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Newton's law of universal gravitation
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Newton's law of universal gravitation states that a particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.[note 1] This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning.[1] It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1686. When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him.
In today's language, the law states that every point mass attracts every other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses, and inversely proportional to the square of the distance between them.[2]
The equation for universal gravitation thus takes the form:
F = G m 1 m 2 r 2 {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }
where F is the gravitational force acting between two objects, m1 and m2 are the masses of the objects, r is the distance between the centers of their masses, and G is the gravitational constant.
The first test of Newton's theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798.[3] It took place 111 years after the publication of Newton's Principia and approximately 71 years after his death.
Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has the product of two charges in place of the product of the masses, and the electrostatic constant in place of the gravitational constant.
Newton's law has since been superseded by Albert Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity in most applications. Relativity is required only when there is a need for extreme precision, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the Sun).
Contents
[hide]
1 History
1.1 Early history
1.2 Plagiarism dispute
1.3 Hooke's work and claims
1.4 Newton's work and claims
1.5 Newton's acknowledgment
1.6 Modern priority controversy
2 Modern form
3 Bodies with spatial extent
4 Vector form
5 Gravitational field
6 Problematic aspects
6.1 Theoretical concerns with Newton's expression
6.2 Observations conflicting with Newton's formula
6.3 Newton's reservations
6.4 Einstein's solution
7 Extensions
8 Solutions of Newton's law of universal gravitation
9 See also
10 Notes
11 References
12 External links
History[edit]
Early history[edit]
A recent assessment (by Ofer Gal) about the early history of the inverse square law is "by the late 1670s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons".
The same author does credit Hooke with a significant and even seminal contribution, but he treats Hooke's claim of priority on the inverse square point as uninteresting since several individuals besides Newton and Hooke had at least suggested it, and he points instead to the idea of "compounding the celestial motions" and the conversion of Newton's thinking away from "centrifugal" and towards "centripetal" force as Hooke's significant contributions.
Newton himself gave credit in his Principia to two persons: Bullialdus[4] (he wrote without proof that there was a force on the earth towards the sun), and Borelli[5] (wrote that all planets were attracted towards the sun). Whiteside wrote that the main influence was Borelli, because Newton had a copy of his book.[6]
Plagiarism dispute[edit]
In 1686, when the first book of Newton's Principia was presented to the Royal Society, Robert Hooke accused Newton of plagiarism by claiming that he had taken from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". At the same time (according to Edmond Halley's contemporary report) Hooke agreed that "the Demonstration of the Curves generated thereby" was wholly Newton's.[7]
In this way, the question arose as to what, if anything, Newton owed to Hooke. This is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy.
Hooke's work and claims[edit]
Robert Hooke published his ideas about the "System of the World" in the 1660s, when he read to the Royal Society on March 21, 1666, a paper "On gravity", "concerning the inflection of a direct motion into a curve by a supervening attractive principle", and he published them again in somewhat developed form in 1674, as an addition to "An Attempt to Prove the Motion of the Earth from Observations".[8] Hooke announced in 1674 that he planned to "explain a System of the World differing in many particulars from any yet known", based on three "Suppositions": that "all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers" [and] "they do also attract all the other Celestial Bodies that are within the sphere of their activity";[9] that "all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a straight line, till they are by some other effectual powers deflected and bent..."; and that "these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers". Thus Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, together with a principle of linear inertia.
Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses.[10] He also did not provide accompanying evidence or mathematical demonstration. On the latter two aspects, Hooke himself stated in 1674: "Now what these several degrees [of attraction] are I have not yet experimentally verified"; and as to his whole proposal: "This I only hint at present", "having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e. "prosecuting this Inquiry").[8] It was later on, in writing on 6 January 1679|80[11] to Newton, that Hooke communicated his "supposition ... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance."[12] (The inference about the velocity was incorrect.[13])
Hooke's correspondence with Newton during 1679–1680 not only mentioned this inverse square supposition for the decline of attraction with increasing distance, but also, in Hooke's opening letter to Newton, of 24 November 1679, an approach of "compounding the celestial motions of the planets of a direct motion by the tangent & an attractive motion towards the central body".[14]
Newton's work and claims[edit]
Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hooke's 1679 letter.[15] Newton also pointed out and acknowledged prior work of others,[16] including Bullialdus,[4] (who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and Borelli[5] (who suggested, also without demonstration, that there was a centrifugal tendency in counterbalance with a gravitational attraction towards the Sun so as to make the planets move in ellipses). D T Whiteside has described the contribution to Newton's thinking that came from Borelli's book, a copy of which was in Newton's library at his death.[6]
Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, he would still have some rights to it in view of his demonstrations of its accuracy. Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center. According to Newton, while the 'Principia' was still at pre-publication stage, there were so many a-priori reasons to doubt the accuracy of the inverse-square law (especially close to an attracting sphere) that "without my (Newton's) Demonstrations, to which Mr Hooke is yet a stranger, it cannot believed by a judicious Philosopher to be any where accurate."[17]
This remark refers among other things to Newton's finding, supported by mathematical demonstration, that if the inverse square law applies to tiny particles, then even a large spherically symmetrical mass also attracts masses external to its surface, even close up, exactly as if all its own mass were concentrated at its center. Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles.[18] In addition, Newton had formulated, in Propositions 43-45 of Book 1[19] and associated sections of Book 3, a sensitive test of the accuracy of the inverse square law, in which he showed that only where the law of force is calculated as the inverse square of the distance will the directions of orientation of the planets' orbital ellipses stay constant as they are observed to do apart from small effects attributable to inter-planetary perturbations.
In regard to evidence that still survives of the earlier history, manuscripts written by Newton in the 1660s show that Newton himself had, by 1669, arrived at proofs that in a circular case of planetary motion, "endeavour to recede" (what was later called centrifugal force) had an inverse-square relation with distance from the center.[20] After his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. According to Newton scholar J. Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s. The lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis.[21] This background shows there was basis for Newton to deny deriving the inverse square law from Hooke.
Newton's acknowledgment[edit]
On the other hand, Newton did accept and acknowledge, in all editions of the Principia, that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the solar system. Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1.[22] Newton also acknowledged to Halley that his correspondence with Hooke in 1679-80 had reawakened his dormant interest in astronomical matters, but that did not mean, according to Newton, that Hooke had told Newton anything new or original: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it ..."[16]
Modern priority controversy[edit]
Since the time of Newton and Hooke, scholarly discussion has also touched on the question of whether Hooke's 1679 mention of 'compounding the motions' provided Newton with something new and valuable, even though that was not a claim actually voiced by Hooke at the time. As described above, Newton's manuscripts of the 1660s do show him actually combining tangential motion with the effects of radially directed force or endeavour, for example in his derivation of the inverse square relation for the circular case. They also show Newton clearly expressing the concept of linear inertia—for which he was indebted to Descartes' work, published in 1644 (as Hooke probably was).[23] These matters do not appear to have been learned by Newton from Hooke.
Nevertheless, a number of authors have had more to say about what Newton gained from Hooke and some aspects remain controversial.[24] The fact that most of Hooke's private papers had been destroyed or have disappeared does not help to establish the truth.
Newton's role in relation to the inverse square law was not as it has sometimes been represented. He did not claim to think it up as a bare idea. What Newton did was to show how the inverse-square law of attraction had many necessary mathematical connections with observable features of the motions of bodies in the solar system; and that they were related in such a way that the observational evidence and the mathematical demonstrations, taken together, gave reason to believe that the inverse square law was not just approximately true but exactly true (to the accuracy achievable in Newton's time and for about two centuries afterwards – and with some loose ends of points that could not yet be certainly examined, where the implications of the theory had not yet been adequately identified or calculated).[25][26]
About thirty years after Newton's death in 1727, Alexis Clairaut, a mathematical astronomer eminent in his own right in the field of gravitational studies, wrote after reviewing what Hooke published, that "One must not think that this idea ... of Hooke diminishes Newton's glory"; and that "the example of Hooke" serves "to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated".[27][28]
Modern form[edit]
In modern language, the law states the following:
Every point mass attracts every single other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:[2]
F = G m 1 m 2 r 2 {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }
where:
F is the force between the masses;
G is the gravitational constant (6.674×10−11 N · (m/kg)2);
m1 is the first mass;
m2 is the second mass;
r is the distance between the centers of the masses.
Error plot showing experimental values for big G.
