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*Gρ*,

Φ(**r**) = −*G *∫ d*M*
|**r**−**r**'|^{−1}
= −*GM*/*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;
Pereira a1903 [from its empirical basis to the theory].

**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 *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];
Hansen et al PRL(19)-a1807 [action principle, gravitational time dilation];
Banerjee & Mukherjee PRD(18)-a1810 [geometric].

**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];
Ferroglia & Fiolhais AJP(20)dec [tidal locking, pedagogical].

@ __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 LRR(15)-a1507 [terrestrial gravity fluctuations];
news sn(19)nov [measuring gravity with trapped atoms];
> s.a. Eötvös Experiment; fifth force.

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

main page
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