Classical Mechanics and Poisson Structures

Abstract

In this chapter, we will briefly recall the Hamiltonian formulation of classical mechanics, focusing in particular on its algebraic aspects. In this framework, a classical system will be described by a commutative algebra of functions (classical observables) with the Poisson bracket as a Lie bracket. We will discuss in detail the properties of the Poisson bracket and introduce the tensor formulation of Poisson structures on manifolds, as the Poisson bracket plays a fundamental role in classical mechanics and in deformation quantization. We will mainly focus on the algebraic rather than geometrical properties of Poisson manifolds, the latter being less important for the theory of deformation quantization. Furthermore, we will introduce the reader to the basic notions needed for the for-mulation of the formality theory, i.e. formal power series, formal Poisson structures and equivalence classes of formal Poisson structures. 2.1 Hamiltonian Mechanics and Poisson Brackets This section aims to give a brief introduction to classical mechanics, starting with Newton's laws and heading towards the Hamiltonian approach, with a particular attention to the role of the Poisson bracket. The interested reader is referred to the classical literature on the subject, as e.g. [1, 2, 6] for an exhaustive treatment. We start by discussing the motion of a point particle of mass m in the Euclidean space R n . The position of the particle is described by the vector q := (q 1 , . . . , q n) ∈ R n . The vector q is generally parametrized by the variable t ∈ R. We say that q(t) is the position of the particle at time t. The velocity v(t) of the particle at time t is defined as v(t) := ˙ q(t) = d q d t (t), (2.1)