1. ## Exact differential

Hi. Given equations of constraint for the motion of a particle in the form:

$\sum_{i=1}^{n} g_{i}(x_{1},...,x_{n} ) dx_{i} = 0$

will be holonomic only if exists a function $f(x_{1},..,x_{n})$ such that:

$\frac{\partial fg_{i}}{\partial x_{j}} = \frac{\partial fg_{j}}{\partial x_{i}}$ for all $i \neq j$.

In the particular case of:

$dx - a sin(\theta) d\phi =0$
$dy + a cos (\theta) d\phi = 0$

I should find an integrating factor of the form $f(x,y,\theta,\phi)$, but I don't know wich are exactly the $g_{i}$ functions. There are four $g_{i}$.

2. Well, your problem is that those two equations are NOT of the form $\sum_{i=1}^{n} g_{i}(x_{1},...,x_{n} ) dx_{i} = 0$ and can be put in that form in a number of different ways. For example, just adding them gives $dx+ dy+ a(cos(\theta)- sin(\theta))d\phi= 0$. In that case $g_x= 1$, $g_y= 1$, $g_\theta= 0$, and $g_\phi= a(cos(\theta)- sin(\theta))$.
If you were to subtract the two equations rather than add, you would get $dx- dy- a(sin(\theta)+ cos(\theta))d\phi$ and that would a different f.
Personally, what I would do is integrate $dx= a sin(\theta)d\phi$ and $dy= -acos(\theta)d\phi$, treating $\theta$ as a constant. That will give x and y as functions of $\theta$ and $\phi$.