User:Tohline/SR/PoissonOrigin
Origin of the Poisson Equation
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[EFE, §8, p. 15] A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations. The virial method … is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account. The virial equations of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations. In this context, Chandrasekhar's focus is on two of the four principal governing equations that serve as the foundation of our entire H_Book, namely, the
The
Poisson Equation
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<math>\nabla^2 \Phi = 4\pi G \rho</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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