Editing Gravitational field (section)

== Classical mechanics ==

In classical mechanics, a gravitational field is a physical quantity.<ref>{{cite book |author-link=Richard Feynman |first=Richard |last=Feynman |title=The Feynman Lectures on Physics |volume= II |publisher=Addison Wesley Longman |year=1970 |isbn=978-0-201-02115-8 |url=https://feynmanlectures.caltech.edu/II_01.html#Ch1-S2 |quote="A 'field' is any physical quantity which takes on different values at different points in space."}}</ref> A gravitational field can be defined using [[Newton's law of universal gravitation]]. Determined in this way, the gravitational field {{math|'''g'''}} around a single particle of mass {{math|''M''}} is a [[vector field]] consisting at every point of a [[Vector (geometry)|vector]] pointing directly towards the particle. The magnitude of the field at every point is calculated by applying the universal law, and represents the force per unit mass on any object at that point in space. Because the force field is conservative, there is a scalar potential energy per unit mass, {{math|Φ}}, at each point in space associated with the force fields; this is called [[gravitational potential]].<ref>{{cite book|title=Dynamics and Relativity |first1=J. R. |last1=Forshaw |first2=A. G. |last2=Smith |publisher=Wiley |date=2009 |isbn=978-0-470-01460-8}}{{page needed|date=October 2017}}</ref> The gravitational field equation is<ref>{{cite book |url=https://archive.org/details/encyclopediaofph00lern |title=Encyclopaedia of Physics |date=1991 |publisher=[[Wiley-VCH]] |isbn=978-0-89573-752-6 |editor1-last=Lerner |editor1-first=R. G. |editor1-link=Rita G. Lerner |edition=2nd |editor2-last=Trigg |editor2-first=G. L. |url-access=registration}} p. 451</ref>
<math display="block">\mathbf{g}=\frac{\mathbf{F}}{m}=\frac{d^2\mathbf{R}}{dt^2}=-GM\frac{\mathbf{R}}{\left|\mathbf{R}\right|^3} = -\nabla\Phi ,</math>
where {{math|'''F'''}} is the [[gravitational force]], {{math|''m''}} is the mass of the [[test mass|test particle]], {{math|'''R'''}} is the radial vector of the test particle relative to the mass (or for Newton's second law of motion which is a time dependent function, a set of positions of test particles each occupying a particular point in space for the start of testing), {{math|''t''}} is [[time]], {{math|''G''}} is the [[gravitational constant]], and {{math|∇}} is the [[del operator]].

This includes Newton's law of universal gravitation, and the relation between gravitational potential and field acceleration. {{math|{{sfrac|d<sup>2</sup>'''R'''|d''t''<sup>2</sup>}}}} and {{math|{{sfrac|'''F'''|''m''}}}} are both equal to the [[gravitational acceleration]] {{math|'''g'''}} (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass<ref>{{cite book|title=Essential Principles of Physics |first1=P. M. |last1=Whelan |first2=M. J. |last2=Hodgeson |edition=2nd |date=1978 |publisher=John Murray |isbn=978-0-7195-3382-2}}{{page needed|date=October 2017}}</ref>). The negative signs are inserted since the force acts antiparallel to the displacement. The equivalent field equation in terms of mass [[density]] {{math|''ρ''}} of the attracting mass is:
<math display="block">\nabla\cdot\mathbf{g}=-\nabla^2\Phi=-4\pi G\rho</math>
which contains [[Gauss's law for gravity]], and [[Poisson's equation#Newtonian gravity|Poisson's equation for gravity]]. Newton's law implies Gauss's law, but not vice versa; see ''[[Gauss's law for gravity#Relation to Newton's law|Relation between Gauss's and Newton's laws]]''.

These classical equations are [[differential equation|differential]] [[equations of motion]] for a [[test particle]] in the presence of a gravitational field, i.e. setting up and solving these equations allows the motion of a test mass to be determined and described.

The field around multiple particles is simply the [[Vector (geometry)#Addition and subtraction|vector sum]] of the fields around each individual particle. A test particle in such a field will experience a force that equals the vector sum of the forces that it would experience in these individual fields. This is<ref>{{cite book|title=Classical Mechanics |edition=2nd |first=T. W. B. |last=Kibble |series=European Physics Series |publisher=[[McGraw Hill]] |location=UK |date=1973 |isbn=978-0-07-084018-8}}{{page needed|date=October 2017}}</ref>
<math display="block">\mathbf{g}
= \sum_{i}\mathbf{g}_i
= \frac{1}{m}\sum_{i}\mathbf{F}_i
= - G\sum_{i}m_i\frac{\mathbf{R}-\mathbf{R}_i}{\left|\mathbf{R}-\mathbf{R}_i\right|^3}
= - \sum_{i}\nabla\Phi_i ,</math>
i.e. the gravitational field on mass {{math|''m<sub>j</sub>''}} is the sum of all gravitational fields due to all other masses ''m''<sub>''i''</sub>, except the mass {{math|''m<sub>j</sub>''}} itself. {{math|'''R'''<sub>''i''</sub>}} is the position vector of the gravitating particle {{math|''i''}}, and {{math|'''R'''}} is that of the test particle.