An analysis of stable points
Math and physics are two subjects so entangled with each other, that its hard to tell which topic relates to one subject or the other. Take today’s topic-stable points. Of course, this applies to math, where stable points can be defined as points on a function where the slope is zero, but the second derivative is positive. In the context of physics, it simply refers to points where the force on a particle is zero, so the velocity doesn’t change. In this article, we will explore stable points in 2-dimensional force fields.
Let’s get started. The first thing we want to do is to define an arbitrary conservative force field
A quick recap on conservative forces. These are forces with a potential associated with them. In a conservative field, the work done on a particle in going from A to B depends only on the endpoints A and B and is thus path independent. One can also say that these forces conserve the total energy of the system.
Now let’s do some math. The first thing to do is to write out F in terms of its individual components. This field is represented in Cartesian coordinates. Then we’ll assume that the origin of this system is a stable point, and finally make a linear approximation of the individual components of force around that point.
As you can see, if we assume that the points x and y in a vector field F are extremely close to the stable origin, we can represent F as a matrix vector product. Now the matrix K has 2 important properties.
Now what do we do? Do we solve equation 1? No. That would be stupid. And tedious. Instead, we can simply analyze the matrix vector product to understand how a particle moves near a stable point. The first thing we realize is that because of the properties of K, a particle moving near a stable point would spiral inwards. This is because both partial derivatives are negative, which implies that the force vectors cause an inward acceleration. Here’s a representation.
As you can see, in this vector field, the arrows tend to point inward, and the force on a particle will cause it to spiral towards the origin.
Now we get to something beautiful about equation 1. It looks a lot like the equation for a 1-dimensional simple harmonic oscillator but extended to 2 dimensions. Indeed, you can see in the figure above that there are points where the force arrows point directly inwards to the origin, which cause the particle to execute simple harmonic motion, and not spiral down. There are two lines in the figure along which a particle can execute SHM if initially placed there. Unsurprisingly, these coincide with the eigenvectors of the matrix K, and their respective eigenvalues. In fact, we find that F = kr, and as you can see, where k is the eigenvalue corresponding to the eigenvector r. as you can see, k must be negative for SHM to occur. For the graph above, the eigenvectors lie on the lines y=x and y=-x, and the respective eigenvalues are -3 and -1. So, if we want to generalize this observation for any matrix K, do we solve for individual eigenvalues? No! we use the important fact that the product of eigenvalues of any matrix is the determinant of that matrix. And we know that, near a stable point, the eigenvalues of a matrix are both negative, so the product, the determinant, must be positive! Thus, we get
We can rewrite the inequality (2) in terms of the potential energy U, so that we get
Now this is actually a theorem in mathematics, known as the second derivative test! It says that if this inequality is true, then the point (0,0), or any point where this inequality is being tested at, could either be a stable point (which is what we’re interested in for this article) or an unstable point.
So, we have just proved a mathematical statement using physics! But of course, that wasn’t the point of this article. The point was to analyze how a particle moves in 2 dimensions around a stable point in a potential, a point where no force acts on an object. We found that if a particle is placed at a point very close to the stable point, it will tend to spiral towards the origin. more importantly, we also found that if a particle is placed along an eigenvector, then it will execute simple harmonic motion along that line. That’s it for today!