Assuming SI units, F is measured in newtons (N), m1 and m2 in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6989667400000000000♠6.674×10−11 N m2 kg−2.[29] The value of the constant G was first accurately determined from the results of the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798, although Cavendish did not himself calculate a numerical value for G.[3] This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's Principia and 71 years after Newton's death, so none of Newton's calculations could use the value of G; instead he could only calculate a force relative to another force.
Bodies with spatial extent[edit]
Gravitational field strength within the Earth
Gravity field near the surface of the Earth – an object is shown accelerating toward the surface
If the bodies in question have spatial extent (as opposed to being point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies.
In this way, it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its centre.[2] (This is not generally true for non-spherically-symmetrical bodies.)
For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance r0 from the center of the mass distribution:[30]
The portion of the mass that is located at radii r < r0 causes the same force at r0 as if all of the mass enclosed within a sphere of radius r0 was concentrated at the center of the mass distribution (as noted above).
The portion of the mass that is located at radii r > r0 exerts no net gravitational force at the distance r0 from the center. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at r0, cancel each other out.
As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration anywhere within the hollow sphere.
Furthermore, inside a uniform sphere the gravity increases linearly with the distance from the center; the increase due to the additional mass is 1.5 times the decrease due to the larger distance from the center. Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than 2/3 of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary. The gravity of the Earth may be highest at the core/mantle boundary.
Vector form[edit]
Gravity field surrounding Earth from a macroscopic perspective.
Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.
F 21 = − G m 1 m 2 | r 12 | 2 r ^ 12 {\displaystyle \mathbf {F} _{21}=-G{m_{1}m_{2} \over {\vert \mathbf {r} _{12}\vert }^{2}}\,\mathbf {\hat {r}} _{12}}
where
F21 is the force applied on object 2 exerted by object 1,
G is the gravitational constant,
m1 and m2 are respectively the masses of objects 1 and 2,
|r12| = |r2 − r1| is the distance between objects 1 and 2, and
r ^ 12 = d e f r 2 − r 1 | r 2 − r 1 | {\displaystyle \mathbf {\hat {r}} _{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert }}} is the unit vector from object 1 to 2.
It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that F12 = −F21.
Gravitational field[edit]
Main article: Gravitational field
The gravitational field is a vector field that describes the gravitational force which would be applied on an object in any given point in space, per unit mass. It is actually equal to the gravitational acceleration at that point.
It is a generalisation of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). For 2 objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write r instead of r12 and m instead of m2 and define the gravitational field g(r) as:
g ( r ) = − G m 1 | r | 2 r ^ {\displaystyle \mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}} }
so that we can write:
F ( r ) = m g ( r ) . {\displaystyle \mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} ).}
This formulation is dependent on the objects causing the field. The field has units of acceleration; in SI, this is m/s2.
Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent. This has the consequence that there exists a gravitational potential field V(r) such that
g ( r ) = − ∇ V ( r ) . {\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}
If m1 is a point mass or the mass of a sphere with homogeneous mass distribution, the force field g(r) outside the sphere is isotropic, i.e., depends only on the distance r from the center of the sphere. In that case
V ( r ) = − G m 1 r . {\displaystyle V(r)=-G{\frac {m_{1}}{r}}.}
the gravitational field is on, inside and outside of symmetric masses.
As per Gauss Law, field in a symmetric body can be found by the mathematical equation:
∂ V {\displaystyle \partial V} g ( r ) ⋅ d A = − 4 π G M enc , {\displaystyle \mathbf {g(r)} \cdot d\mathbf {A} =-4\pi GM_{\text{enc}},}
where ∂ V {\displaystyle \partial V} is a closed surface and M enc {\displaystyle M_{\text{enc}}} is the mass enclosed by the surface.
Hence, for a hollow sphere of radius R {\displaystyle R} and total mass M {\displaystyle M} ,
| g ( r ) | = { 0 , if r < R G M r 2 , if r ≥ R {\displaystyle |\mathbf {g(r)} |={\begin{cases}0,&{\mbox{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\mbox{if }}r\geq R\end{cases}}}
For a uniform solid sphere of radius R {\displaystyle R} and total mass M {\displaystyle M} ,
| g ( r ) | = { G M r R 3 , if r < R G M r 2 , if r ≥ R {\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\mbox{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\mbox{if }}r\geq R\end{cases}}}
Problematic aspects[edit]
Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities φ/c2 and (v/c)2 are both much less than one, where φ is the gravitational potential, v is the velocity of the objects being studied, and c is the speed of light.[31] For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since
Φ c 2 = G M s u n r o r b i t c 2 ∼ 10 − 8 , ( v E a r t h c ) 2 = ( 2 π r o r b i t ( 1 y r ) c ) 2 ∼ 10 − 8 {\displaystyle {\frac {\Phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8}}
where rorbit is the radius of the Earth's orbit around the Sun.
