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Classical mechanics discusses the behaviour of macroscopic bodies under the influence of forces (without necessarily specifying the origin of these forces). If it's possible, USE MORE SPECIFIC TAGS like [newtonian-mechanics], [lagrangian-formalism], and [hamiltonian-formalism].
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How to check $\mathbf v' · \mathbf V$ and $\mathbf v'^2$ are time derivatives of some other ...
From Landau, Lifshitz Mechanics p.127
$$
\renewcommand{\vec}[1]{\mathbf{#1}}
L'=\frac{1}{2}m(\vec{v}'^2+\vec{v'}\cdot\vec{V}+\vec{V}^2)-U
$$
He states that "$\vec{V}^2(t)$ can be written as the tota …