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When I learned about saddle points I had this expression 'rt-(s^2)', where r=Dx, t=Dy, s=Dxy=Dyx. And the intuition behind why it is so was also clear.

In an electric/magnetic field, in Earnshaw' theorem stable equilibrium can't exist - in some direction it has to be unstable (that's why saddle points I guess). But how does $\nabla^2U = 0$ indicate so?

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$C^2$-solutions to Laplace's equation are by definition harmonic functions, which in turn satisfy a strong maximum principle: The maximum/minimum in the interior is only possible for constant functions. In other words: any stationary/critical point for a non-constant function is a saddle point.

See also e.g. this & this related Phys.SE posts.

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