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Let $\{q_i,p_i, i=1,...,n\}$ be coordinate and their conjugate momentum.

Let $\xi_k, k=1,...,2n$ be generalized coordinates which equal to $\{q_i,p_i, i=1,...,n\}$

Suppose the matrix $$w_{ij}:=\sum\limits_{k=1}^n \left({\partial \xi_i\over \partial q_k}{\partial \xi_j\over \partial p_k}-{\partial \xi_j\over \partial q_k}{\partial \xi_i\over \partial p_k}\right)$$

What then does the inverse of $w$, i.e. $w^{-1}$, mean?

This question is related to this question. I thought I might get a better response if I focus on a smaller problem.

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Um, if $w^{-1}$ exists, it means the matrix such that $w^{-1}w = I$? What kind of meaning do you want? A physical interpretation? An interpretation in terms of the standard symplectic form on the cotangent bundle? –  Neal Oct 15 '12 at 15:50
@Neal : Thanks for commenting! I am wondering what the inverse of $w$ would be , explicitly. –  Daphne P Oct 15 '12 at 15:57

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