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\frac{d\vec {p}}{dt}=\vec {F}.

N :

\sum_{n=1}^{N} \frac{\vec{dp_n}}{dt}=\sum_{n=1}^{N}\sum_{m=1}^{N}\ \vec{F}_{n,m}, \qquad m\ne n, \qquad\qquad (1)

, n- m-. , \vec {F}_{a,b} \vec {F}_{b,a} , \vec{F}_{a,b} = -\vec{F}_{b,a}. (1) , :

\sum_{n=1}^{N} \frac{d\vec{p}_n}{dt}=0

\!\qquad \frac {d}{dt}\sum_{n=1}^{N}\vec{p}_n=0.

, , , :

\sum_{n=1}^{N}\vec{p}_n=\overrightarrow {\mathrm{const}} \qquad\! ( ).

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\mathcal L \equiv \mathcal L(q_i, \dot q_i, t), q_i\,, \dot q_i t. q , \dot q_i \equiv \frac{\partial q_i}{\partial t}. , q_i=\vec r_a, \ \dot q_i = \vec v_a \!a- . , - , : \vec r_a \to \vec r_a + \vec{\xi}, \vec{\xi} \equiv \overrightarrow {\mathrm{const}}. :

\delta \mathcal L = \sum_{a}\frac{\partial\mathcal L}{\partial \vec r_a} \delta  \vec r_a = \vec{\xi}\ \sum_{a} \frac{\partial \mathcal L}{\partial \vec r_a},

. , : \delta \mathcal L =0. , \vec \xi  , :

\sum_{a} \frac{\partial \mathcal L}{\partial \vec r_a}=0.

\frac{d}{dt}\frac{\partial \mathcal L}{\partial \dot q_i}-\frac{\partial \mathcal L}{\partial q_i}=0:

\sum_{a} \frac{\partial \mathcal L}{\partial \vec r_a} = \sum_{a}\frac{d}{dt}\frac{\partial \mathcal L}{\partial \vec v_a} =  \frac{d}{dt}\sum_{a}\frac{\partial \mathcal L}{\partial \vec v_a} = 0 .

, , ,  . :

\vec P = \sum_{a}\frac{\partial \mathcal L}{\partial \vec v_a} =  \overrightarrow {\mathrm{const}}. .

, : \mathcal L = \frac{mv^2}{2}, , :

\vec P = \sum_a m_a \vec v_a = \overrightarrow {\mathrm{const}}.

: \mathcal L = -mc^2\sqrt{1-\frac{v^2}{c^2}},

\vec P = \sum_a \frac{m_a \vec v_a}{\sqrt{1-\frac{v^2}{c^2}}} = \overrightarrow {\mathrm{const}}.

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T^\mu_{\nu;\mu}=0,

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