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| In [[celestial mechanics]], '''Jacobi's integral''' (named after [[Carl Gustav Jacob Jacobi]]) is the only known conserved quantity for the [[circular restricted three-body problem]] problem <ref name=BnF>[http://visualiseur.bnf.fr/StatutConsulter?N=VERESS3-1201640420309&B=1&E=PDF&O=NUMM-90217 Bibliothèque nationale de France]. {{cite journal|last=Jacobi|first=Carl G. J.|title=Sur le movement d'un point et sur un cas particulier du problème des trois corps|journal=Comptes Rendus de l'Académie des Sciences de Paris|year=1836|volume=3|pages=59-61}}</ref>; unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible. The integral has been used to derive numerous solutions in special cases.
| | 58 year old Zoologist Dominic from Port Coquitlam, has interests for instance bird watching, property developers new condo in singapore - [http://www.gzza.com/zzss/blog/member.asp?action=view&memName=AntoniaLapine425597 Click To See More], singapore and tea tasting. Has finished a great around the world tour that covered visiting the Armenian Monastic Ensembles of Iran. |
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| == Definition ==
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| ===Synodic system===
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| [[Image:ThreeBodyProblem Synodic.png|thumb|right|Co-rotating system]]
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| One of the suitable coordinate systems used is the so-called ''synodic'' or co-rotating system, placed at the [[barycentre]], with the line connecting the two masses ''μ''<sub>1</sub>, ''μ''<sub>2</sub> chosen as ''x''-axis and the length unit equal to their distance. As the system co-rotates with the two masses, they remain '''stationary''' and positioned at (−''μ''<sub>2</sub>, 0) and (+''μ''<sub>1</sub>, 0)<sup>1</sup>.
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| In the (''x'', ''y'')-coordinate system, the Jacobi constant is expressed as follows:
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| :<math>C_J=n^2 (x^2+y^2) + 2 \left(\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2}\right) - \left(\dot x^2+\dot y^2+\dot z^2\right)</math>
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| where:
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| *<math>n=\frac{2\pi}{T}</math> is the [[mean motion]] ([[orbital period]] T)
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| *<math>\mu_1=Gm_1\,\!,\mu_2=Gm_2\,\!</math>, for the two masses ''m''<sub>1</sub>, ''m''<sub>2</sub> and the [[gravitational constant]] ''G''
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| *<math>r_1\,\!,r_2\,\!</math> are distances of the test particle from the two masses
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| Note that the Jacobi integral is minus twice the total energy per unit mass in the rotating frame of reference: the first term relates to [[Centrifugal force|centrifugal]] [[potential energy]], the second represents [[gravitational potential]] and the third is the [[kinetic energy]]. In this system of reference, the forces that act on the particle are the two gravitational attractions, the centrifugal force and the Coriolis force. Since the first three can be derived from potentials and the last one is perpendicular to the trajectory, they are all conservative, so the energy measured in this system of reference (and hence, the Jacobi integral) is a constant of motion. For a direct computational proof, see below.
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| ===Sidereal system===
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| [[Image:ThreeBodyProblem Sideral.png|x2thumb|right|Inertial system.]]
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| In the inertial, sidereal co-ordinate system (''ξ'', ''η'', ''ζ''), the masses are orbiting the [[barycentre]]. In these co-ordinates the Jacobi constant is expressed by:
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| :<math>C_J=2 \left(\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2}\right) + 2n\left(\xi \dot \eta- \eta \dot \xi\right) - \left(\dot \xi ^2+\dot \eta ^2+\dot \zeta^2\right).</math>
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| ===Derivation===
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| In the co-rotating system, the accelerations can be expressed as derivatives of a single scalar function
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| : <math>U(x,y,z)=\frac{n^2}{2}(x^2+y^2)+\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2}</math>
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| Using Lagrangian representation of the equations of motion:
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| [Eq.1] <math>\ddot x - 2n\dot y = \frac{\delta U}{\delta x}</math>
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| [Eq.2] <math>\ddot y + 2n\dot x = \frac{\delta U}{\delta y}</math>
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| [Eq.3] <math>\ddot z = \frac{\delta U}{\delta z}</math>
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| Multiplying [Eq.1], [Eq.2] and [Eq.3] by <math>\dot x, \dot y </math> and <math>\dot z </math> respectively and adding all three yields
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| : <math>\dot x \ddot x+\dot y \ddot y +\dot z \ddot z = \frac{\delta U}{\delta x}\dot x + \frac{\delta U}{\delta y}\dot y + \frac{\delta U}{\delta z}\dot z = \frac{dU}{dt} </math>
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| Integrating yields
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| : <math>\dot x^2+\dot y^2+\dot z^2=2U-C_J </math>
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| where ''C''<sub>''J''</sub> is the constant of integration.
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| The left side represents the square of the velocity ''v'' of the test particle in the co-rotating system.
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| <sup>1</sup><small>This co-ordinate system is [[a non-inertial reference frame|non-inertial]], which explains the appearance of terms related to [[centrifugal force|centrifugal]] and [[Coriolis force|Coriolis]] accelerations.</small>
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| == See also ==
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| *[[Rotating reference frame]]
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| *[[Tisserand's Criterion]]
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| ==Notes==
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| {{Reflist}}
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| ==Bibliography==
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| Carl D. Murray and Stanley F. Dermot ''Solar System Dynamics'' [Cambridge, England: Cambridge University Press, 1999], pages 68–71. (ISBN 0-521-57597-4)
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| [[Category:Orbits]]
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58 year old Zoologist Dominic from Port Coquitlam, has interests for instance bird watching, property developers new condo in singapore - Click To See More, singapore and tea tasting. Has finished a great around the world tour that covered visiting the Armenian Monastic Ensembles of Iran.