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Goldstein Derivation 1.6

  1. Jan 29, 2016 #1
    1. The problem statement

    A particle moves in the ##xy## plane under the constraint that its velocity vector is always directed towards a point on the ##x## axis whose abscissa is some given function of time ##f(t)##. Show that for ##f(t)## differentiable but otherwise arbitrary, the constraint is NONHOLONOMIC.

    2. The attempt at a solution
    I have got the differential equation relating the generalised coordinates (in this case they are ##x## and ##y## since the system has two degrees of freedom):

    ## ydx + [f(t)-x]dy = 0 ##

    If the constraint equation is nonholonomic, then the previous integral can not be performed but it's true that it can be written as

    ## \displaystyle \int \frac{dx}{x-f(t)} = \int \frac{dy}{y}##


    which, I think is an integrable expression in terms of logarithmic functions for any ##f(t)##. Some people say that the previous integral can not be performed in general due to the arbitrariness of ##f(t)##. Can anyone explain me why?
     
  2. jcsd
  3. Jan 30, 2016 #2

    PAllen

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    Science Advisor
    Gold Member

    Are you forgetting that t must be considered as an implicit function of x and y in your differential expression? That is, f(t) cannot be considered as independent of x and y.
     
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