Editing Canonical transformation (section)

=== Symplectic condition ===
Sometimes the Hamiltonian relations are represented as:

<math display="block">\dot{\eta}= J \nabla_\eta H
</math>

Where <math display="inline">J := \begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix},</math> 

and <math display="inline">\mathbf{\eta} =
  \begin{bmatrix}
    q_1\\
    \vdots \\
    q_n\\
    p_1\\
    \vdots\\
    p_n\\    
  \end{bmatrix}
</math>. Similarly, let <math display="inline">\mathbf{\varepsilon} =
  \begin{bmatrix}
    Q_1\\
    \vdots \\
    Q_n\\
    P_1\\
    \vdots\\
    P_n\\    
  \end{bmatrix}
</math>.



From the relation of partial derivatives, converting the <math>\dot{\eta}= J \nabla_\eta H
</math> relation in terms of partial derivatives with new variables gives <math>\dot{\eta}=J ( M^T \nabla_\varepsilon H)
</math> where <math display="inline">M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}</math>. Similarly for <math display="inline">\dot{\varepsilon}
</math>,

<math display="block">\dot{\varepsilon}=M\dot{\eta} =M J M^T \nabla_\varepsilon H
</math>


Due to form of the Hamiltonian equations for <math display="inline">\dot{\varepsilon}
</math>,

<math display="block">\dot{\varepsilon}=J \nabla_\varepsilon K  =  J \nabla_\varepsilon H
</math>


where <math display="inline">\nabla_\varepsilon K  =  \nabla_\varepsilon H
</math> can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:<ref> {{Harvnb|Goldstein|Poole|Safko|2007|p=381-384}}</ref>

<math display="block">M J M^T = J
</math>

The left hand side of the above is called the Poisson matrix of <math>\varepsilon
</math>, denoted as <math display="inline">\mathcal P(\varepsilon) = MJM^T
</math>. Similarly, a Lagrange matrix of <math>\eta
</math> can be constructed as <math display="inline">\mathcal L(\eta) = M^TJM   
</math>.<ref name=":0">{{Harvnb|Giacaglia|1972|p=8-9}}</ref> It can be shown that the symplectic condition is also equivalent to <math display="inline">M^T J M = J  
</math> by using the <math display="inline">J^{-1}=-J
</math> property. The set of all matrices <math display="inline">M
</math> which satisfy symplectic conditions form a [[symplectic group]]. The symplectic conditions are equivalent with indirect conditions as they both lead to the equation <math display="inline">\dot{\varepsilon}=  J \nabla_\varepsilon H
</math>, which is used in both of the derivations.