Euler-Lagrange equations from a complex Lagrangian

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.

• There are instances when, even though the action is real, the complex saddle points (or trajectories) have to be taken into account. See e.g. this. – mavzolej Mar 16 at 22:41
• Crossposted from math.stackexchange.com/q/3150619/11127 @S. McGrew: Do you have any physical system in mind? Which? – Qmechanic Mar 17 at 3:17
• For what it's worth, see my answer at physics.stackexchange.com/q/438956 – akhmeteli Mar 17 at 3:29
• @akhmeteli, it's a useful comment. I'm not concerned at this point whether the result is physical or not; just what the result looks like. – S. McGrew Mar 17 at 3:41
• The action in FPI is real. – Qmechanic Mar 17 at 3:57

1. A stationary action principle for a complex action $$S_c=S_1+iS_2\in \mathbb{C}$$ is equivalent to 2 real stationary action principles for the real and imaginary part, $$S_1,S_2\in\mathbb{R}$$. In other words, the EL equations for $$S_c$$ are precisely the EL equations for $$S_1$$ and the EL equations for $$S_2$$. It may be possible to organize the EL equations for $$S_c$$ as complex equations, especially if the Lagrangian is holomorphic.
2. In the Feynman path integral $$Z$$, the action $$S$$ is real, at least in the Minkowskian formulation. However, when evaluating the semiclassical approximation via the method of steepest descent, one typically deforms the contour of integration into the complex plane, which may lead to complex contributions to $$Z$$. Stationary points in the complex plane may or may not have a straight-forward physical interpretation as solutions to the (analytic continued) EL equations.