In situations where either dimensionless parameter is large, then general relativity must be used to describe the system. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity.
Theoretical concerns with Newton's expression[edit]
There is no immediate prospect of identifying the mediator of gravity. Attempts by physicists to identify the relationship between the gravitational force and other known fundamental forces are not yet resolved, although considerable headway has been made over the last 50 years (See: Theory of everything and Standard Model). Newton himself felt that the concept of an inexplicable action at a distance was unsatisfactory (see "Newton's reservations" below), but that there was nothing more that he could do at the time.
Newton's theory of gravitation requires that the gravitational force be transmitted instantaneously. Given the classical assumptions of the nature of space and time before the development of General Relativity, a significant propagation delay in gravity leads to unstable planetary and stellar orbits.
Observations conflicting with Newton's formula[edit]
Newton's Theory does not fully explain the precession of the perihelion of the orbits of the planets, especially of planet Mercury, which was detected long after the life of Newton.[32] There is a 43 arcsecond per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th century.
The predicted angular deflection of light rays by gravity that is calculated by using Newton's Theory is only one-half of the deflection that is actually observed by astronomers. Calculations using General Relativity are in much closer agreement with the astronomical observations.
In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey Newton's law of universal gravitation. Astrophysicists, however, explain this spectacular phenomenon in the framework of Newton's laws, with the presence of large amounts of dark matter.
Newton's reservations[edit]
While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" that his equations implied. In 1692, in his third letter to Bentley, he wrote: "That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it."
He never, in his words, "assigned the cause of this power". In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity (although he invented two mechanical hypotheses in 1675 and 1717). Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science. He lamented that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer has yet to be found. And in Newton's 1713 General Scholium in the second edition of Principia: "I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses.... It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies."[33]
Einstein's solution[edit]
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These objections were explained by Einstein's theory of general relativity, in which gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. In Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. This allowed a description of the motions of light and mass that was consistent with all available observations. In general relativity, the gravitational force is a fictitious force due to the curvature of spacetime, because the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime.
Extensions[edit]
Newton was the first to consider in his Principia an extended expression of his law of gravity including an inverse-cube term of the form
F = G m 1 m 2 r 2 + B m 1 m 2 r 3 {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}+B{\frac {m_{1}m_{2}}{r^{3}}}\ } , B {\displaystyle B} a constant
attempting to explain the Moon's apsidal motion. Other extensions were proposed by Laplace (around 1790) and Decombes (1913):[34]
F ( r ) = k m 1 m 2 r 2 exp  ( − α r ) {\displaystyle F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\exp(-\alpha r)} (Laplace)
F ( r ) = k m 1 m 2 r 2 ( 1 + α r 3 ) {\displaystyle F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\left(1+{\alpha \over {r^{3}}}\right)} (Decombes)
In recent years, quests for non-inverse square terms in the law of gravity have been carried out by neutron interferometry.[35]
Solutions of Newton's law of universal gravitation[edit]
Main article: n-body problem
The n-body problem is an ancient, classical problem[36] of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem — from the time of the Greeks and on — has been motivated by the desire to understand the motions of the Sun, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.[37] The n-body problem in general relativity is considerably more difficult to solve.
The classical physical problem can be informally stated as: given the quasi-steady orbital properties (instantaneous position, velocity and time)[38] of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.[39]
The two-body problem has been completely solved, as has the restricted three-body problem.[40]
See also[edit]
Book: Isaac Newton
Physics portal
Bentley's paradox
Gauss's law for gravity
Jordan and Einstein frames
Kepler orbit
Newton's cannonball
Newton's laws of motion
Static forces and virtual-particle exchange
Notes[edit]
Jump up ^ It was shown separately that large, spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers.
