Editing Canonical transformation (section)

=== Limitations on the four types of generating functions ===
Considering <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> as an example, using generating function of second kind: <math display="inline">{p}_i = \frac{\partial G_{2}}{\partial {q}_i}
 </math> and <math display="inline">{Q}_i = \frac{\partial G_{2}}{\partial {P}_i} </math>, the first set of equations consisting of variables <math display="inline">\mathbf{p} </math>, <math display="inline">\mathbf{q} </math> and <math display="inline">\mathbf{P} </math> has to be inverted to get <math display="inline">\mathbf{P}(\mathbf q, \mathbf p) </math>. This process is possible when the matrix defined by <math display="inline">a_{ij}=\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j} </math> is non-singular using the [[inverse function theorem]], and can be restated as the following relation.<ref>{{Harvnb|Johns|2005|p=438}}</ref>

<math display="block">\left|\begin{array}{l l l}{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{1}\partial q_{1}}}}}&{{\cdots}}&{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{1}\partial q_{n}}}}}\\ {\quad \vdots} & {\ddots}&{\quad \vdots}\\{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{n}\partial q_{1}}}}}&{{\cdots}}&{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{n}\partial q_{n}}}}}\end{array}\right|{\neq0}</math>

Hence, restrictions are placed on generating functions to have the matrices: <math display="inline">\left[\frac{\partial^2 G_1}{\partial Q_j\partial q_i} \right] </math>, <math display="inline">\left[\frac{\partial^2 G_2}{\partial P_j\partial q_i} \right]  </math>, <math display="inline">\left[\frac{\partial^2 G_3}{\partial p_j\partial Q_i} \right]  </math> and <math display="inline">\left[\frac{\partial^2 G_4}{\partial p_j\partial P_i} \right]  </math>, being non-singular.<ref>{{Harvnb|Lurie|2002|p=547}}</ref><ref>{{Harvnb|Sudarshan|Mukunda|2010|p=58}}</ref> These conditions also correspond to local invertibility of the coordinates. From these restrictions, it can be stated that type 1 and type 4 generating functions always have a non-singular <math display="inline">\left[\frac{\partial Q_i(\mathbf q,\mathbf p)}{\partial p_j} \right]  </math> matrix whereas type 2 and type 3 generating functions always have a non-singular <math display="inline">\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right]  </math> matrix. Hence, the canonical transformations resulting from these four generating functions alone are not completely general.<ref>{{Harvnb|Johns|2005|p=437-439}}</ref>