I'm still pretty new to this differential business and I have a couple of question concerning this problem I've come across. We're given a differential form $\alpha=p_1 dx_1+p_2 dx_2-H(p_1,p_2)dt$ defined on $\mathbb{R}^5=\{(p_1,x_1,p_2,x_2,t)\}$ with $H(p_1,p_2)$ a globally defined smooth function dependent only on $p_1$ and $p_2$. First, I'd like to find $d\alpha$. This is fairly straightforward, but I'm having some notational difficulties. I should have $ d\alpha=dp_1\wedge dx_1+dp_2\wedge dx_2 - [\text{something}]\wedge dt $ where the "something'' above should be the total derivative of $H(p_1,p_2)$. Is there a good notation for expressing the total derivative of a general function in two variables, namely $H(p_1,p_2)$?
If $\alpha$ is the differential of a globally defined smooth function $f$, then I would have that $d\alpha=0$. Since this isn't the case, $\alpha$ is not the differential of a globally defined smooth function.
Finally, I'd like to consider the restriction of $d\alpha$ to the $3$-dimensional plane $\{p_1=4,p_2=5\}$. Then $ d\alpha = 0 $ since $\alpha = 4dx_1+5dx_2-H(4,5)dt$. Because $d\alpha=0$, it must be the differential of a smooth function defined on this plane. In order to find this function explicitly, I think I need to integrate $\alpha:$ $ \int 4dx_1+\int 5dx_2 - \int H(4,5)dt = 4x_1+5x_2-H(4,5)t $ Is this correct? Specifically, is the restriction of $d\alpha$ to the plane provided above the differential of $4x_1+5x_2-H(4,5)t$?