I was tasked with finding the path a particle takes through this potential function.
$$U(x)=x^2+xy+y^2$$
I then took the gradient, and this produced a pair of differential equations.
$$\frac{d^2x}{dt^2}=\frac{1}{m}(-2x-y)$$
$$\frac{d^2y}{dt^2}=\frac{1}{m}(-2y-x)$$
I have yet to encounter a problem of this form. It looks simple, but it has proven to be very tricky to solve. I tried a couple substitution techniques that initially seemed promising but inevitably failed. I even used Laplace transforms, but that got very messy and I eventually gave up.
Any thoughts? Thank you!
Set $u=x+y$, $v=x-y$ then \begin{align} \frac{d^2u}{dt^2}&=-\frac{3u}m\\ \frac{d^2v}{dt^2}&=-\frac{v}m \end{align} are easily solved scalar equation. In general you have to decouple the system $m\ddot q=-Aq$ via the eigen-decomposition of the system matrix $A$.