Holonomic constraints integrating factor question

[eyenkay]

Ok, here's the question, have patience with my terrible latex skills...

Homework Statement

The equations of constraint of the rolling disk:
dx - asin(theta)d(phi) = 0 -> 1.
dy + acos(theta)d(phi) = 0 -> 2.
are special cases of general linear diff-eqs of constraint of the form:
$$\sum$$g_i(x1,...,xn)dxi = 0 -> 3.

A constraint condition of this tupe is holonomic only if an integrating function f(x1,...,xn) can be found that turns it into an exact differential. Clearly the fn. must be such that
$$\delta$$(fg_i)/$$\delta$$x_j = $$\delta$$(fg_j)/$$\delta$$x_i -> 4.

for all i$$\neq$$j. Show that no such integrating factor can be found for either of equations 1 or 2.

Homework Equations

above

The Attempt at a Solution

I have found that when I put either equation 1 or 2 into 3, i can cancel the df, and I am left with the original equation, but with the sign backwards (ie 1 becomes dx + ..., and 2 becomes dy - ...). I don't know if this actually means anything...

I am wondering if I can possibly arrange eq 4. so that i have something to integrate, and show that f diverges...? I don't know how to show that there is no such f...

Anybody have any tips or ideas?
Thanks for your time.

 

[NateD]

The best way to approach this problem is to start by writing down the integrating factor that you are trying to find. That is, you need to try and find a function f(x1,...,xn) such that eq 4. is satisfied. Once you have done this, you can then substitute the equations 1 and 2 into eq 4. and try to solve for your integrating factor. You will find that the equations will not be solvable, as the left-hand side of eq 4. will not equal the right-hand side. This will demonstrate that it is impossible to find an integrating factor for either equation 1 or 2.