Goldstein Derivation 1.6: Nonholonomic Constraints in Particle Motion

[DrHouse]

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?

 

[PAllen]

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.