Global Hamilton principal functions of the eikonal equations on $S_2$ and $H_2$

This monograph grew out of a series of lectures given at the XXVI Summer School of Mathematical Physics, Ravello, September 2001, organized by G.N.F.M. (gruppo Nazionale di Fisica Matematica) of I.N.d.A.M. (Istituto Nazionale di Alta Matematica, Roma), at the Department of Mathematics of the University of Torino in the academic years 2000/2001 and 2001/2002, and at the Departmento of Physical Sciences of the University of Napoli, May 2003. The elements of Symplectic Geometry and Analytical Mechanics on which these lectures are based can be found in the literature of the seventies and eighties of the last century. The bibliography is of course far from complete and refers the reader to some of the important contributions. Here, we introduce only the essential notions of symplectic geometry needed for application to the geometrical theory of the Hamilton-Jacobi equation and to the control theory of static systems. Most of these notions are well known, but the way they are assembled and used is new in many respects. A fundamental role in the present approach is played by the notion of generating family and by two operations: the composition of generating families of symplectic relations and the canonical lift from objects on manifolds (submanifolds, relations, mappings, vector fields, etc.) to symplectic objects on the corresponding cotangent bundles. Generating families describe special subsets of cotangent bundles which we call Lagrangian sets. A Lagrangian set is a Lagrangian submanifold (which may be immersed) if the generating family is a Morse family. However, there are physically interesting examples of Lagrangian sets which are not Lagrangian submanifolds. An advantage of considering generating families as fundamental objects is that, while the composition of two symplectic relations may not be a smooth relation, the composition of two generating families is always a smooth function. In other words, the symplectic creed as formulated by A. Weinstein in his article Symplectic geometry (1981) in the form everything is a Lagrangian submanifold, which means that one should try to express objects in symplectic geometry and mechanics in terms of Lagrangian submanifolds, is here replaced by everything has a generating family. The geometrical theory of the Hamilton-Jacobi equation is closely related to Geometrical Optics. The symplectic formulation of Hamiltonian Optics presented here differs from other formulations illustrated in papers and well known reference books cited in the Bibliography and it is, in my opinion, very close to the original ideas of Hamilton. From a geometrical view-point a Hamilton-Jacobi equation is a coisotropic submanifold of a cotangent bundle. A geometrical solution is a Lagrangian set described by a generating family and contained in the coisotropic submanifold. There are two fundametal symplectic relations associated with a Hamilton-Jacobi equation, the characteristic relation and the characteristic reduction. The two corresponding generating families are the Hamilton principal function and the complete solution of the Hamilton-Jacobi equation, respectively. By composing the latter with its transpose we get the former. Since the characteristic relation is a singular Lagrangian submanifold, the Hamilton principal function is necessarily a generating family and not a two-point function as in the classical theory. Cauchy data (or sources of systems of rays), mirror and lenses are represented by symplectic relations thus, by generating families. Then the Cauchy problem and the actions of a lens or of a mirror on a system of rays are translated into the composition of generating families. What is presented here is only a first approach to Geometrical Optics based on the notions of symplectic relation and generating family. We do not cover many important examples of optical phenomena, which can be found in standard reference books (e.g. Synge, Luneburg, Buchdahl) and which probably can be treated within this framework. Perhaps, the use of generating families and symplectic relations does not yield a revolutionary progress in Hamiltonian Optics, but we are obliged to introduce these concepts if, for example, we want to give a global meaning to the Hamilton characteristic function, as shown in Chapters 3 and 4. Symplectic relations and generating families can play an interesting role also in the control theory of static systems, including thermostatic systems. Chapter 5 is devoted to this matter. Our approach is based on the notion of control relation and on an extended version of the virtual work principle for constrained systems with non-controlled degrees of freedom (hidden variables). Several examples of singular phenomena concerning static systems and thermostatics are illustrated. In particular, it is shown how the Maxwell rule follows as a theorem from the extended virtual work principle. Thermostatics of simple and composite systems is here described in the four-dimensional state space, with global coordinates (S, V, P, T), entropy, volume, pressure, absolute temperature, endowed with the natural symplectic structure induced by the first principle of thermodynamics. An outline of the basic tools of calculus on manifolds needed in our discussion is given in Appendix A. A supplementary note (Appendix B) written in collaboration with Franco Cardin (Dipartimento di Matematica Pura e Applicata, Università di Padova), is devoted to the calculus of global principal Hamilton functions for the eikonal equations on the two-dimensional sphere S2 and pseudo-sphere H2.