The local minimality of flows is provided by differential equations, and the global minimality is established by integral variational equations.
This is the general method! But on particle internal level this simplistic approach no longer works as the metric locally is no longer linear (=> curvature of tangent can't vanish!). So you have to find a flux topology that globally satisfies the differential equation, what means for all points on a manifold.
Further particles have 0 degree of freedom what means F(X,t) = F(X,t+P) P the period time. So variation is reduced to harmonic functions that themself again follow F(X,t) = F(X,t+P') P' e.g. equal an internal oscillation of an axes.
If you once study the SOP charge function you can see this oscillation!
You can only solve this problem with topology!