We revisit the variational approach to nonlinear stability of planar flows, which was developed by V. I. Arnold around 1965. In particular, we study the coercivity properties of the quadratic form that describes the second variation of the energy at a radially symmetric vortex with strictly decreasing vorticity profile. We also show that this quadratic form can be used to obtain a new proof of nonlinear stability for the Lamb-Oseen vortices, which are self-similar solutions of the two-dimensional Navier-Stokes equations. This is all joint work with Vladimir Sverak.