Verifying equations of motion of Lagrangian

Question

Given the following Lagrangian of a scalar field $$ L = +i \psi^*\partial_t\psi - \frac{1}{2m} \nabla \psi^* \cdot \nabla \psi, $$ I want to compute the equation(s) of motion. First by varying wrt $\psi^*$ and then wrt $\psi$.

First I say $$\frac{\partial L}{\partial \psi^{*}} = + i\partial_t\psi$$ and $$\partial_{\mu} (\frac{\partial L}{\partial (\partial_{\mu} \psi^*)}) = \partial_{t} (\frac{\partial L}{\partial (\partial_{t} \psi^*)}) + \nabla(\frac{\partial L}{\partial (\nabla \psi^*)}) = 0 - \frac{1}{2m}\nabla^2 \psi. $$ By Euler -Lagrange I get

$$ i \frac{\partial \psi}{\partial t} = - \frac{1}{2m} \nabla^2 \psi. $$

Now if I do the same but vary to $\psi$ instead of $\psi^*$, I get $$ i \psi^* = +\frac{1}{2m} \nabla^2 \psi^{*}, $$ as equation of motion. The only main difference is that $$\frac{\partial L}{\partial \psi} = 0.$$

Could someone verify if the answers are correct please?

The last equation is not a typo.But something feels weird, therefore I'd appreciate a verification. '

EDIT: Solved, my mistake was that I only considered $(\frac{\partial L}{\partial (\partial_{t} \psi)}) = i \psi^*$, instead of $\partial_{t} (\frac{\partial L}{\partial (\partial_{t} \psi)}) = i \frac{\partial \psi^{*}}{\partial t}$.

you should show a little more effort in your exercise or at least show us your thought process behind these equations — Struggling_Student, Nov 14 '19 at 13:41
HINT: Your variation with respect to $\psi$ is not in fact correct. Take a careful look at where that $i \psi^*$ term came from. — Michael Seifert, Nov 14 '19 at 14:24
@MichaelSeifert the $i\psi^*$ came from deriving wrt $\partial_t \psi$. But I really do not see what I'm doing wrong. — user3397129, Nov 14 '19 at 14:26
The term $\partial L/\partial (\partial_{t} \psi)$ doesn't appear in the equations of motion for $\psi$, at least not directly. — Michael Seifert, Nov 14 '19 at 14:30
ahhhhhh, very stupid. It should have been $\partial_t (\partial L / \partial(\partial_t \psi))$ — user3397129, Nov 14 '19 at 14:34
This Lagrangian $$\mathcal{L}\left(\psi,\boldsymbol{\nabla}\psi,\overset{::\centerdot}{\psi},\psi^{\boldsymbol{}},\boldsymbol{\nabla}\psi^{\boldsymbol{}}, \overset{!\centerdot}{\psi^{\boldsymbol{}}}\right)=\mathrm i\hbar\psi^{\boldsymbol{}}\overset{::\centerdot}{\psi}!\boldsymbol{-}\dfrac{\hbar^2}{2m}\boldsymbol{\nabla}\psi!\boldsymbol{\cdot}!\boldsymbol{\nabla}\psi^{\boldsymbol{}} !\boldsymbol{-}!V\psi\psi^{\boldsymbol{}} $$.... — Frobenius, Nov 14 '19 at 17:24
...is one for the Schroedinger equation $$i\hbar \overset{::\centerdot}{\psi}:\boldsymbol{+}\dfrac{\hbar^2}{2m}\nabla^2\psi::\boldsymbol{-}V\left(\mathbf{x},t\right)\psi, \boldsymbol{=} 0$$ See therein Why treat complex scalar field and its complex conjugate as two different fields?. — Frobenius, Nov 14 '19 at 17:25

Verifying equations of motion of Lagrangian

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