Editing Poisson bracket (section)

== Poisson matrix in canonical transformations ==
{{Main|Canonical transformation}}
The concept of Poisson brackets can be expanded to that of matrices by defining the Poisson matrix.

Consider the following canonical transformation:<math display="block">\eta = 
  \begin{bmatrix}
    q_1\\
    \vdots \\
    q_N\\
    p_1\\
    \vdots\\
    p_N\\    
  \end{bmatrix} \quad \rightarrow \quad \varepsilon = 
  \begin{bmatrix}
    Q_1\\
    \vdots \\
    Q_N\\
    P_1\\
    \vdots\\
    P_N\\    
  \end{bmatrix} </math>Defining <math display="inline">M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}</math>, the Poisson matrix is defined as <math display="inline">\mathcal P(\varepsilon) = MJM^T
</math>, where <math>J</math> is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that:<math display="block">\mathcal P_{ij}(\varepsilon) = [MJM^T]_{ij}=\sum_{k=1}^{N} \left( \frac{\partial \varepsilon_i}{\partial \eta_{k}} \frac{\partial \varepsilon_j}{\partial \eta_{N+k}} - \frac{\partial \varepsilon_i}{\partial \eta_{N+k}} \frac{\partial \varepsilon_j}{\partial \eta_k}\right)=\sum_{k=1}^{N} \left( \frac{\partial \varepsilon_i}{\partial q_{k}} \frac{\partial \varepsilon_j}{\partial p_k} - \frac{\partial \varepsilon_i}{\partial p_k} \frac{\partial \varepsilon_j}{\partial q_k}\right)=\{ \varepsilon_i,\varepsilon_j\}_\eta.
</math>

The Poisson matrix satisfies the following known properties:<math display="block">\begin{align}
\mathcal P^T &= - \mathcal P \\
|\mathcal P| &= \frac{1}{|M|^2}\\
\mathcal P^{-1}(\varepsilon)&= -(M^{-1})^T J M^{-1} = - \mathcal L (\varepsilon)\\
\end{align}
</math>

where the <math display="inline">\mathcal L(\varepsilon)
</math> is known as a Lagrange matrix and whose elements correspond to [[Lagrange bracket]]s. The last identity can also be stated as the following:<math display="block">\sum_{k=1}^{2N} \{\eta_i,\eta_k\}[\eta_k,\eta_j] = -\delta_{ij} 
</math>Note that the summation here involves generalized coordinates as well as generalized momentum.

The invariance of Poisson bracket can be expressed as:  <math display="inline">\{ \varepsilon_i,\varepsilon_j\}_\eta=\{ \varepsilon_i,\varepsilon_j\}_\varepsilon = J_{ij}
</math>, which directly leads to the symplectic condition: <math display="inline">MJM^T = J    
</math>.<ref>{{Cite book |last=Giacaglia |first=Giorgio E. O. |title=Perturbation methods in non-linear systems |date=1972 |publisher=Springer |isbn=978-3-540-90054-2 |series=Applied mathematical sciences |location=New York Heidelberg |pages=8–9}}</ref>