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Доведення 1Edit

Зв'язок між швидкостями відносно різних ІСВ.

Приймається, що вісь \ O_{x} співнапрямлена з вектором відносної швидкості ІСВ (на загальність не впливає, викладки спрощуються):

\ v'^{2} = v_{x}'^{2} + v_{y}'^{2} + v_{z}'^{2} = \frac{v^{2} + u^{2} - 2v_{x}u - (v_{y}^{2} + v_{z}^{2})\frac{u^{2}}{c^{2}}}{(1 - \frac{v_{x}u}{c^{2}})^{2}} \Rightarrow v'^{2} - c^{2} = \frac{v^{2} + u^{2} - 2v_{x}u - (v_{y}^{2} + v_{z}^{2})\frac{u^{2}}{c^{2}}}{(1 - \frac{v_{x}u}{c^{2}})^{2}} - c^{2} =

 = \frac{v^{2} + u^{2} - 2v_{x}u - (v_{y}^{2} + v_{z}^{2})\frac{u^{2}}{c^{2}} - c^{2} + 2v_{x}u - \frac{v_{x}u}{c^{2}}}{(1 - \frac{v_{x}u}{c^{2}})^{2}} = \frac{v^{2} + u^{2} - \frac{v^{2}u^{2}}{c^{2}} - c^{2} - \frac{v_{x}^{2}u^{2}}{c^{2}}}{(1 - \frac{v_{x}u}{c^{2}})^{2}} = \frac{c^{2}(\frac{u^{2}}{c^{2}} - 1) + v^{2}(1 - \frac{u^{2}}{c^{2}})}{(1 - \frac{v_{x}u}{c^{2}})^{2}} =

\ = \frac{(v^{2} - c^{2})(1 - \frac{u^{2}}{c^{2}})}{(1 - \frac{v_{x}u}{c^{2}})^{2}} \Rightarrow 1 - \frac{v'^{2}}{c^{2}} = \frac{(1 - \frac{v^{2}}{c^{2}})(1 - \frac{u^{2}}{c^{2}})}{(1 - \frac{v_{x}u}{c^{2}})^{2}} .

У загальному випадку

\ 1 - \frac{\mathbf {v}{'}^{2}}{c^{2}} = \frac{(1 - \frac{\mathbf {v}^{2}}{c^{2}})(1 - \frac{\mathbf {u}^{2}}{c^{2}})}{(1 - \frac{(\mathbf v \cdot \mathbf u)}{c^{2}})^{2}} .

Доведення 2Edit

Перетворення Лоренца для компонент 3-сили.

\ \frac{dE'}{\sqrt{1 - \frac{v'^{2}}{c^{2}}}dt'} = \frac{(\mathbf F' \cdot \mathbf v')}{\sqrt{1 - \frac{v'^{2}}{c^{2}}}} = \frac{dE - dp_{x}u}{\sqrt{1 - \frac{u^{2}}{c^{2}}}\sqrt{1 - \frac{v^{2}}{c^{2}}}dt} = \frac{(\mathbf F \cdot \mathbf v) - F_{x}u}{\sqrt{1 - \frac{v^{2}}{c^{2}}}\sqrt{1 - \frac{u^{2}}{c^{2}}}} \Rightarrow \frac{(\mathbf F' \cdot \mathbf v')}{\sqrt{1 - \frac{v'^{2}}{c^{2}}}} = \frac{(\mathbf F \cdot \mathbf v) - F_{x}u}{\sqrt{1 - \frac{u^{2}}{c^{2}}}\sqrt{1 - \frac{v^{2}}{c^{2}}}} ,

\ \frac{dp_{x}'}{\sqrt{1 - \frac{v'^{2}}{c^{2}}}dt'} = \frac{F_{x}'}{\sqrt{1 - \frac{v'^{2}}{c^{2}}}} = \frac{dp_{x} - \frac{dE u}{c^{2}}}{\sqrt{1 - \frac{u^{2}}{c^{2}}}\sqrt{1 - \frac{v^{2}}{c^{2}}}dt} = \frac{F_{x} - \frac{(\mathbf F \cdot \mathbf v) u}{c^{2}}}{\sqrt{1 - \frac{u^{2}}{c^{2}}}\sqrt{1 - \frac{v^{2}}{c^{2}}}} \Rightarrow \frac{F_{x}'}{\sqrt{1 - \frac{v'^{2}}{c^{2}}}} = \frac{F_{x} - \frac{(\mathbf F \cdot \mathbf v) u}{c^{2}}}{\sqrt{1 - \frac{u^{2}}{c^{2}}}\sqrt{1 - \frac{v^{2}}{c^{2}}}}.

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