References[edit]
Jump up ^ Isaac Newton: "In [experimental] philosophy particular propositions are inferred from the phenomena and afterwards rendered general by induction": "Principia", Book 3, General Scholium, at p.392 in Volume 2 of Andrew Motte's English translation published 1729.
^ Jump up to: a b c - Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. Preceded by A Guide to Newton's Principia, by I.Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4
^ Jump up to: a b The Michell-Cavendish Experiment, Laurent Hodges
^ Jump up to: a b Bullialdus (Ismael Bouillau) (1645), "Astronomia philolaica", Paris, 1645.
^ Jump up to: a b Borelli, G. A., "Theoricae Mediceorum Planetarum ex causis physicis deductae", Florence, 1666.
^ Jump up to: a b D T Whiteside, "Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664-1684", Journal for the History of Astronomy, i (1970), pages 5-19; especially at page 13.
Jump up ^ H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), giving the Halley-Newton correspondence of May to July 1686 about Hooke's claims at pp.431-448, see particularly page 431.
^ Jump up to: a b Hooke's 1674 statement in "An Attempt to Prove the Motion of the Earth from Observations" is available in online facsimile here.
Jump up ^ Purrington, Robert D. (2009). The First Professional Scientist: Robert Hooke and the Royal Society of London. Springer. p. 168. ISBN 3-0346-0036-4. Extract of page 168
Jump up ^ See page 239 in Curtis Wilson (1989), "The Newtonian achievement in astronomy", ch.13 (pages 233-274) in "Planetary astronomy from the Renaissance to the rise of astrophysics: 2A: Tycho Brahe to Newton", CUP 1989.
Jump up ^ Calendar (New Style) Act 1750
Jump up ^ Page 309 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #239.
Jump up ^ See Curtis Wilson (1989) at page 244.
Jump up ^ Page 297 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #235, 24 November 1679.
Jump up ^ Page 433 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #286, 27 May 1686.
^ Jump up to: a b Pages 435-440 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #288, 20 June 1686.
Jump up ^ Page 436, Correspondence, Vol.2, already cited.
Jump up ^ Propositions 70 to 75 in Book 1, for example in the 1729 English translation of the Principia, start at page 263.
Jump up ^ Propositions 43 to 45 in Book 1, in the 1729 English translation of the Principia, start at page 177.
Jump up ^ D T Whiteside, "The pre-history of the 'Principia' from 1664 to 1686", Notes and Records of the Royal Society of London, 45 (1991), pages 11-61; especially at 13-20.
Jump up ^ See J. Bruce Brackenridge, "The key to Newton's dynamics: the Kepler problem and the Principia", (University of California Press, 1995), especially at pages 20-21.
Jump up ^ See for example the 1729 English translation of the Principia, at page 66.
Jump up ^ See page 10 in D T Whiteside, "Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664-1684", Journal for the History of Astronomy, i (1970), pages 5-19.
Jump up ^ Discussion points can be seen for example in the following papers: N Guicciardini, "Reconsidering the Hooke-Newton debate on Gravitation: Recent Results", in Early Science and Medicine, 10 (2005), 511-517; Ofer Gal, "The Invention of Celestial Mechanics", in Early Science and Medicine, 10 (2005), 529-534; M Nauenberg, "Hooke's and Newton's Contributions to the Early Development of Orbital mechanics and Universal Gravitation", in Early Science and Medicine, 10 (2005), 518-528.
Jump up ^ See for example the results of Propositions 43-45 and 70-75 in Book 1, cited above.
Jump up ^ See also G E Smith, in Stanford Encyclopedia of Philosophy, "Newton's Philosophiae Naturalis Principia Mathematica".
Jump up ^ The second extract is quoted and translated in W.W. Rouse Ball, "An Essay on Newton's 'Principia'" (London and New York: Macmillan, 1893), at page 69.
Jump up ^ The original statements by Clairaut (in French) are found (with orthography here as in the original) in "Explication abregée du systême du monde, et explication des principaux phénomenes astronomiques tirée des Principes de M. Newton" (1759), at Introduction (section IX), page 6: "Il ne faut pas croire que cette idée ... de Hook diminue la gloire de M. Newton", [and] "L'exemple de Hook" [serve] "à faire voir quelle distance il y a entre une vérité entrevue & une vérité démontrée".
Jump up ^ Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028  . Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633. Direct link to value..
