Editing Canonical transformation (section)

=== Generalized use of generating functions ===

In other words, since {{math|('''Q''', '''P''')}} and {{math|('''q''', '''p''')}} are each {{math|2''N''}} independent functions, it follows that to have generating function of the form <math display="inline">G_{1}(\mathbf{q}, \mathbf{Q}, t) </math> and <math>G_{4}(\mathbf{p}, \mathbf{P}, t)</math> or <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> and <math>G_{3}(\mathbf{p}, \mathbf{Q}, t)</math>, the corresponding Jacobian matrices <math display="inline">\left[\frac{\partial Q_i}{\partial p_j} \right]  </math> and <math display="inline">\left[\frac{\partial P_i}{\partial p_j} \right]  </math> are restricted to be non singular, ensuring that the generating function is a function of {{math|2''N'' + 1}} independent variables. However, as a feature of canonical transformations, it is always possible to choose {{math|2''N''}} such independent functions from sets {{math|('''q''', '''p''')}} or {{math|('''Q''', '''P''')}}, to form a generating function representation of canonical transformations, including the time variable. Hence, it can be proven that every finite canonical transformation can be given as a closed but implicit form that is a variant of the given four simple forms.<ref>{{Harvnb|Sudarshan|Mukunda|2010|pages=58-60}}</ref>

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!Proof
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Consider taking a full set of generalized coordinates <math display="inline"> \{q_{1}, q_{2}, \ldots, q_{N-1}, q_{N} \} </math> and adding to the set, while preserving local invertibility of coordinates in the set, as many transformed coordinates as possible, labelled <math display="inline">\{Q_{1}, Q_{2}, \ldots, Q_{k} \}</math> without loss of generality.

It can be shown that the set, <math display="inline">\{q_{1}, \ldots, q_{N}, Q_{1}, \ldots, Q_{k}, P_{k+1}, \ldots, P_{N} \}</math> is a set of locally independent coordinates. Proof of local invertibility of the set of coordinates is given by proving non singularity of <math display="inline">\frac{\partial(Q_1,\ldots,Q_k,P_{k+1},\ldots,P_N)}{\partial(p_1,\ldots,p_N)}</math> or the non existence of a non trivial null eigenvector such that <math display="inline">\sum_a\epsilon_a\frac{\partial Q_a}{\partial p_s}+\sum_b \eta_b\frac{\partial P_b}{\partial p_s}=0,\, \forall s</math> where the index <math display="inline">a=1,\ldots,k</math> and <math display="inline">b=k+1,\ldots,N</math>.

Letting <math display="inline"> Q_b=f_b(q_s,Q_a) </math> and assuming the existence of a null eigenvector in the following derivation:

<math display="inline"> \eta_{b'}=\sum_a\epsilon_a \{Q_{b'},Q_a\}+\sum_b\eta_b\{Q_{b'},P_b\} =\sum_s \frac{\partial f_{b'}}{\partial q_s}(\sum_a\epsilon_a\frac{\partial Q_a}{\partial p_s}+\sum_b \eta_b\frac{\partial P_b}{\partial p_s})=0 </math>

Hence all <math display="inline">\eta_b=0</math>. By condition of local invertibility it follows that for the remaining part of the equation, <math display="inline"> \sum \frac{\partial Q_a}{\partial p_i}\epsilon_i= \delta Q_a(p_1,\ldots,p_N) =0\implies \epsilon_i =0 \quad \forall\, a=1,\ldots,k </math> thereby showing that the only null eigenvector <math display="inline">\frac{\partial(Q_1,\ldots,Q_k,P_{k+1},\ldots,P_N)}{\partial(p_1,\ldots,p_N)}</math> is the trivial vector implying that it is a non singular matrix. Hence it is shown that it is possible to take sets such as <math display="inline">\{q_{1}, \ldots, q_{N}, Q_{1}, \ldots, Q_{k},P_{k+1}, \ldots, P_{N} \}</math> that is a combination of new and old coordinates that preserves the {{math|2''N''}} independent variables property which can be used to interpret any coordinate transform as arising from a generating function on these set of coordinates.
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