research

three-body problem

Consider a system of three point masses $m_1 \geq m_2 \geq m_3$ moving in three dimensions, subject to their mutual gravity, e.g., Earth–Moon–spacecraft. This is the three-body problem (3BP). Famously, this system has no general closed-form solution; but there are some simplifications we can apply.

Suppose $m_3 \ll m_2$. Then the motion of $m_1$ and $m_2$ (the primaries) are unaffected by the motion of $m_3$. With this, the system can be parameterized with the parameter $\mu = \frac{m_2}{m_1+m_2}$ and becomes the restricted 3BP (R3BP).

The evolution of two masses under their mutual gravity is well understood to be the conic sections. If the primaries move in circular orbits, then we recover the circular R3BP (CR3BP), where—after some frame changes—the motion of $m_3$ is expressed by the equations $$\begin{aligned} \ddot{x} &= 2\dot{y} + x + \partial_x U, \\ \ddot{y} &= -2\dot{x} + y + \partial_y U, \\ \ddot{z} &= \partial_z U, \end{aligned}$$ where $U = \frac{(1-\mu)}{r_{13}} + \frac{\mu}{r_{23}}$ is the potential function, $r_{i3}$ is the distance from mass $i$ to $m_3$, and $m_1$ and $m_2$ are fixed at $(-\mu, 0, 0)$ and $(1-\mu, 0, 0)$, respectively.

The rich dynamical structure of the CR3BP manifests as periodic, quasi-periodic, and chaotic motion. My research focuses on understanding this structure.