Jump up ^ Equilibrium State
Jump up ^ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. New York: W. H.Freeman and Company. ISBN 0-7167-0344-0 Page 1049.
Jump up ^ - Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and the Earth.)
Jump up ^ - The Construction of Modern Science: Mechanisms and Mechanics, by Richard S. Westfall. Cambridge University Press. 1978
Jump up ^ http://physicsessays.org/doi/abs/10.4006/1.3038751?journalCode=phes[dead link]
Jump up ^ http://journals.aps.org/prc/abstract/10.1103/PhysRevC.75.015501
Jump up ^ Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the n-body problem, especially Ms. Kovalevskaya's ~1868-1888, twenty-year complex-variables approach, failure; Section 1: The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics (Chapter 1, the motion of a rigid body about a fixed point (Euler and Poisson equations); Chapter 2, Mathematical Exterior Ballistics), good precursor background to the n-body problem; Section 2: Celestial Mechanics (Chapter 1, The Uniformization of the Three-body Problem (Restricted Three-body Problem); Chapter 2, Capture in the Three-Body Problem; Chapter 3, Generalized n-body Problem).
Jump up ^ See References sited for Heggie and Hut. This Wikipedia page has made their approach obsolete.
Jump up ^ Quasi-steady loads refers to the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, a steady-state condition refers to a system's state being invariant to time; otherwise, the first derivatives and all higher derivatives are zero.
Jump up ^ R. M. Rosenberg states the n-body problem similarly (see References): Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move? Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces first before the motions can be determined.
Jump up ^ A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary n can be approximated via Taylor series, but in practice such an infinite series must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the n-body problem may be solved using numerical integration, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book Gravitational N-body Simulations listed in the References.
External links[edit]
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Mary, Did You Know?
From Wikipedia, the free encyclopedia
Jump to: navigation, search
A depiction of the Nativity of Jesus by Francesco Londonio (c. 1750)
"Mary, Did You Know?" is a Christmas song with lyrics written by Mark Lowry and music written by Buddy Greene. It was originally recorded by Christian recording artist Michael English on his self-titled debut solo album in 1991 (English and Lowry were both members of the Gaither Vocal Band at the time). It reached No. 6 on CCM Magazine's AC Chart.[1] Lowry would record the song several times himself, most notably with the Gaither Vocal Band on their 1998 Christmas album Still the Greatest Story Ever Told.
The song has since gone on to become a modern Christmas classic, being recorded by many artists over the years across multiple genres.
Contents
[hide]
1 Kenny Rogers & Wynonna Judd version
2 Clay Aiken version
3 Cee Lo Green version
4 Pentatonix version
5 Other notable versions
6 Musical
7 Criticism
8 References
9 External links
Kenny Rogers & Wynonna Judd version[edit]
"Mary, Did You Know?"
Single by Kenny Rogers & Wynonna Judd
from the album The Gift
Released
December 1996
Format
CD single
Genre
Country
Length
3:53
Label
Magnatone
Songwriter(s)
Mark Lowry, Buddy Greene
Producer(s)
Brent Maher, Jim McKell
Kenny Rogers singles chronology
"I'll Be There for You"
(1992) "Mary, Did You Know?"
(1996) "The Greatest"
(1999)
"I'll Be There for You"
(1992) "Mary, Did You Know?"
(1996) "The Greatest"
(1999)
A duet version recorded by Wynonna Judd and Kenny Rogers on Rogers' holiday album The Gift peaked at No. 55 on the Billboard Hot Country Singles & Tracks chart in 1997.[2]
Chart positions
Chart (1997)
Peak
position
US Hot Country Songs (Billboard)[3] 55
Clay Aiken version[edit]
"Mary, Did You Know?"
Single by Clay Aiken
from the album Merry Christmas with Love
Released
2005
Format
Digital download
Recorded
2004
Genre
Contemporary Christian music
Length
3:21
Label
RCA
Songwriter(s)
Mark Lowry, Buddy Greene
Producer(s)
Walter Afanasieff
Clay Aiken singles chronology
"Have Yourself a Merry Little Christmas"
(2005) "Mary, Did You Know?"
(2005) "Without You"
(2006)
"Have Yourself a Merry Little Christmas"
(2005) "Mary, Did You Know?"
