Path of current functional

Current follows the path of least resistance or shortest path. I just want to prove this or rather reproduce it using calculus of variations. I just want to show it in a fancy way. I want help to form the FUNCTIONAL for it.
Useful equations:
I=dq/dt=nqvA
R=rho*l/A
Where v is drift velocity

Any suggestion (may be using different equations and parameters)?
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 Recognitions: Science Advisor The current does not actually take the path of least resistance. It takes all available paths. In general: $$J = \sigma E$$ Where J is current density, E is electric field, and σ is the electrical conductivity.
 It prefers path of least resistance. Or it maximum current follows the path of least resistance upon division at a point.