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Hamiltonian of planar circular restricted 3-body problem

…in rotating coordinates

Translation of homework done for the course Computational Astrodynamics.1

The coordinates of inertial and rotating rotating reference frame are connected

\[(\xi, \eta) = R(t)(t) (x, y) = x(\cos t, \sin t) + y(-\sin t, \cos t)\] \[(x, y) = R^{-1}(t) (\xi, \eta) = \xi(\cos t, -\sin t) + \eta(\sin t, \cos t)\]

and

\[(\dot{\xi}, \dot{\eta}) = (\dot{x} - y, \dot{y} + x) \cos t - (\dot{y} + x, y - \dot{x}) \sin t\] \[(\dot{x}, \dot{y}) = (\dot{\xi} + \eta, \dot{\eta} - \xi) \cos t - (\xi - \dot{\eta}, \dot{\xi} + \eta) \sin t\]

For the large bodies, from normalization (usage of non-dimensional coordinates)

\[x_2 - x_1 = 1\]

and from the conservation of momentum (constant velocity of the center of mass)

\[(1 - \mu) x_1 + \mu x_2 = 0\]

Solving the system

\[(x_1, x_2) = (-\mu, 1 - \mu)\]

In the planar circular restricted problem the orbits of them are

\[(\xi_1, \eta_1) = x_1 (\cos t, \sin t) \quad (\xi_2, \eta_2) = x_2 (\cos t, \sin t)\]

and the distances of the small body from the previous is

\[r_1^2 = (\xi + \mu \cos t)^2 + (\eta + \mu \sin t)^2\] \[r_2^2 = (\xi - (1 - \mu) \cos t)^2 + (\eta - (1 - \mu) \sin t)^2\]

Utilizing the conversion relationships the distances are written

\[r_1^2 = (x + \mu)^2 + y^2\] \[r_2^2 = (x - (1 - \mu))^2 + y^2\]

The gravitional potential is written

\[V = -\frac{1 - \mu}{r_1} - \frac{\mu}{r_2}\]

and then the Langrangian is written

\[L = \frac{1}{2}(\dot{\xi}^2 + \dot{\eta}^2) - V\]

Utilizing the convertion relationships, it is written

\[L = \frac{1}{2}((\dot{x} - y)^2 + (\dot{y} + x)^2) - V\]

The momenta are calculated differentiating this to the generalized velocities

\[(p_x, p_y) = (\partial_\dot{x}, \partial_\dot{y}) L = (\dot{x} - y, \dot{y} + x)\]

And therefore the Langrangian is written

\[L = \frac{1}{2}(p_x^2 + p_y^2) - V\]

The Hamiltonian is found from Legendre transformation

\[H = \dot{x} p_x + \dot{y} p_y - L\]

Replacing and adding/subtracting the kinetical energy

\[H = \frac{1}{2}(p_x^2 + p_y^2) + p_x (\dot{x} - p_x) + (\dot{y} - p_y)p_y + V\]

Finally

\[H = \frac{1}{2}(p_x^2 + p_y^2) + p_x y - x p_y - \frac{1 - \mu}{r_1} - \frac{\mu}{r_2}\]
  1. Post date reflects original date; translation done 06/04/2021.