Equilibrium point

Question

How would I find the equilibrium points for a function with two variables? V(x,y). I know for one variable it is V' set equal to zero, solve for x. How would I do this with two variables? I took the partial derivative then subbed one variable into the other equation but ended up with zero equals zero since the variables canceled out. What does this mean? Is there a different way to approach this?

Answer 1

I'm assuming you mean that $V$ is a scalar function of two variables meaning $V:\mathbb{R^2}\rightarrow\mathbb{R} \; .$ In Cartesian coordinates, it is as simple as setting the partial derivatives equal to zero just as you said. We typically write this in terms of the gradient of the function $V$ as

$$\nabla V = \begin{bmatrix} \frac{\partial V}{\partial x} & \frac{\partial V}{\partial y} \end{bmatrix} = \begin{bmatrix} 0 & 0 \end{bmatrix} \; .$$

Now that we are in two dimensions, we solve this system of two equations with two variables for $x$ and $y.$ This idea generalizes to any function $f:\mathbb{R^n}\rightarrow\mathbb{R}\;\;$ as we just have to solve the system $\nabla f = \vec{0} \; .$