Is every constraint involving only two coordinates integrable?

Question

There is a footnote on Goldstein's Classical Mechanics (3rd ed., page 15) which says the following:

In principle, an integrating factor can always be found for a first-order differential equation of constraint in systems involving only two coordinates and such constraints are therefore holonomic.

I am trying to show this but I can only do that for linear constraints. In that case the constraint can be written as $$\frac{dy}{dx}+a(x)y=b(x),\tag1$$ and after multiplying both sides by an integrating factor $e^{\left(\int a(x)dx\right)}$, one readily gets the integrable equation $$d\left(e^{\left(\int a(x)dx\right)}y\right)=b(x)e^{\left(\int a(x)dx\right)}dx.$$

The point is I cannot write constraints such as $$dy+\left[a(x)y^2+b(x)\right]dx=0,\tag2$$ in the form of $(1)$. Is there something missing in Goldstein's claim? If not, how to prove it for non-linear constraints?

Answer 1

(Assuming the usual continuity and nonsingularity conditions) A 1st order differential constraint of two variables $x$ and $y$ can be written as $$\begin{matrix}\omega = M(x,y)dx + N(x,y)dy=0& [1] \end{matrix}$$ Rearrange [1] to $$\begin{matrix}\frac{dy}{dx} = -\frac{M(x,y)}{N(x,y)} & [2] \end{matrix}$$ that leads to the form of a standard ordinary 1st order differential equation in one dependent variable $y$ and one independent variable $x$, and one that has a solution with a given initial condition $x_0,y_0$ in the implicit form of $F(x,y)=k$ for some $k$ that depends on the initial condition. This implicit equation is a solution to [1] and then is equivalent to $dF=0$ or in coordinates $$\begin{matrix}\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy=0 & [3]\end{matrix}$$ But [1] and [3] can exist simultaneously iff $\frac{\frac{\partial F}{\partial x}}{M(x,y)}=\frac{\frac{\partial F}{\partial y}}{N(x,y)}=\lambda(x,y)$ for some $\lambda$ that is just the sought for integrating multiplier, i.e. $$\begin {matrix}dF=\lambda\omega & [4]\end {matrix}$$

Answer 2

Comments to the post (v2):

1. Be aware that Goldstein later in eq. (2.20') discusses more systematically semi-holonomic constraints that are allowed to depend on time $t$. However on page 15 Goldstein assumes implicitly that there is no explicit $t$-dependence, i.e. there are only 2 generalized coordinates, say $x$ and $y$, without time $t$. The semi-holonomic constraint next kills 1 d.o.f.

2. Goldstein is a physics (rather than a math) textbook. The reader is expected to figure out the mathematimatical shortcomings by himself! E.g. there are implicit regularity condition (such as, e.g., differentiability). See also this and this related Phys.SE posts.

3. In particular, there can be global obstructions. What Goldstein is trying to convey is the fact that an inexact differential $$\omega~=~f(x,y)\mathrm{d}x+g(x,y)\mathrm{d}y\tag{A}$$ on a 2-dimensional manifold $M$ (for a point $p\in M$ with $\omega_p\neq 0$ non-vanishing) has an integrating factor $\lambda$ (in a sufficiently small open neighborhood $U\ni p$) that makes the one-form $\left. \lambda\omega\right|_U$ exact. The condition $$\left(f\frac{\partial }{\partial y}- g\frac{\partial }{\partial x}\right)\ln\lambda~=~ \frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\tag{B}$$ for $\ln\lambda$ is a linear first-order PDE in 2 variables, which one may show has local solutions. An exact differential corresponds to a holonomic constraint.