This article summarizes equations in the theory of gravitation.
A common misconception occurs between centre of mass and centre of gravity.
They are defined in similar ways but are not exactly the same quantity.
Centre of mass is the mathematical description of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts.
They are equal if and only if the external gravitational field is uniform.
Centre of gravity for a set of discrete masses:
r
c o g
=
1
|
(
i
)
i
)
1
(
{\displaystyle {\begin{aligned}\mathbf {r} _{\mathrm {cog} }&={\frac {1}{M\left|\mathbf {g} \left(\mathbf {r} _{i}\right)\right|}}\sum _{i}\mathbf {m} _{i}\left|\mathbf {g} \left(\mathbf {r} _{i}\right)\right|\\&={\frac {1}{M\left|\mathbf {g} \left(\mathbf {r} _{\mathrm {cog} }\right)\right|}}\sum _{i}\mathbf {r} _{i}m_{i}\left|\mathbf {g} \left(\mathbf {r} _{i}\right)\right|\end{aligned}}\,\!}
Centre of gravity for a continuum of mass:
ρ
{\displaystyle {\begin{aligned}\mathbf {r} _{\mathrm {cog} }&={\frac {1}{M\left|\mathbf {g} \left(\mathbf {r} _{\mathrm {cog} }\right)\right|}}\int \left|\mathbf {g} \left(\mathbf {r} \right)\right|\mathrm {d} \mathbf {m} \\&={\frac {1}{M\left|\mathbf {g} \left(\mathbf {r} _{\mathrm {cog} }\right)\right|}}\int \mathbf {r} \left|\mathbf {g} \left(\mathbf {r} \right)\right|\mathrm {d} ^{n}m\\&={\frac {1}{M\left|\mathbf {g} \left(\mathbf {r} _{\mathrm {cog} }\right)\right|}}\int \mathbf {r} \rho _{n}\left|\mathbf {g} \left(\mathbf {r} \right)\right|\mathrm {d} ^{n}x\end{aligned}}\,\!}
In the weak-field and slow motion limit of general relativity, the phenomenon of gravitoelectromagnetism (in short "GEM") occurs, creating a parallel between gravitation and electromagnetism.
The gravitational field is the analogue of the electric field, while the gravitomagnetic field, which results from circulations of masses due to their angular momentum, is the analogue of the magnetic field.
It can be shown that a uniform spherically symmetric mass distribution generates an equivalent gravitational field to a point mass, so all formulae for point masses apply to bodies which can be modelled in this way.
General classical equations.