(2005) "Without You"
(2006)
"Mary, Did You Know?" is Clay Aiken's fifth single for his Christmas album Merry Christmas with Love. Aiken performed the song on The Tyra Banks Show.[4]
Chart positions
Chart (2005)
Peak
position
U.S. Billboard Hot Adult Contemporary Tracks 35
U.S. Billboard Hot Christian Songs 32
Cee Lo Green version[edit]
"Mary, Did You Know?"
Song by Cee Lo Green
from the album Cee Lo's Magic Moment
Released
2012
Format
Digital download
Recorded
2012
Genre
R&B
Length
3:54
Label
RCA
Songwriter(s)
Mark Lowry, Buddy Greene
Producer(s)
Adam Anders, Peer Åström
"Mary, Did You Know?" is the eleventh track on Cee Lo's Christmas album Cee Lo's Magic Moment. On December 15, 2012 the song appeared on the US R&B Songs chart at number 22. This version is the slowest of all. It also served as the closing credits song to the 2014 20th Century Fox movie Son of God.
Chart positions
Chart (2012)
Peak
position
U.S. Billboard R&B Songs[5] 11
U.S. Billboard R&B/Hip-Hop Songs[5] 44
Pentatonix version[edit]
"Mary, Did You Know?"
Song by Pentatonix
from the album That's Christmas to Me
Released
October 17, 2014
Format
Digital download
Recorded
2012
Genre
Ambient, a cappella
Length
3:23
Label
RCA
Songwriter(s)
Mark Lowry, Buddy Greene
Music video
"Mary, Did You Know?" on YouTube
"Mary, Did You Know?" is the sixth track on Pentatonix's second Christmas album, That's Christmas to Me.[6] The song both debuted and peaked at No. 26 on the weekly Billboard Hot 100 chart for the week ending December 6, 2014.
Chart positions
Chart (2014)
Peak
position
US Billboard Hot 100[7] 26
Other notable versions[edit]
1993: Kathy Mattea on her Christmas album Good News, which won a Grammy Award for Best Southern, Country or Bluegrass Gospel Album
1995: Vickie Winans recorded the song for her self-titled album
1997: Gary Chapman on his Christmas album This Gift
1998: Donny Osmond on his album Christmas at Home
1999: Natalie Cole on her album The Magic of Christmas
1999: Reba McEntire on her Christmas album The Secret of Giving
2000: Lea Salonga on her album The Christmas Album
2001: Rescue on their album The First Christmas
2001: Hayley Westenra on her second album, My Gift to You, released in New Zealand and Australia, by Universal New Zealand label; also included as a bonus song on a special UK version of her first international album, Pure (2003); and performed it on her 2004 PBS Great Performances special, Live from New Zealand, which was released on DVD.
2004: Glen Campbell recorded the song on his album Love Is the Answer: 24 Songs of Faith, Hope and Love.
2004: The Katinas on their album Family Christmas
2005: Jeff Fenholt on his album "Down from His Glory" ... and more
2006: Stan Whitmire on his album Christmas Around the Piano
2007: Kimberley Locke on her album Christmas
2008: Kutless on BEC Recordings album X Christmas
2008: MC6 A Cappella on their album Christmas
2009: Jerry Douglas on his album Jerry Christmas
2010: Aled Jones on his album Aled's Christmas Gift
2010: Jessica Simpson on her album Happy Christmas
2011: Rascal Flatts released a version of the song as a single, which appears on the 2012 compilation A Very Special Christmas: 25 Years Bringing Joy to the World
2012: Don Moen on his album Christmas: A Season of Hope (a duet with its composer, Buddy Greene)
2012: Jeremy Camp on his album Christmas: God with Us
2012: Scotty McCreery on his album Christmas with Scotty McCreery
2013: Mary J. Blige on her album A Mary Christmas
2013: Linda Eder on her album Christmas Where You Are
2013: The Oak Ridge Boys on their album Christmas Time's A-Comin'
2014: Sanam Puri as solo cover
2014: The slowest version of this song played in the credits of the film Son of God
2015: India.Arie on Christmas with Friends; a duet with Gene Moore, Jr.
2015: Jordan Smith on The Voice's season nine finals. This version debuted and peaked at No. 24 on the Billboard Hot 100 chart and at No. 2 on the Billboard Holiday 100 chart on the week ending January 2, 2016.
