Editing Canonical transformation (section)

===Type 2 generating function===
The type 2 generating function <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> depends only on the old [[generalized coordinates]] and the new generalized momenta <math display="inline">G \equiv  G_{2}(\mathbf{q}, \mathbf{P}, t)-\mathbf{Q} \cdot \mathbf{P}</math> where the <math>-\mathbf{Q} \cdot \mathbf{P}</math> terms represent a [[Legendre transformation]] to change the right-hand side of the equation below. 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{Q} \cdot \dot{\mathbf{P}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{2}}{\partial t} + \frac{\partial G_{2}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial G_{2}}{\partial \mathbf{P}} \cdot \dot{\mathbf{P}}</math>

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

<math display="block">\begin{align}
\mathbf{p} &= \frac{\partial G_{2}}{\partial \mathbf{q}} \\
\mathbf{Q} &= \frac{\partial G_{2}}{\partial \mathbf{P}} \\
K &= H + \frac{\partial G_{2}}{\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_{2}}{\partial \mathbf{q}}</math> define relations between the new generalized momenta {{math|'''P'''}} and the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. Ideally, one can invert these relations to obtain formulae for each {{math|''P<sub>k</sub>''}} as a function of the old canonical coordinates. Substitution of these formulae for the {{math|'''P'''}} coordinates into the ''second'' set of {{mvar|N}} equations <math display="inline">\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}}</math> yields analogous formulae for the new generalized coordinates {{math|'''Q'''}} 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_{2}}{\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_{2} \equiv \mathbf{g}(\mathbf{q}; t) \cdot \mathbf{P}</math> where {{math|'''g'''}} is a set of {{mvar|N}} functions. This results in a point transformation of the generalized coordinates <math display="inline">\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}} = \mathbf{g}(\mathbf{q}; t)</math>.