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Symplectic Geometry

2022; Springer Nature; Linguagem: Inglês

10.1007/978-3-031-05122-7_3

1617-9692

Nima Moshayedi,

Homotopy and Cohomology in Algebraic Topology

Symplectic geometry is mainly motivated by the study of classical mechanics and dynamical systems. It can only be described for even-dimensional geometries where it serves as a way of measuring 2-dimensional objects. The word symplectic comes form the greek word for complex, which indicates that this theory is naturally associated to the field of complex numbers. In fact, it describes a generalization of complex geometry since the structure is not as strict as one can associated to each symplectic structure a certain compatible (almost) complex structure that is unique up to homotopy which is not integrable. This means that symplectic structures provide a more flexible structure than complex ones. In classical mechanics, we usually describe the dynamics of a mass particle through the coordinates which are given by position q i and momentum p i. Hence, the prototypical space where the particle moves is given by $$\mathbb {R}^6$$ , since we have 3 space coordinates and to each one the corresponding momentum coordinate. In general, one can consider the space $$M:=\mathbb {R}^{2n}$$ for n ≥ 1. The symplectic structure is then defined as an area 2-form $$\omega =\sum _{i=1}^n{\mathrm {d}} q_i\land {\mathrm {d}} p_i$$ . The dynamical information is usually encoded in a function H ∈ C ∞(M) called the Hamiltonian. It is convenient to extract a vector field $$X_H\in \mathfrak {X}(M)$$ out of H in order to express the dynamics in terms of the flow lines of this vector field, i.e. to consider a differential equation with respect to the change of H. In particular, one would like to consider a map ω: TM → T ∗ M, or equivalently an element of T ∗ M ⊗ T ∗ M, such that $${\mathrm {d}} H=\iota _{X_H}\omega =\omega (X_H,\enspace )$$ . Additionally, one would like the choice of X H for each H to be unique in this way. It is easy to see that this will require ω to be non-degenerate. Moreover, one would like to have H such that it does not change along flow lines which means that dH(X H) = 0. This would imply that ω(X H, X H) = 0 and thus one requires ω to be alternating. This is the reason why we want ω to be a 2-form. Note that this also implies that the underlying space has to be even-dimensional since every skew-symmetric linear map for odd dimensions is singular. Finally, one also would like that ω does not change under flow lines. Mathematically, this is expressed as the vanishing of the Lie derivative $$L_{X_H}\omega =0$$ . Using Cartan’s magic formula ( 2.19 ), we get $$\displaystyle L_{X_H}\omega ={\mathrm {d}}\iota _{X_H}\omega +\iota _{X_H}{\mathrm {d}}\omega ={\mathrm {d}}({\mathrm {d}} H)+\iota _{X_H}{\mathrm {d}}\omega ={\mathrm {d}}\omega (X_H). $$ Hence, if we require dω(X H) = 0 for the vector field induced by different Hamiltonians H, we require ω to be closed, i.e. dω = 0. The aim of this chapter is to introduce the most important concepts of symplectic geometry in a precise way by first considering the linear setting and then move to the global case. We will then move to more sophisticated methods which will be relevant to several aspects of quantization as we will see later on.