Angular Momentum 10-9-2020

Two particles of masses $m_1$ and $m_2$ are in the $xy$ plane. The particles are free to move under the influence of their mutual gravitation. At time $t = 0$ , their positions and velocities are:

$(x_1, y_1 ) = (0,0) \\ (x_2, y_2 ) = (1,0) \\ (\dot{x}_1, \dot{y}_1) = (1,0) \\ (\dot{x}_2, \dot{y}_2) = (-0.5,1)$

Let $\vec{P}_1 = (x_1,y_1)$ and $\vec{P}_2 = (x_2,y_2)$ be the positions of the particles, and let $\vec{P}$ be a reference point. Define two displacement vectors:

$\vec{r}_1 = \vec{P}_1 - \vec{P} \\ \vec{r}_2 = \vec{P}_2 - \vec{P}$

Let the velocities of the particles be $\vec{v}_1 = (\dot{x}_1, \dot{y}_1)$ and $\vec{v}_2 = (\dot{x}_2, \dot{y}_2)$ . The magnitudes of the angular momenta of the particles are:

$L_1 = |\vec{r}_1 \times m_1 \, \vec{v}_1| \\ L_2 = |\vec{r}_2 \times m_2 \, \vec{v}_2|$

The quantity $L_1$ has a local maximum between $t = 0$ and $t = 1$ . What is its value?

Details and Assumptions:
1) $m_1 = m_2 = 1$
2) Universal gravitational constant $G = 1$
3) $\vec{P} = (P_x, P_y) = (0,3)$
4) $|\cdot|$ denotes the magnitude of a vector, and $\times$ denotes the vector cross product