So, I am reading a paper on Quantum Brachistochrome and on the second page they say that they are doing a variation w.r.p. $<\phi|$, (which is a lagrange mulriplier) of the following action:
$$ S(\psi, H, \phi, \lambda) = \int dt [\frac{\sqrt{< \dot{\psi}|(1-P)| \dot{\psi}>}}{\Delta E} + (i<\dot{\phi}|\psi> + <\phi|H|\psi> + c.c.) + \lambda (Tr \tilde{H}^2/2 - \omega ^2)]\tag{1}$$
To get:
$$ i |\dot{\psi}> = H | \psi >.\tag{2}$$
I'm super confused what do they mean by that and how its used. The variational principle I have seen concerns the approximations for minimal energy states and this doesn't seem like it. Am I missing some mathematical concept here, or is the variational principle somehow used here? (extra question: what is the c.c. here?)
