Hamiltonian Mechanics of Discrete Particle Systems

In the previous chapter, we discussed briefly the fundamental nature of the symplectic structure of theories in optics in order to illustrate the underlying uniformity, physical consistency, and mathematical simplicity inherent to a symplectic mathematical formulation of the governing equations. Hence the main emphasis of chapter 2 was to “discover” the symplectic structure in the physical theories of optics and to see how this structure is interconnected with and implies fundamental theorems in optics, such as Fermat's principle and Hamilton's equations. In the present chapter, we continue our efforts to present a coherent description of symplectic transformations and their applications to physical systems; however, here we switch our emphasis from the underlying symplectic structure of the dynamical equations to the physical integrity of the Poisson bracket and the canonical equations which find their roots in Hamilton's principle of least action and the calculus of variations. Hence we intend to cover ground in this chapter which we neglected in the previous one, and, in so doing, to gradually begin to move towards the applications of the extended bracket formalism at which this book is aimed. In order to apply Hamilton's principle of least action, we first need to study a simple problem of the calculus of variations, following Bedford [1985, §1.1]. Let x be a real variable (x∊ R) on the closed interval x1≤ x≤ x2, denoted [x1 ,x2] .