Newtonian Gravitation |

**Theory** > s.a. equivalence principle; models of spacetime structure.

* __Idea__: Bodies interact
through a gravitational force *F*_{g} = *G* *m*_{g} *m'*_{g}/*r*^{2}
acting at a distance, and accelerations are proportional
to forces with proportionality constant equal to the inertial mass, *a* = *F*/*m*_{i}.

* __Masses__: The force is
proportional to the gravitational masses, but as Newton knew *m*_{g} ∝ *m*_{i},
which makes gravitation similar to "inertial" forces,
in that the acceleration is independent of each body's individual characteristics.

* __Potential__: If we define Φ(**r**)
= *U*(*r*)/*r* for a test body, then the potential Φ satisfies the Poisson equation ∇^{2}Φ =
4π*G**ρ*,

Φ(**r**)
= –*G *∫ d*M * |**r**–**r**'|^{–1} = –*G**M*/*r* – *G* **D** · **r**/*r*^{3} – \(1\over2\)*G* *Q*_{ij} *r*^{i }*r*^{j} / *r*^{5} +
... ,

where **D** = ∫ d*M* **r**'
is the dipole moment of the mass distribution (always zero with respect to
the center of mass), and *Q*_{ij}
= ∫ d*M* [3* r _{i}'r_{j}'*
–

*

*U*_{self}
= \(1\over2\)∫ *ρ*(**r**)
Φ(**r**)
d^{3}*r* = (1/8π*G*)
* *∫ (∇Φ)^{3} d^{3}*r* +
* *∫ *ρ*(**r**) Φ(**r**)
d^{3}*r* .

@ __References__: Deser AJP(05)aug-gq/04 [from
field theory]; Counihan EJP(07)
[basic principles]; Yurtsever et al a1004 [inverse problem]; in Ohanian & Ruffini 13; in Poisson & Will 14.

**Special Topics** > s.a. cosmology;
gravitational constant; Newton's
Theorem; teaching [weightlessness, tides].

* __Tidal forces__: A mass *m* located
at (*x*, *y*, *z*) with respect to a frame centered
at a point a distance *r*_{0} away
from a center of attraction of mass *M*, feels a tidal force

*F*_{x} = –*x* (*GMm*/*r*_{0}^{3})
, *F*_{y} = –*y* (*GMm*/*r*_{0}^{3})
, *F*_{z} =
2*z* (*GMm*/*r*_{0}^{3})
;

In general relativity the expression is more complicated, uses the equation
for geodesic deviation.

* __Other formulations__:
A (slightly generalized) geometric version is the Newton-Cartan
theory.

__N____-body problem__: Volchan a0803 [Sundman-Weierstraß theorem of total collapse]; Farrés et al CMDA(13)-a1208 [high-precision symplectic integrators for the Solar System].

@ __Tidal forces__: Masi AJP(07)feb
[compressive radial]; Efroimsky & Williams CMDA(09)-a0803 [tidal torques]; > s.a. Love
Number.

@ __Other situations__: Odagaki & Kawai AJP(98)aug
[many-particle
statistics]; Beig & Schmidt PRS(03)gq/02 [self-gravitating
extended bodies]; Teixeira phy/03 [infinite
straight line of mass]; Buchert PLA(06)gq/05 [self-gravitating
dust]; Ridgely EJP(11) [in material media].

@ __Other formulations and issues__: De Pietri et al gq/92,
CQG(95)gq/94,
CQG(95)gq/94 [generalization];
Nardone JPA(98)
[regularization]; Natário GRG(06)gq/04 [initial-value
form, and warp drive].

**Phenomenology** > s.a. cosmological
models [Newtonian cosmology]; phenomenology
of gravity; modifications and tests
of newtonian gravity.

@ __Effects__: Abramowicz et al GRG(97) [curvature of space and perihelion
precession].

@ __Specific objects__: Dittrich a1609 [Dirichlet's massive homogeneous ellipsoid].

@ __Measurement__: Kulikov JMO(06)qp/05 [transparency
of cold atoms]; Charrière et al PRA(12) [local *g* measurements]; Graney PT(12)sep [Giovanni Battista Riccioli]; Harms a1507-LRR [terrestrial gravity fluctuations]; > s.a. Eötvös Experiment; fifth
force.

> __Related topics__: see critical
collapse; Newton's Theorem [shell theorem]; orbits
in newtonian gravity.

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sep 2016