Caution

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Dynamics

Canonical dynamics

In their canonical form, the equations generated by the Hamiltonian \(H(x,y,z,p_x,p_y,p_z)\) are written as follows.

\[\begin{split}\dot{x} &= p_x + y \\ \dot{y} &= p_y - x \\ \dot{z} &= p_z \\ \dot{p}_x &= +p_y - \frac{(1-\mu)(x+\mu)}{r_1(x,y,z)^3} - \frac{\mu (x + \mu - 1)}{r_2(x,y,z)^3} \\ \dot{p}_y &= -p_x - \frac{(1-\mu)y}{r_1(x,y,z)^3} - \frac{\mu y}{r_2(x,y,z)^3} \\ \dot{p}_z &= -\frac{(1-\mu)z}{r_1(x,y,z)^3} - \frac{\mu z}{r_2(x,y,z)^3}\end{split}\]

Note

Integrating in canonical coordinates preserves the Hamiltonian better than other coordinates because the canonical variables are symplectic.

Position/velocity dynamics

In their position/velocity form, the equations of motion using the state \((x,y,z,\dot{x},\dot{y},\dot{z})\) in second-order form are written as follows.

\[\begin{split}\ddot{x} &= +2\dot{y} + x - \frac{(1-\mu)(x+\mu)}{r_1(x,y,z)^3} - \frac{\mu (x + \mu - 1)}{r_2(x,y,z)^3} \\ \ddot{y} &= -2\dot{x} + y - \frac{(1-\mu)y}{r_1(x,y,z)^3} - \frac{\mu y}{r_2(x,y,z)^3} \\ \ddot{z} &= -\frac{(1-\mu)z}{r_1(x,y,z)^3} - \frac{\mu z}{r_2(x,y,z)^3}\end{split}\]