Jacobi integral

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In celestial mechanics, Jacobi's integral represents a solution to the circular restricted three-body problem of celestial mechanics.

Jacobi integral is the only known integral for the 3-body restricted problem; unlike in 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.

1 Definition
2 See also
3 References

[edit] Definition

[edit] Synodic system

Co-rotating system.

One of the suitable co-ordinates system used is so called synodic or co-rotating system, placed at the barycentre, with the line connecting the two masses μ₁, μ₂ 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 (-μ₂,0) and (+μ₁,0)¹.

In the co-ordinate system $x\,\!,y\,\!$ , the Jacobi constant is expressed as follows:

$C_J=n^2(x^2+y^2)+2\cdot (\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2})-(\dot x^2+\dot y^2+\dot z^2)$

where:

$n=\frac{2\pi}{T}$ is the mean motion (orbital period T)
$\mu_1=Gm_1\,\!,\mu_2=Gm_2\,\!$ , for the two masses m₁, m₂ and the gravitational constant G
$r_1\,\!,r_2\,\!$ are distances of the test particle from the two masses

Note that are the second term represents gravitational potential and the third $\dot x^2+\dot y^2+\dot z^2=v^2$ the kinetic energy (per unit mass).

[edit] Sidereal system

Inertial system.

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 :

$C_J=2 \cdot(\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2})+ 2n(\xi \dot \eta- \eta \dot \xi) - (\dot \xi ^2+\dot \eta ^2+\dot \zeta^2)$

[edit] Derivation

In the co-rotating system, the accelerations can be expressed as derivatives of a single scalar function $U(x,y,z)=\frac{n^2}{2}(x^2+y^2)+\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2}$

[Eq.1] $\ddot x - 2n\dot y = \frac{\delta U}{\delta x}$

[Eq.2] $\ddot y + 2n\dot x = \frac{\delta U}{\delta y}$

[Eq.3] $\ddot z = \frac{\delta U}{\delta z}$

Multiplying [Eq.1] , [Eq.2] and [Eq.3] par $\dot x, \dot y$ and $\dot z$ respectively and adding all three yields

$\dot x \ddot x+\dot y \ddot y +\dot z \ddot z = \frac{\delta U}{\delta x}\dot x + \frac{\delta U}{\delta y}\dot y + \frac{\delta U}{\delta z}\dot z = \frac{dU}{dt}$