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Canonical transformations, generating function

  1. Feb 29, 2012 #1


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    1. The problem statement, all variables and given/known data
    Given the generating function [itex]F=\sum _i f_i (q_j,t)P_i[/itex],
    1)Find the corresponding canonical transformations.
    2)Show that the transformations of generalized coordinates are canonical transformations.
    3)What meaning does the canonical transformation originated by the generating function [itex]\Phi (q,P)=\alpha qP[/itex] has?

    2. Relevant equations
    [itex]p_i=\frac{\partial F }{\partial q_i}[/itex], [itex]P_i=-\frac{\partial F }{\partial Q_i}[/itex], [itex]H'=H+\frac{\partial F }{\partial t}[/itex].

    3. The attempt at a solution
    I don't know how to start. The notation confuses me, particularly the j. Should the sum be a sum over i and j?

    Edit:1)[itex]Q_i=\frac{\partial F }{\partial P_i}=f_i(q_j,t)[/itex]. A canonical transformation is such that [itex]\dot Q_i=\frac{\partial H'}{\partial P_i}[/itex] and [itex]\dot P_i =-\frac{\partial H'}{\partial Q_i}[/itex].
    Therefore I guess I must verify that [itex]\frac{\partial ^2 F}{\partial t \partial P_i}=\dot Q_i[/itex] and that... oh well I'm totally confused.
    Last edited: Feb 29, 2012
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