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  1. Is it possible for a system to have holonomic and non-holonomic constraints at the same time?

  2. If so, in this scenario does it make sense to talk about a set of independent 'generalized coordinates'?

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  1. Yes. E.g. take your favorite $d$-dimensional non-holonomic system and pretend that it lives in $d+1$ dimensions by imposing a holonomic constraint that the extra dimension is a constant.

  2. No, not in the conventional sense of the word generalized coordinates $(q^1, \ldots, q^n)$. They are by definition constructed out of a larger set of coordinates [typically $({\bf r}_1, \ldots, {\bf r}_N)$] by imposing a set of holonomic constraints only. If we now impose further [e.g. non-holonomic$^\dagger$] constraints, the generalized coordinates $(q^1, \ldots, q^n)$ wont be completely independent anymore.

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$^\dagger$ Semiholonomic constraints can be implemented with the help of Lagrange multipliers, cf. e.g. my Phys.SE answer here.

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Why not?

A holomonic constraint in classical mechanics are relations between the position variables (and possibly time)

And this means that a constraint can be written as

$f(q_1,...,q_n,t) =0

Where of course the variables $q_1,...,q_n$ are the usual generalised position variables.

Thus a non-holonomic constraint is one that includes a dependence upon a generalised velocity.

Since we can have more than one constraint, it's obviously possible to have a holonomic and non-holonomic constraint.

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