unknown manipulation to deal with variation and obtain energy balance equation?

Question

I'm studying a paper relevant to using d'Alembert's principle to describe the motion of fluid. The authors shows an interesting manipulation to obtain energy balance equation, which makes me confused. (see [1]) They start with $$\delta L_{sys}+\delta W-\int_{CV}\frac{\partial}{\partial t}(\rho\textbf{U})\cdot\delta\textbf{r}dv+\int_{CS}\rho\textbf{U}\cdot\delta\textbf{r}(\textbf{U}-\textbf{v})\cdot\textbf{n}ds=0$$ and assume configuration is not prescribed at $t_1$ and $t_2$, so $\delta\textbf{r}≠0.$ ($L_{sys}$ is the Lagrangian of the system) Instead, they use an operator given by $$\delta()=dt\frac{d()}{dt}$$ so the equation becomes $$dt\bigg[ \frac{d}{dt}(L_{sys})+\frac{d}{dt}(W)-\int_{CV}\frac{\partial}{\partial t}(\rho\textbf{U})\cdot\textbf{U}dv+\int_{CS}\rho\textbf{U}\cdot\textbf{U}(\textbf{U}-\textbf{v})\cdot\textbf{n}ds\bigg]=0$$ where $\textbf{U}=d\textbf{r}/dt$. Does anyone know this manipulation? I didn't find this in textbook but I want to understand its physical meaning. Hope someone can explain it, thanks so mush.
[1]https://link.springer.com/book/10.1007/978-3-030-26133-7 (chapter 4.3.3)