Why is Laplacian ubiquitous?

What I am asking here is a moral question. Mathematically moral, don't bother physics.

I mean, Euler's number is ubiquitous because, among all the exponentials, it alone is its own derivative with all the consequences we know.

I know that the Laplacian contains information about curvature, mean of function etc. so it is what you want in mean curvature flow, or geometrical like fields with the same flavor. It is not my field but it seems to me that it i pops out even in combinatoric arguments in Lie algebras. One can think billions of interesting facts, you got it.

So it seems that there is something fundamental that I can't reach and nobody has ever pointed out to me.

What are your thoughts?

The (negative) Laplacian is $-\text{div} \nabla$, and the integration by parts formula tells us that $-\text{div}$ is the adjoint of the gradient operator $\nabla$ (in settings where the boundary term vanishes). So the Laplacian has the familiar pattern $A^T A$, which recurs throughout linear algebra and math. This suggests that the (negative) Laplacian is a symmetric positive definite operator, so we would hope that there exists a basis of eigenfunctions for the Laplacian. This motivates the topic of eigenvalues of the Laplacian.

Gilbert Strang emphasizes the ubiquity of $A^T A$ in his linear algebra and applied math books.