User:Tohline/SR/PoissonOrigin
Origin of the Poisson Equation
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The
Poisson Equation
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<math>\nabla^2 \Phi = 4\pi G \rho</math> |
is derived straightforwardly from Isaac Newton's inverse-square law of gravitation. In presenting this derivation we follow closely the presentation found in §2.1 of [BT87]
According to Isaac Newton's inverse-square law of gravitation, the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho({\vec{x'}}) d^3x'</math>, as follows:
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<math>~\vec{a}(\vec{x})</math> |
<math>~=</math> |
<math>~ \int \biggl[\frac{(\vec{x'} - \vec{x})}{|\vec{x'} - \vec{x}|}\biggr] G\rho(\vec{x'}) d^3 x' </math> |
Drawn from Other Wiki Pages
It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of <math>~\vec{v}</math> to represent the inertial velocity field. More importantly, he adopts a different variable name and a different sign convention to represent the gravitational potential, specifically,
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<math>~ - \Phi = \mathfrak{B} </math> |
<math>~=</math> |
<math>~ G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math> |
Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [EFE] into this H_Book.
… which expresses simply the conservation of the angular momentum of the system. The symmetric part of the tensor expression gives what is generally referred to as (see [EFE] for details) the,
Tensor Virial Equation
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<math>~\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math> |
<math>~=</math> |
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math> |
See Also
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