Editing Canonical transformation (section)

===Type 1 generating function===
The type 1 generating function {{math|''G''<sub>1</sub>}} depends only on the old and new generalized coordinates <math display="inline">G \equiv G_{1}(\mathbf{q}, \mathbf{Q}, t)</math>. To derive the implicit transformation, we expand the defining equation above
<math display="block"> \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{1}}{\partial t} + \frac{\partial G_{1}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial G_{1}}{\partial \mathbf{Q}} \cdot \dot{\mathbf{Q}}</math>

Since the new and old coordinates are each independent, the following {{math|2''N'' + 1}} equations must hold

<math display="block">\begin{align}
\mathbf{p} &= \frac{\partial G_{1}}{\partial \mathbf{q}} \\
\mathbf{P} &= -\frac{\partial G_{1}}{\partial \mathbf{Q}} \\
K &= H + \frac{\partial G_{1}}{\partial t}
\end{align}</math>

These equations define the transformation {{math|('''q''', '''p''') → ('''Q''', '''P''')}} as follows: The ''first'' set of {{mvar|N}} equations <math display="inline">\ \mathbf{p} = \frac{\ \partial G_{1}\ }{ \partial \mathbf{q} }\ </math> define relations between the new [[generalized coordinates]] {{math|'''Q'''}} and the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. Ideally, one can invert these relations to obtain formulae for each {{math|''Q<sub>k</sub>''}} as a function of the old canonical coordinates. Substitution of these formulae for the {{math|'''Q'''}} coordinates into the ''second'' set of {{mvar|N}} equations <math display="inline">\mathbf{P} = -\frac{\partial G_{1}}{\partial \mathbf{Q}}</math> yields analogous formulae for the new generalized momenta {{math|'''P'''}} in terms of the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. We then invert both sets of formulae to obtain the ''old'' [[canonical coordinates]] {{math|('''q''', '''p''')}} as functions of the ''new'' [[canonical coordinates]] {{math|('''Q''', '''P''')}}. Substitution of the inverted formulae into the final equation <math display="inline">K = H + \frac{\partial G_{1}}{\partial t}</math> yields a formula for {{mvar|K}} as a function of the new [[canonical coordinates]] {{math|('''Q''', '''P''')}}.

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let <math display="inline">G_{1} \equiv \mathbf{q} \cdot \mathbf{Q}</math>. This results in swapping the generalized coordinates for the momenta and vice versa 

<math display="block">\begin{align}
\mathbf{p} &= \frac{\partial G_{1}}{\partial \mathbf{q}} = \mathbf{Q} \\
\mathbf{P} &= -\frac{\partial G_{1}}{\partial \mathbf{Q}} = -\mathbf{q}
\end{align}</math>

and {{math|1=''K'' = ''H''}}. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.