Попытка вывести новые уравнения движения в гравитационном поле

Last modified by Alexey Popov on 2019/08/23 13:50

$\mathcal L = KE - PE - const$

$const = \rho_n c^2$

$\rho_n = m_n\left(\frac{m_n c}{\hbar}\right )^3$

$\mathcal L= \frac{1}{2}\rho \mathbf v^2 + \frac{1}{2}\rho_n \mathbf v^2 + \frac{1}{2}\frac{1}{c^2}w\mathbf v^2 - \rho c^2 - w - \rho_n c^2$

$\mathcal L = \frac{1}{2}\frac{1}{\rho_n}\rho \mathbf v^2 + \frac{1}{2} \mathbf v^2 + \frac{1}{2}\frac{1}{\rho_n c^2}w\mathbf v^2 - \frac{1}{\rho_n}\rho c^2 - \frac{1}{\rho_n}w -c^2$

$\nabla \mathcal L - \frac{\partial }{\partial t}\frac{\partial \mathcal L}{\partial \mathbf v} = 0$

$\nabla \mathcal L = \frac{1}{2}\frac{1}{\rho_n}\nabla (\rho \mathbf v^2) + \frac{1}{2}\nabla (\mathbf v^2) + \frac{1}{2}\frac{1}{\rho_n c^2}\nabla (w\mathbf v^2) - \frac{ c^2}{\rho_n}\nabla \rho - \frac{1}{\rho_n}\nabla w$

$\frac{\partial \mathcal L}{\partial \mathbf v} = \frac{1}{\rho_n}\rho \mathbf v + \frac{1}{\rho_n c^2}w\mathbf v + \mathbf v$

$\frac{\partial }{\partial t}\frac{\partial \mathcal L}{\partial \mathbf v} = \frac{1}{\rho_n}\frac{\partial \rho \mathbf v}{\partial t} + \frac{1}{\rho_n c^2}\frac{\partial w\mathbf v}{\partial t} + \frac{\partial \mathbf v}{\partial t}$

$\nabla \mathcal L - \frac{\partial }{\partial t}\frac{\partial \mathcal L}{\partial \mathbf v} =$

$= \frac{1}{2}\nabla (\mathbf v^2) + \frac{1}{2}\frac{1}{\rho_n}\nabla (\rho \mathbf v^2) + \frac{1}{2}\frac{1}{\rho_n c^2}\nabla (w\mathbf v^2) - \frac{ c^2}{\rho_n}\nabla \rho - \frac{1}{\rho_n}\nabla w - \frac{1}{\rho_n}\frac{\partial \rho \mathbf v}{\partial t} - \frac{1}{\rho_n c^2}\frac{\partial w\mathbf v}{\partial t} - \frac{\partial \mathbf v}{\partial t}= 0$

$\mathbf a - 2\frac{\partial \mathbf v}{\partial t} + \mathbf v \times (\nabla \times \mathbf v) = \frac{ c^2}{\rho_n}\nabla \rho + \frac{1}{\rho_n}\nabla w +\frac{1}{\rho_n}\frac{\partial \rho \mathbf v}{\partial t} + \frac{1}{\rho_n c^2}\frac{\partial w\mathbf v}{\partial t} - \frac{1}{2}\frac{1}{\rho_n}\nabla (\rho \mathbf v^2)-\frac{1}{2}\frac{1}{\rho_n c^2}\nabla (w\mathbf v^2)$