Editing Conservative vector field (section)

==Conservative forces==

[[File:Conservative fields.png|thumb|upright=1.5|Examples of potential and gradient fields in physics:{{unbulleted list
| {{color box|yellow}} Scalar fields, scalar potentials:{{unbulleted list
 | style=margin-left:1.6em;
 | '''V<sub>G</sub>''', gravitational potential
 | '''W<sub>pot</sub>''', (gravitational or electrostatic) potential energy
 | '''V<sub>C</sub>''', Coulomb potential
 }}| {{color box|cyan}} Vector fields, gradient fields:{{unbulleted  list
 | style=margin-left:1.6em;
 | '''a<sub>G</sub>''', gravitational acceleration
 | '''F''', (gravitational or electrostatic) force
 | '''E''', electric field strength
 }}}}
]]

If the vector field associated to a force <math>\mathbf{F}</math> is conservative, then the force is said to be a [[conservative force]].

The most prominent examples of conservative forces are gravitational force (associated with a gravitational field) and electric force (associated with an electrostatic field). According to [[Newton's law of universal gravitation|Newton's law of gravitation]], a [[gravitational force]] <math>\mathbf{F}_{G}</math> acting on a mass <math>m</math> due to a mass <math>M</math> located at a distance <math>r</math> from <math>m</math>, obeys the equation

<math display="block">\mathbf{F}_{G} = - \frac{G m M}{r^2} \hat{\mathbf{r}},</math>

where <math>G</math> is the [[gravitational constant]] and <math>\hat{\mathbf{r}}</math> is a ''unit'' vector pointing from <math>M</math> toward <math>m</math>. The force of gravity is conservative because <math>\mathbf{F}_{G} = - \nabla \Phi_{G}</math>, where

<math display="block">\Phi_{G} ~ \stackrel{\text{def}}{=} - \frac{G m M}{r}</math>

is the [[gravitational potential energy]]. In other words, the gravitation field <math>\frac{\mathbf{F}_{G}}{m}</math> associated with the gravitational force <math>\mathbf{F}_{G}</math> is the [[gradient]] of the gravitation potential <math>\frac{\Phi_{G}}{m}</math> associated with the gravitational potential energy <math>\Phi_{G}</math>. It can be shown that any vector field of the form <math>\mathbf{F}=F(r) \hat{\mathbf{r}}</math> is conservative, provided that <math>F(r)</math> is integrable.

For [[conservative force]]s, ''path independence'' can be interpreted to mean that the [[work done]] in going from a point <math>A</math> to a point <math>B</math> is independent of the moving path chosen (dependent on only the points <math>A</math> and <math>B</math>), and that the work <math>W</math> done in going around a simple closed loop <math>C</math> is <math>0</math>:

<math display="block">W = \oint_{C} \mathbf{F} \cdot d{\mathbf{r}} = 0.</math>

The total [[conservation of energy|energy]] of a particle moving under the influence of conservative forces is conserved, in the sense that a loss of potential energy is converted to the equal quantity of kinetic energy, or vice versa.