# Lagrangian Mechanics: semi-holonomic constraints

By switching to a different set of coordinates, can you make problem with semi holonomic constraints into a problem with holonomic constraints? If so, then when can you do this? I wold like to know if this is possible for all semi-holonomic problems, some semi-holonomic problems or no semi-holonomic problems.

My intuition is that it probably works for some specific cases, but not in general. However, I don't know the reason why. Would be very nice with some sort of proof.

(Constraints on the form: $$f=f(q_i,\dot q_i,t)$$.)

1. A non-holonomic$$^1$$ constraint is by definition a constraint that is not holonomic, e.g. on the form $$f(q,\dot{q},t)=0$$ or an inequality.
2. A semi-holonomic constraint $$\omega~\equiv~\sum_{j=1}^na_j(q,t)~\mathrm{d}q^j+a_0(q,t)\mathrm{d}t~=~0$$ is equivalent to a holonomic constraint iff there exist an integrating factor $$\lambda(q,t)\neq 0$$ and a one-form $$\eta$$ such that $$\lambda\omega+ f\eta~\equiv~\mathrm{d}f ,$$ cf. my related Phys.SE answer here.
$$^1$$ If you are using the 3rd edition of Goldstein, be aware of erratum.
• I have Pearson New International edition Third edition from 2014. But the last correction made on the errata homepage was in 2010. ( astro.physics.sc.edu/Goldstein ) I checked one of the revisions ( a $\delta$ corrected to $\partial$) and it was indeed corrected in my edition. Guess it is updated then. Sep 13, 2022 at 6:21