Skew symmetric scalar product

Recall the relation between skew-symmetric scalar products and exterior differential 2-forms. Consider the form... 

Here a = WijdxiAdxj and sgrad / = ((sgrad /)i,..., (sgrad /)2n) Thus, the Poisson bracket of the functions f and g is a skew-symmetric scalar product (with respect to the form a ) of their skew gradients. If we denote by the coefficients of the matrix reciprocal to the matrix (c< y), then the Poisson bracket can be written in the local coordinates xi,..., X2n on Af as... 

In line with this theory, a symplectic geometry in a linear space TxM s constructed whose properties are determined by setting a skew-symmetric ( ) nondegenerate scalar product (, ). This cannot be done in all cases but only when the dimension of the manifold M is even. Then an even-dimensional space TxM transforms into a symplectic space In what follows, we shall assume for convenience that a symplectic space is modelled on a Euclidean space (on which, therefore, two forms are simultaneously given, namely, symmetric and skew-symmetric). 

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