Can Calculus of Variations Prove the Path of Least Resistance for Current Flow?

[sodaboy7]

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)?

 

[K^2]

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.

 

[sodaboy7]

It prefers path of least resistance. Or it maximum current follows the path of least resistance upon division at a point.