# Equations of motion for a Weyl spinor in the context of SUSY

I'm learning supergravity from the textbook of Antoine Van Proeyen (this is from page 114).

Suppose I'm given a Lagrangian $$\mathcal{L} = - \partial^{\mu} \bar{Z} \partial_{\mu} Z - \bar{\chi} \gamma^{\mu} \partial_{\mu} P_{L} \chi + \bar{F} F + (F + \bar{F})(mZ + gZ^2) - \bar{\chi} P_{L} (m + 2 g Z) \chi.$$

Here $$\chi$$ is a Majorana spinor, $$P_L$$ is the projection operator, $$Z$$ is a complex scalar field, $$F$$ is a complex scalar auxiliary field.

I don't understand how one derives the equations of motion for the field $$\chi$$ and $$\bar{\chi}$$, when projection operators are involved. Do I just calculate $$\partial_{\sigma} \frac{ \partial \mathcal{L}}{\partial (\partial_{\sigma} P_L \chi)} = \frac{\partial \mathcal{L}}{\partial P_L \chi} ~?$$ Also, are $$P_L \chi$$ and $$P_R \chi$$ are considered independent fields? I would get $$\frac{\partial \mathcal{L}}{\partial P_L \chi} = - \bar{\chi} (m + 2g Z)$$ and $$\partial_{\sigma} \frac{ \partial \mathcal{L}}{\partial (\partial_{\sigma} P_L \chi)} = - \partial_{\sigma} \bar{\chi} \gamma^{\sigma}.$$

And if I want to compute the equations of motions for $$\bar{\chi}$$, how do I do that?