Электромагнитная гравитация

Last modified by Alexey Popov on 2019/06/20 13:47

$\rho \nabla \cdot \mathbf S = - \frac{\partial^2 \rho^2}{\partial t^2} + \frac{1}{2}\Delta \mathbf J \cdot \mathbf J + \nabla \cdot \left (\mathbf J \times \left ( \nabla \times \mathbf J \right )\right )$

$\mathbf J = \rho \mathbf v$

$\mathbf S = \rho \mathbf a$

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

$\rho = const$

$\nabla \cdot \mathbf a = \frac{1}{2}\Delta \mathbf v \cdot \mathbf v + \nabla \cdot \left ( \mathbf v \times \left ( \nabla \times \mathbf v \right ) \right )$

$\nabla \times \mathbf B = \mu_0 \mathbf J + \frac{1}{c^2}\frac{\partial \mathbf E}{\partial t}$

$\mathbf J \cdot (\nabla \times \mathbf B) = \mu_0 \mathbf J \cdot \mathbf J + \frac{1}{c^2}\mathbf J \cdot \frac{\partial \mathbf E}{\partial t}$

$\nabla \cdot (\mathbf B \times \mathbf J) + \mathbf B \cdot (\nabla \times \mathbf J) = \mu_0 \mathbf J \cdot \mathbf J + \frac{1}{c^2} \frac{\partial \mathbf J \cdot\mathbf E}{\partial t}- \frac{1}{c^2}\mathbf E \cdot \frac{\partial \mathbf J}{\partial t}$

$\nabla \cdot (\mathbf E \times \mathbf B)) = \frac{1}{2}\frac{\partial \mathbf E^2 + \mathbf B^2}{\partial t} + \mathbf J \cdot \mathbf E$

$\frac{1}{2}\nabla \mathbf E^2 = (\mathbf E \cdot \nabla)\mathbf E + \mathbf E \times (\nabla \times \mathbf E) = (\mathbf E \cdot \nabla)\mathbf E - \mathbf E \times \frac{\partial \mathbf B}{\partial t} = (\mathbf E \cdot \nabla)\mathbf E - \frac{\partial \mathbf E \times \mathbf B}{\partial t}+ \frac{\partial \mathbf E}{\partial t} \times \mathbf B = (\mathbf E \cdot \nabla)\mathbf E - \frac{\partial \mathbf E \times \mathbf B}{\partial t}+ (\nabla \times \mathbf B) \times \mathbf B - \mathbf J \times \mathbf E = \frac{1}{2}\nabla (\mathbf E \cdot \mathbf E) - \mathbf E \times (\nabla \times \mathbf E) - \frac{\partial \mathbf E \times \mathbf B}{\partial t}+ (\nabla \times \mathbf B) \times \mathbf B - \mathbf J \times \mathbf E$

$\frac{1}{2}\nabla \mathbf B^2 = (\mathbf B \cdot \nabla)\mathbf B + \mathbf B \times (\nabla \times \mathbf B) = (\mathbf B \cdot \nabla)\mathbf B + \mathbf B \times \frac{\partial \mathbf E}{\partial t} + \mathbf B \times \mathbf J = (\mathbf B \cdot \nabla)\mathbf B + \frac{\partial \mathbf B \times \mathbf E}{\partial t} - \frac{\partial \mathbf B}{\partial t}\times \mathbf E + \mathbf B \times \mathbf J = \frac{1}{2}\nabla (\mathbf B \cdot \mathbf B) - \mathbf B \times (\nabla \times \mathbf B) + \frac{\partial \mathbf B \times \mathbf E}{\partial t} + (\nabla \times \mathbf E )\times \mathbf E + \mathbf B \times \mathbf J$

$\nabla(\nabla \cdot (\mathbf E \times \mathbf B)) = \frac{\partial \mathbf B \times \mathbf J}{\partial t} + \nabla(\mathbf J \cdot \mathbf E)$

$\Delta(\mathbf B \times \mathbf J) + \nabla \times \nabla \times (\mathbf B \times \mathbf J)+ \nabla \square\frac{1}{2}(\mathbf B \cdot \mathbf B) = \mu_0 \nabla (\mathbf J \cdot \mathbf J) + \frac{1}{c^2} \frac{\partial^2 \mathbf B \times \mathbf J}{\partial t^2}- \frac{1}{2}\frac{1}{c^2}\nabla \square(\mathbf E \cdot \mathbf E)- \frac{1}{c^2}\nabla \rho^2 - \frac{\partial \nabla(\nabla \cdot (\mathbf E \times \mathbf B))}{\partial t }+\frac{1}{c^2}\nabla (\nabla \cdot (\rho \mathbf E))$

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Created by Alexey Popov on 2019/06/07 14:51