- Kepler problem
In

classical mechanics ,**Kepler’s problem**is a special case of thetwo-body problem , in which the two bodies interact by acentral force **F**that varies in strength as the inverse square of the distance "r" between them. The force may be either attractive or repulsive. The "problem" to be solved is to find the position or speed of the two bodies over time given their masses and initial positions and velocities. Using classical mechanics, the solution can be expressed as aKepler orbit using sixorbital elements .The Kepler problem is named after

Johannes Kepler , who proposedKepler's laws of planetary motion (which are part ofclassical mechanics and solve the problem for the orbits of the planets) and investigated the types of forces that would result in orbits obeying those laws (called "Kepler's inverse problem").cite book | last=Goldstein | first=H. | authorlink=Herbert Goldstein | year=1980 | title=Classical Mechanics | edition=2^{nd}edition | publisher=Addison Wesley]**Applications**The Kepler problem arises in many contexts, some beyond the physics studied by Kepler himself. The Kepler problem is important in

celestial mechanics , since Newtonian gravity obeys aninverse square law . Examples include a satellite moving about a planet, a planet about its sun, or two binary stars about each other. The Kepler problem is also important in the motion of two charged particles, sinceCoulomb’s law ofelectrostatics also obeys aninverse square law . Examples include thehydrogen atom,positronium andmuonium , which have all played important roles as model systems for testing physical theories and measuring constants of nature.Facts|date=August 2008The Kepler problem and the

simple harmonic oscillator problem are the two most fundamental problems inclassical mechanics . They are the "only" two problems that have closed orbits, i.e., return to their starting point with the same velocity (Bertrand's theorem ). The Kepler problem has often been used to develop new methods in classical mechanics, such asLagrangian mechanics ,Hamiltonian mechanics , theHamilton-Jacobi equation , andaction-angle coordinates .Facts|date=August 2008 The Kepler problem also conserves theLaplace-Runge-Lenz vector , which has since been generalized to include other interactions. The solution of the Kepler problem allowed scientists to show that planetary motion could be explained entirely by classical mechanics and Newton’s law of gravity; the scientific explanation of planetary motion played an important role in ushering in the Enlightenment.The

Kepler problem in general relativity can be solved to produce more accurate predictions than by using classical mechanics, especially in the case of strong gravitational fields.**Mathematical definition**The

central force **F**that varies in strength as the inverse square of the distance "r" between them::$mathbf\{F\}\; =\; frac\{k\}\{r^\{2\; mathbf\{hat\{r$

where "k" is a constant and $mathbf\{hat\{r$ represents the

unit vector along the line between them. [*cite book | last = Arnold | first = VI | authorlink = Vladimir Arnold | year = 1989 | title = Mathematical Methods of Classical Mechanics, 2nd ed. | publisher = Springer-Verlag | location = New York | pages = 38 | id = ISBN 0-387-96890-3*] The force may be either attractive ("k"<0) or repulsive ("k">0). The correspondingscalar potential (thepotential energy of the non-central body) is::$V(r)\; =\; frac\{k\}\{r\}$

**olution of the Kepler problem**The equation of motion for the radius $r$ of a particle of mass $m$ moving in a central potential $V(r)$ is given by Lagrange's equations

:$mfrac\{d^\{2\}r\}\{dt^\{2\; -\; mr\; omega^\{2\}\; =\; mfrac\{d^\{2\}r\}\{dt^\{2\; -\; frac\{L^\{2\{mr^\{3\; =\; -frac\{dV\}\{dr\}$

where $omega\; equiv\; frac\{d\; heta\}\{dt\}$ and the

angular momentum $L\; =\; mr^\{2\}omega$ is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force $frac\{dV\}\{dr\}$ equals the centripetal force requirement $mr\; omega^\{2\}$, as expected.The definition of

angular momentum allows a change of independent variable from $t$ to $heta$:$frac\{d\}\{dt\}\; =\; frac\{L\}\{mr^\{2\; frac\{d\}\{d\; heta\}$

giving the new equation of motion that is independent of time

:$frac\{L\}\{r^\{2\; frac\{d\}\{d\; heta\}\; left(\; frac\{L\}\{mr^\{2\; frac\{dr\}\{d\; heta\}\; ight)-\; frac\{L^\{2\{mr^\{3\; =\; -frac\{dV\}\{dr\}$

This equation becomes quasilinear on making the change of variables $u\; equiv\; frac\{1\}\{r\}$ and multiplying both sides by $frac\{mr^\{2\{L^\{2$

:$frac\{d^\{2\}u\}\{d\; heta^\{2\; +\; u\; =\; -frac\{m\}\{L^\{2\; frac\{d\}\{du\}\; V(1/u)$

For an inverse-square force law such as the gravitational or electrostatic potential, the

potential can be written:$V(mathbf\{r\})\; =\; frac\{k\}\{r\}\; =\; ku$ The orbit $u(\; heta)$ can be derived from the general equation

:$frac\{d^\{2\}u\}\{d\; heta^\{2\; +\; u\; =\; -frac\{m\}\{L^\{2\; frac\{d\}\{du\}\; V(1/u)\; =\; -frac\{km\}\{L^\{2$

whose solution is the constant $-frac\{km\}\{L^\{2$ plus a simple sinusoid

:$u\; equiv\; frac\{1\}\{r\}\; =\; -frac\{km\}\{L^\{2\; left\; [\; 1\; +\; e\; cos\; left(\; heta\; -\; heta\_\{0\}\; ight)\; ight]$

where $e$ (the

**eccentricity**) and $heta\_\{0\}$ (the**phase offset**) are constants of integration.This is the general formula for a

conic section that has one focus at the origin; $e=0$ corresponds to acircle , $e<1$ corresponds to an ellipse, $e=1$ corresponds to aparabola , and $e>1$ corresponds to ahyperbola . The eccentricity $e$ is related to the totalenergy $E$ (cf. theLaplace-Runge-Lenz vector ):$e\; =\; sqrt\{1\; +\; frac\{2EL^\{2\{k^\{2\}m$

Comparing these formulae shows that $E<0$ corresponds to an ellipse (all solutions which are closed orbits are ellipses), $E=0$ corresponds to a

parabola , and $E>0$ corresponds to ahyperbola . In particular, $E=-frac\{k^\{2\}m\}\{2L^\{2$ for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius).For a repulsive force ("k">0) only $e>1$ applies.

**ee also***

Kepler orbit

*Kepler's laws of planetary motion

*Kepler problem in general relativity

*Laplace-Runge-Lenz vector

*Bertrand's theorem

*Hamilton-Jacobi equation

*Action-angle coordinates **References**

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