that linear dynamical systems are interesting on their own.

Some Examples of Linear and Nonlinear Physical Systems and Their Dynamical Equations

This chapter is devoted to the discussion of a few simple examples of dynamics by using elementary means. The purpose of that is twofold, on one side after the discussion of these examples we will have a catalogue of systems to test the ideas we would be introducing later on; on the other hand this collection of simple systems will help to illustrate how geometrical ideas actually are born from dynamics.

The Classical Formulations of Dynamics of Hamilton and Lagrange

The present chapter is perhaps the place where our discourse meets more neatly the classic textbooks on the subject. Most classical books concentrate on the description of the formalisms developed by Lagrange and Euler on one side, and Hamilton and Jacobi on the other and commonly called today the Lagrangian and the Hamiltonian formalism respectively. The approach taken by many authors is that of postulating that the equations of dynamics are derived from variational principles (a route whose historical episodes are plenty of lights and shadows [Ma84]).

The Geometry of Hermitean Spaces: Quantum Evolution

The basic geometrical structures arising in Quantum Mechanics are analyzed as in previous chapters, that is, we ask when a given dynamics possesses simultaneously invariant symplectic and metric stuctures. After solving this inverse problem in the linear case, we will discuss some of its implications.