# Physical importance of centrifugal force and Coriolis force in the context of the restricted 3-body problem

In the textbook Galactic Dynamics (Binney and Tremaine), the restricted three-body problem to find the trajectory of a test mass that orbits in the combined gravitational field of two masses $$M$$ and $$m$$ is discussed in Sec. 8.3.1. The two masses orbit their common center-of-mass with angular velocity $$\Omega=\sqrt{\frac{G(M+m)}{R_0^3}}$$ where $$R_0$$ is the separation between the masses and $$G$$ is the gravitational constant.

Now, a coordinate system is chosen to be centered at the center-of-mass and that rotates at angular velocity $$\Omega$$, so that the gravitational field is stationary in that coordinate system. The two-dimensional rotating potential (discussed in Sec. 3.3.2) is used to study the orbits of the test particle.

The conserved Jacobi integral in the rotating frame can be written as $$E_J=\frac{1}{2}|\dot{\mathbf{x}}|^2+\Phi-\frac{1}{2}|\mathbf{\Omega}\times\mathbf{x}|^2=\frac{1}{2}|\dot{\mathbf{x}}|^2+\Phi_\mathscr{eff}$$

where the effective potential $$\Phi_\mathscr{eff}$$ is given by

$$\Phi_\mathscr{eff}(\mathbf{x})=\Phi(\mathbf{x})-\frac{1}{2}|\mathbf{\Omega}\times\mathbf{x}|^2=\Phi(\mathbf{x})-\frac{1}{2}\left[|\mathbf{\Omega}|^2|\mathbf{x}|^2-(\mathbf{\Omega}\cdot\mathbf{x})^2\right]$$

Here, we can easily observe that the effective potential is the sum of the gravitational and a repulsive centrifugal potential.

Then, we can obtain the following Hamilton's equations: $$\mathbf{\dot{p}}=-\nabla\Phi-(\mathbf{\Omega}\times\mathbf{p})$$ $$\mathbf{\dot{x}}=\mathbf{p}-(\mathbf{\Omega}\times\mathbf{x})$$

Eliminating $$\mathbf{p}$$ between these equations we find $$\mathbf{\ddot{x}}=-\nabla\Phi-\left[2\mathbf{\Omega}\times\mathbf{\dot{x}}\right]-\left[\mathbf{\Omega\times(\Omega\times x)}\right]$$

From this equation, we define

Coriolis force: $$\mathbf{F}_\mathscr{cor}=-2\mathbf{\Omega}\times\mathbf{\dot{x}}$$

Centrifugal force: $$\mathbf{F}_\mathscr{cf}=-\mathbf{\Omega\times(\Omega\times x)}$$

The following are my questions associated with the problem:

Question 1: The effective potential (as written above) is a sum of the gravitational potential and a repulsive centrifugal potential. There is no term that corresponds to the Coriolis force. However, the Coriolis force appears in the equations of motion that depends on $$\mathbf{\dot{x}}$$. So I think that the Coriolis term arises naturally due to the behavior of the Hamilton's equation rather than the exact form of $$\Phi_{\mathscr{eff}}$$. Is this correct or am I wrong?

Question 2: Since both centrifugal and Coriolis forces appear in the equations of motion obtained from the effective potential $$\Phi_\mathscr{eff}$$, both these forces should affect the test particle motion in the restricted three-body problem. I am interested to know how these two forces affect the test particle motion at different Lagrangian points (in particular, $$L_1$$, that lies midway between the two masses and is important in modelling accretion disks). In most textbooks, only the effect of the centrifugal force is considered.

NOTE: My question is different from the following other questions on Physics.SE: How are the Lagrange points determined? , Why are $L_4$ and $L_5$ lagrangian points stable? , Why are the Lagrangian points $L_1$, $L_2$ & $L_3$ unstable?