First Integral of 2 by 2 system

Question

I have to show that in a linear 2 by 2 system if its a node, sink or source then it does not have first integrals. I tried considering a function $f(x,y)$, if its constant at a solution then $f(\phi)=c$ thus its directional derivative is $0$ along the curve, but i reached nowhere, does anyone have an idea on how to show it?

Answer 1

I don't think it matters if the system is linear or not. Suppose the system is defined by the vector field

$\vec X(x, y) = \begin{pmatrix} X_x(x, y) \\ X_y(x, y) \end{pmatrix}; \tag 1$

thus it may be written

$\dot x = X_x(x, y), \tag 2$

$\dot y = X_y(x, y); \tag 3$

then if $\phi_t$ is the flow of the system, and the point $p$ is a sink, we have

$\displaystyle \lim_{t \to \infty} \phi_t(x) = p \tag 4$

for all $x \in U$, $U$ being some neighborhood of $p$; now if $f(x, y)$ is a first integral, it satisfies

$\vec X(f) = \nabla f(x, y) \cdot \vec X(x, y) = 0, \tag 5$

which implies $f(x, y)$ is constant along the integral curves of $\vec X$, since

$\dfrac{df(\phi_t(x_0, y_0))}{dt} = \dfrac{\partial f(\phi_t(x_0, y_0))}{\partial x} \dot x + \dfrac{\partial f(\phi_t(x_0, y_0))}{\partial y} \dot y$ $= \dfrac{\partial f(\phi_t(x_0, y_0))}{\partial x} X_x(x, y) + \dfrac{\partial f(\phi_t(x_0, y_0))}{\partial y} X_y(x, y)$ $= \nabla f(\phi_t(x_0, y_0)) \cdot \vec X(\phi_t(x_0, y_0)) = 0, \tag 6$

which may be interpreted as saying that $f(\phi_t(x_0, y_0))$ is constant with respect to $t$; next, by virtue of (4) we see that

$f(x_0, y_0) = f(\phi_t(x_0, y_0)) = \displaystyle \lim_{t \to \infty} f(\phi_t(x_0, y_0)) = f(p) \tag 7$

for any $(x_0, y_0) \in U$, which shows that $f(x,y)$ is constant, at least on $U$; therefore we cannot have a non-trivial first integral of motion in the region of a sink of $\vec X(x, y)$; in the event that $p$ is a source, essentially the same argument applies but with $t \to -\infty$; we have thus shown that a system cannot be possessed of a non-trivial first integral in the presence of a node.

In the event $\vec X$ has a center or a saddle, however, the situation is different. For example, the integral curves of the system

$\dot x = y, \; \dot y = -x \tag 8$ are seen to be the circles $x^2 + y^2 = r^2$, since

$\dot{r^2} = 2x\dot x + 2y \dot y = 2(xy - yx) = 0, \tag 9$

and here $r^2$ (or indeed simply $r$) is a constant of the motion; here $p = (0, 0)$ is a center; and for the system

$\dot x = y, \; \dot y = x \tag{10}$

the quantity

$h = x^2 - y^2 \tag{11}$

is constant on the trajectories:

$\dot h = 2(x \dot x - y \dot y) = 2(xy - yx) = 0, \tag{12}$

which are hyperbolas; here $(0, 0)$ is a saddle, as the reader may easily confirm, since (10) is a linear differential equation with matrix

$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \tag{13}$

the eigenvalues of which are $\pm 1$.