I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a complex-valued action, so am looking for E-L equations that correspond to (a) constant amplitude of the action, (b) constant phase of the action, or (c) both.
The papers I've found mostly avoid the issue by allowing complex valued field variables within the Lagrangian but ensuring that the Lagrangian itself is real-valued.
This paper might be relevant: Non-standard complex Lagrangian dynamics
Any advice will be welcome.