Introduction to symplectic mechanics

Zaidni Azeddine (1)

1 UM6P - Université Mohammed VI Polytechnique [Ben Guerir]

The aim of the present lecture note is to provide some basic notions of Hamiltonian mechanics and their relationship with symplectic geometry. The introduction is devoted to present the historical appearance of Lagrangian and Hamiltonian mechanics by using simple examples. The second section discusses the basic notions on manifolds as the phase space of a dynamical system is naturally a manifold. This manifold has a special structure known as symplectic structure. The mathematical objects we use to treat the symplectic structure are essentially the differential forms. The qualities of these forms will be discussed in sections three and four, with important concepts such as exterior and interior product, exterior differentiation, integration of forms, and so on. In section five, we shall discuss the features of a symplectic manifold as well as some useful concepts for dealing with the Hamiltonian structure, such as the Lie derivative and Poisson bracket. In section six, we present a numerical scheme for Hamilton's equations that can provide meaningful simulations in phase space. Section seven provides several examples from various disciplines that demonstrate the usefulness of the Hamiltonian formalism. This technique is fundamental and applicable in a variety of physics fields, including classical mechanics, fluid mechanics, plasma physics, optics, quantum mechanics, chemistry, and so on.

Domaines: Physique mathématique [math-ph]; Géométrie symplectique [math.SG]; Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]; Science non linéaire [physics]; Physique [physics]

Zaidni Azeddine. Introduction to symplectic mechanics. Master. Morocco. 2022. ⟨hal-03611773⟩