…in rotating coordinates
Translation of homework done for the course Computational Astrodynamics.^{1}
The coordinates of inertial and rotating reference frame are connected
\[ (\xi, \eta) = R (t) (x, y) = x (\cos t, \sin t) + y ( \sin t, \cos t) \]
\[ (x, y) = R^{ 1} (t) (\xi, \eta) = \xi (\cos t,  \sin t) + \eta (\sin t, \cos t) \]
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 nondimensional 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 gravitational potential is written
\[ V =  \frac{1  \mu}{r_{1}}  \frac{\mu}{r_{2}} \]
and then the Lagrangian is written
\[ L = \frac{1}{2} (\dot{\xi}^2 + \dot{\eta}^2)  V \]
Utilizing the conversion 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 Lagrangian 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
\[ H = \frac{1}{2} (p_{x}^2 + p_{y}^2) + p_{x} y  x p_{y}  \frac{1  \mu}{r_{1}}  \frac{\mu}{r_{2}} \]

Post date reflects original date; translation done 06/04/2021. ↩