Вывод всеобщих уравнений движения

Last modified by Alexey Popov on 2019/07/24 19:58

$\mathbf E = - \nabla \varphi -\frac{\partial \mathbf A}{\partial t} + \nabla \times \mathbf K$

$-\mu_0 \mathbf M = - \nabla \times \mathbf E -\frac{\partial \nabla \times\mathbf A}{\partial t} + \nabla \times(\nabla \times \mathbf K)$

$\mathbf a + \frac{D\mathbf v}{Dt} + \nabla \times (\mathbf r \times \mathbf a) = 0$

$\mathbf a = \frac{\partial \mathbf v}{\partial t} + \frac{1}{2}\nabla (\mathbf v \cdot \mathbf v) - \mathbf v \times (\nabla \times \mathbf v) + \nabla \times (\mathbf r \times \mathbf a)$

$\boldsymbol \omega = -\frac{1}{c^2}\frac{\partial (\mathbf r \times \mathbf a)}{\partial t} + \nabla \times \mathbf v - \nabla (\mathbf r \cdot \boldsymbol \omega)$

$\nabla \times \mathbf a = \frac{\partial \nabla \times \mathbf v}{\partial t} - \nabla \times (\mathbf v \times (\nabla \times \mathbf v)) + \nabla \times (\nabla \times (\mathbf r \times \mathbf a))$

$\nabla \times \boldsymbol \omega = -\frac{1}{c^2}\frac{\partial \nabla \times (\mathbf r \times \mathbf a)}{\partial t} + \nabla \times (\nabla \times \mathbf v)$

$\frac{\partial\mathbf a}{\partial t} = \frac{\partial^2 \mathbf v}{\partial t^2} + \frac{1}{2}\frac{\partial\nabla (\mathbf v \cdot \mathbf v)}{\partial t} - \frac{\partial(\mathbf v \times (\nabla \times \mathbf v))}{\partial t} + \nabla \times \frac{\partial(\mathbf r \times \mathbf a)}{\partial t}$

$\frac{\partial\boldsymbol \omega}{\partial t} = -\frac{1}{c^2}\frac{\partial^2 (\mathbf r \times \mathbf a)}{\partial t^2} + \nabla \times \frac{\partial\mathbf v}{\partial t} - \nabla \frac{\partial(\mathbf r \cdot \boldsymbol \omega)}{\partial t}$

$\nabla \times \mathbf a = \frac{\partial\boldsymbol \omega}{\partial t} + \frac{1}{c^2}\frac{\partial^2 (\mathbf r \times \mathbf a)}{\partial t^2} + \nabla \frac{\partial(\mathbf r \cdot \boldsymbol \omega)}{\partial t} - \nabla \times (\mathbf v \times \boldsymbol \omega) + \nabla \times (\nabla \times (\mathbf r \times \mathbf a))$

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Created by Alexey Popov on 2019/07/24 19:58