Friday, 15 April 2016

classical mechanics - Independent canonical coordinate variables?


In Goldstein's Classical Mechanics (2nd ed.) on section 9-1 page 382, there is a discussion about finding a canonical transformation (qi,pi)(Qj(qi,pi,t),Pj(qi,pi,t)) from a given generating function F=F1(qi,Qi,t). The following is written on the page:



Equation (9-11) then takes the form,


pi˙qiH=Pi˙QiK+dF1dt,=Pi˙QiK+F1t+F1qi˙qi+F1Qi˙Qi


Since the old and the new coordinates, qi and Qi , are separately independent, Eq. (9-13) can hold identically only if the coefficients of ˙qi and ˙Qi each vanish:



I don't understand the bold text above. How are qi and Qi separately independent? I thought Qi was, in general, a function of the original canonical coordinate variables qi, thereby making it explicitly dependent on qi (not independent). Could somebody explain how these two canonical coordinate variables are separately independent?



Answer




Let us suppress t-dependence in this answer for simplicity. A canonical transformation1 (q,p)(Q(q,p),P(q,p))

can be viewed as a graph, thereby yielding a 2n-dimensional submanifold M embedded inside a 4n-dimensional total manifold N. The total manifold N has 4n local coordinates (q,p,Q,P). The submanifold M can in certain cases be viewed as a graph of a function (q,Q)(p(q,Q),P(q,Q)),
where q and Q are independent variables, and where the functions p(q,Q) = F1(q,Q)q,
P(q,Q) = F1(q,Q)P,
comes from an F1-type generating function F1(q,Q).


References:



  1. H. Goldstein, Classical Mechanics.


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1 Whatever that means.


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