Understanding the Math Behind Homework Equations

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Homework Statement



This is not really a problem but I was going over my lecture notes and I see \mathscr{H}=\frac{1}{2}\left(\pi^{2} + \vec{\nabla}\phi \cdot \vec{\nabla}\phi + m^{2}\phi^{2}\right) and \frac{\partial\mathscr{H}}{\partial\phi} = -\nabla^{2}\phi + m^{2}\phi

Homework Equations


The Attempt at a Solution



I would think that \frac{\partial\mathscr{H}}{\partial\phi} = \nabla^{2}\phi + m^{2}\phi. But I don't know where the minus sign is coming from.
 
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I just found a vector identity \vec{\nabla}\phi\cdot\vec{\nabla}\phi = \vec{\nabla}\cdot\left(\phi\vec{\nabla}\phi\right) - \phi\vec{\nabla}^{2}\phi. I now see how the result follow.

EDIT: I'm confused again. will the phi derivative of the first term vanish?
 
If by "the first term" you mean the \pi^2 term, then yes, the \phi derivative will kill that term; reason being \phi doesn't appear in that term, only its time derivative.

Edit: Sorry, I understand what you meant by "first term" now. The first term vanishes at infinity so you can ignore it.
 
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The 3-divergence in the Hamiltonian density (i.e. the first term in the RHS of the equality in post #2) can be discarded, since by integration of full space gives 0, because scalar fields are normally taken from the Schwartz space of test functions.
 
dextercioby said:
The 3-divergence in the Hamiltonian density (i.e. the first term in the RHS of the equality in post #2) can be discarded, since by integration of full space gives 0, because scalar fields are normally taken from the Schwartz space of test functions.

Sonny Liston said:
If by "the first term" you mean the \pi^2 term, then yes, the \phi derivative will kill that term; reason being \phi doesn't appear in that term, only its time derivative.

Edit: Sorry, I understand what you meant by "first term" now. The first term vanishes at infinity so you can ignore it.
That makes sense. I also realized that it can be integrated by parts using the same reasoning of vanishing at the boundaries. Thanks!
 
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