Editing Canonical transformation (section)

=== Symplectic condition ===
Applying transformation of co-ordinates formula for <math> \nabla_\eta H = M^T \nabla_\varepsilon H   
</math>, in Hamiltonian's equations gives:

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

Similarly for <math display="inline">\dot{\varepsilon}
</math>:      

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

or:      

<math display="block">\dot{\varepsilon}=J \nabla_\varepsilon K  =  J \nabla_\varepsilon H + J \nabla_\varepsilon 
 \left( \frac{\partial G}{\partial t}\right)
</math>      

Where the last terms of each equation cancel due to <math display="inline">J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t}  </math> condition from canonical transformations. Hence leaving the symplectic relation: <math display="inline">M J M^T = J
</math> which is also equivalent with the condition <math display="inline">M^T J M = J
</math>. It follows from the above two equations that the symplectic condition implies the equation <math display="inline">J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t}  </math>, from which the indirect conditions can be recovered. Thus, symplectic conditions and indirect conditions can be said to be equivalent in the context of using generating functions.