1.3 Notations and Preliminaries

In this section, we introduce notations and some important definitions that will be used frequently in the book.

The notation will be standard most of the times except where stated otherwise. Moreover, **N** will denote the set of natural numbers, while **Z** will denote the set of integers. Similarly,ℜ, ℜ^{n} will denote respectively, the real line and the *n-*dimensional real vector space, *t* ∈ ℜ will denote the time parameter.

$\mathcal{X}$, *M*, *N*,… will denote *differentiable-manifolds* of dimension *n* which are locally Euclidean and $TM\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\cup}_{x\in M}{T}_{x}M,\text{\hspace{0.17em}}{T}^{\star}M\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\cup}_{x\in M}{T}_{x}^{\star}M$ will denote respectively the *tangent and cotangent bundles* of *M* with dimensions 2*n*. Moreover, *π* and *π*^{⋆} will denote the natural projections *T M* → *M* and *T*

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