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I've been looking without success for references describing a generalization 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 tor (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.

Any advice will be welcome.

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