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I don't understand the meaning of "up to total derivatives".

It was used during a lecture on superfluid. It says as follows:

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Lagrangian for complex scalar field ##\phi## is

$$

\mathcal{L}=\frac12 (\partial_\mu \phi)^* \partial^\mu \phi - \frac12 m^2 |\phi|^2 -\lambda |\phi|^4.

$$

Take non-relativistic limit:

$$

\phi(x)=\dfrac{1}{\sqrt{2m}}e^{-imt}\varphi(t,x).

$$

Then, lagrangian for non-relativistic complex scalar field ##\mathcal{L}_{NR}## can be written as follows:

$$

\mathcal{L}_{NR}=\partial_t\phi^* \partial_t \phi - \nabla \phi^* \cdot \nabla \phi - m^2|\phi|^2 -\lambda|\phi|^4\\

=\dfrac{1}{2m}(im\varphi^*+\dot{\varphi}^*)(-im\varphi+\dot{\varphi})-\dfrac{1}{2m}\nabla\varphi^*\cdot \nabla \varphi -\dfrac{m}{2}|\varphi|^2-\dfrac{\lambda}{4m^2}|\varphi|^4.

$$

In non-relativistic limit,

$$

\partial_t\sim \nabla^2,

$$

therefore, we only consider first order of ##\partial_t##.

$$

\mathcal{L}_{NR}=\dfrac{1}{2m}[im(-im)\varphi^*\varphi+im\varphi^*\dot{\varphi}-im\dot{\varphi}^*\varphi]-\dfrac{1}{2m}\nabla\varphi^*\cdot \nabla \varphi -\dfrac{m}{2}|\varphi|^2-\dfrac{\lambda}{4m^2}|\varphi|^4\\

\simeq \dfrac{i}{2}(\varphi^*\dot{\varphi}-\dot{\varphi}^*\varphi)-\dfrac{1}{2m}\nabla\varphi^*\cdot \nabla \varphi -\dfrac{\lambda}{4m^2}|\varphi|^4\\

$$

Now,up to total derivatives,

$$

\mathcal{L}_{NR}\simeq \dfrac{i}{2}(\varphi^*\dot{\varphi}-\dot{\varphi}^*\varphi)-\dfrac{1}{2m}\nabla\varphi^*\cdot \nabla \varphi -\dfrac{\lambda}{4m^2}|\varphi|^4\\

=\varphi^*\left( i\partial_t+\dfrac{\nabla^2}{2m} \right) \varphi -\dfrac{\lambda}{4m^2}|\varphi|^4.

$$

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I don't understand the last part of this. Drop total derivatives?

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# I What does it mean: "up to total derivatives"

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