2015: Peter Hollens published a six-voice a cappella version on Youtube[8]
2015: The Braxtons on their album Braxton Family Christmas
2015: Danny Gokey on his album Christmas Is Here
2016: Voctave on their album The Spirit of the Season (feat. composer Mark Lowry)
2016: Beckah Shae on her album Let It Snow
2017: Cabin 5[9] as a single
2017: Zara Larsson on the soundtrack of the CGI-animated Christmas film The Star
Musical[edit]
The song was also used as the basis for a stage musical, also titled Mary, Did You Know and written by Bruce Greer, that won the 1999 Dove Award for Musical of the Year.[10]
Criticism[edit]
"Mary, Did You Know?" remains a popular piece of Christmas music, although some criticize it for what they perceive as a lack of Scriptural or theological depth.[11][12] For example, Lutheran writer Holly Scheer, in addressing the rhetorical-question title of the song writes that those familiar with "the biblical account of Christ’s conception and birth shouldn’t need to ask if Mary knew, because the Bible plainly tells us she did."[13] On the other hand the Roman Catholic author Karl Erickson implored people to praise "Mary, Did You Know?" as a "beautiful expression of God’s love for all of us" and responded to criticism of the piece, stating that "Mary, Did You Know is a song and not a theological essay. Even as simple Christmas music, however, it does a fine job conveying the beauty and profound mystery of the Incarnation. Yes, there is some poetic license taken within the lyrics, but much less than is taken within innumerable other songs we are asked to sing regularly—Sing a New Church, for example."[14]
References[edit]
Jump up ^ Brothers, Jeffrey Lee (2003). Hot hits : adult contemporary charts, 1978-2001 (1st ed.). Bloomington, IN: 1stBooks Library. p. 86. ISBN 9781410732941.
Jump up ^ https://www.allmusic.com/artist/p93355/charts-awards/billboard-singles
Jump up ^ "Kenny Rogers Chart History (Hot Country Songs)". Billboard.
Jump up ^ Clay Aiken - Mary Did You Know - The Tyra Banks Show on YouTube
^ Jump up to: a b https://www.billboard.com/artist/298842/cee+lo+green/chart?f=1230
Jump up ^ Official Video - Mary, Did You Know? - Pentatonix on YouTube
Jump up ^ "Pentatonix Chart History (Hot 100)". Billboard. Retrieved December 4, 2014.
Jump up ^ Mary, did you know - Peter Hollens (Youtube Video)
Jump up ^ https://open.spotify.com/embed/track/1mdbai6o6xmqhazzABTxP3
Jump up ^ Cusic, Don (2010). Encyclopedia of contemporary Christian music: pop, rock, and worship. ABC-CLIO. pp. 141–. ISBN 978-0-313-34425-1. Retrieved 7 June 2011.
Jump up ^ McTeigue SJ, Fr. Robert. "The Problem With "Mary Did You Know"". Aleteia. Aleteia. Retrieved 27 November 2017.
Jump up ^ "Mary, Did You Know?". EWTN. EWTN. Retrieved 27 November 2017.
Jump up ^ Scheer, Holly (21 December 2016). "Why 'Mary Did You Know' Is The Most Biblically Illiterate Christmas Tune". The Federalist. Retrieved 27 November 2017.
Jump up ^ Erickson, Karl (30 December 2015). "What Did Mary Know?". Catholic 365. Retrieved 27 November 2017.
External links[edit]
Lyrics of this song at MetroLyrics
[show]
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[show]
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/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: Guidance on how to use and edit Wikipedia

values: About the project, what you can do, where to find things

values: A list of recent changes in the wiki [ctrl-alt-r]

values: portal navigation p-interaction p-interaction-label

values: List of all English Wikipedia pages containing links to this page [ctrl-alt-j]

values: Recent changes in pages linked from this page [ctrl-alt-k]

values:
/wiki/Wikipedia:File_Upload_Wizard
Upload files [ctrl-alt-u]
u

values: A list of all special pages [ctrl-alt-q]

values: Permanent link to this revision of the page

values: Link to connected data repository item [ctrl-alt-g]

values: Printable version of this page [ctrl-alt-p]

values:
uls-settings-trigger
Language settings
0
button
true

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Mary_Did_You_Know rootpage-Mary_Did_You_Know skin-vector action-view