Suppose you have two conjugate variables $q$ and $p$ that are canonically transformed into two other variables $Q$ and $P$. What needs to hold true for these variables in terms of Poisson brackets? I am being asked to confirm that $Q$ and $P$ are canonically conjugate through Poisson brackets.
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$\begingroup$ Canonically transformed in what sense? $\endgroup$– Qmechanic ♦Oct 30, 2021 at 17:27
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1 Answer
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A canonical trasformation conserve the poisson braket, so if the variables q,p are conjugate they satisfy {p,q} = 1 , also the trasformed variables have to satisfy the same condition
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$\begingroup$ But isn't that always the case? If you have {Q,P}=(dQ/dQ)*(dP/dP)-(dQ/dP)*(dP/dQ)=1 isnt that always one? I feel like I need to use the relations between the old and new variables. $\endgroup$– thecarnyOct 30, 2021 at 13:32
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$\begingroup$ If you make a canonical trasformation is always true. But It's not true for a general trasformation. $\endgroup$ Oct 30, 2021 at 13:35