Defining Poisson Brackets: Analytic Functions in Multiple Variables

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Poisson brackets can be defined for analytic functions in multiple variables by utilizing fundamental properties of these brackets. The analytic nature of the functions allows for their representation through power expansions, which can be simplified using established rules of Poisson brackets. The discussion highlights the need for resources on Taylor expansions in multiple variables, particularly for functions of two variables. Participants suggest searching for "calculus of several/many variables" and refer to Wikipedia for information on Taylor series. The conversation emphasizes the lack of specific references for combining Taylor series with Poisson brackets.
stefano colom
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l know you can define poisson brackets between two analytic function in several variables f(q1,q2,q3,..,p1,p2,p3,..) and g (q1,q2,q3,..,p1,p2,p3,..) only by foundamental poisson brackets and their proprieties.how is it possible?
 
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If the function is analytic, this means it is equal to its power expansion. If you substitute this Taylor expansion in the Poisson bracket you want to calculate, you can then use simple rules to simplify it (like P.B. of sum is sum of P.B. etc) to few basic P.B.s, like that for q1,p1.
 
how it is the power expansion of the function f(x,y)?
l know the power expansion of f(x), but the one of f(x,y)? can anyone tell me a website where l can find more informations about that?thank you
 
can you suggest me a site where l can find how to use taylor series with Poisson Brackets?
 
Unfortunatly I do not know any reference for this topic.
 
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