Coriolis Theorem (Transport Theorem)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

The Coriolis theorem (a.k.a. transport theorem) relates the time derivative of any vector in an inertial frame $\mathcal I$ to its derivative in a frame $\mathcal R$ that rotates with angular velocity $\vec\omega$ relative to $\mathcal I$ . For any vector $\vec q$ ,

\left(\frac{\mathrm d\vec q}{\mathrm dt}\right)_{\!\mathcal I} = \left(\frac{\mathrm d\vec q}{\mathrm dt}\right)_{\!\mathcal R} + \vec\omega\times\vec q.

$\vec\omega$ is the instantaneous angular velocity of $\mathcal R$ relative to $\mathcal I$ . The vector itself is frame-independent, but its components and the operation "differentiate with respect to time" both depend on the frame — the Coriolis theorem is the transformation rule for the latter. It is the foundation on which equations of motion are derived in every rotating reference frame used in astrodynamics: the synodic frame, the Moon-fixed frame, the Earth-fixed frame, and so on.

Sketch of Derivation

Let $\hat e_1, \hat e_2, \hat e_3$ be the unit basis vectors of $\mathcal R$ , so $\vec q=q_1\hat e_1+q_2\hat e_2+q_3\hat e_3$ . Differentiating in $\mathcal I$ :

\left(\frac{\mathrm d\vec q}{\mathrm dt}\right)_{\!\mathcal I} = \sum_i \dot q_i\hat e_i + \sum_i q_i\!\left(\frac{\mathrm d\hat e_i}{\mathrm dt}\right)_{\!\mathcal I}.

The first sum is the derivative in $\mathcal R$ (where $\hat e_i$ are constants); in the second sum the basis vectors rotate in $\mathcal I$ with $\mathrm d\hat e_i/\mathrm dt=\vec\omega\times\hat e_i$ . Substitution gives the Coriolis theorem.

Two Applications: Coriolis and Centrifugal Accelerations

Apply the theorem to the position vector $\vec r$ of a body at the origin of $\mathcal R$ . The first application gives the velocity transformation

\vec v_I = \vec v_R + \vec\omega\times\vec r,

where $\vec v_I, \vec v_R$ are the time derivatives of $\vec r$ in $\mathcal I, \mathcal R$ . Differentiating once more (allowing $\vec\omega$ to vary in time) and rearranging gives the acceleration transformation:

\vec a_I = \vec a_R + 2\,\vec\omega\times\vec v_R + \dot{\vec\omega}\times\vec r + \vec\omega\times(\vec\omega\times\vec r).

The three extra terms on the right are:

Coriolis acceleration $2\vec\omega\times\vec v_R$ : present only when the body moves relative to $\mathcal R$ ; direction perpendicular to $\vec v_R$ .
Euler acceleration $\dot{\vec\omega}\times\vec r$ : present only when the frame's angular velocity changes; vanishes for uniform rotation.
Centrifugal acceleration $\vec\omega\times(\vec\omega\times\vec r)$ : always directed outward from the rotation axis; magnitude $\omega^2\rho$ , where $\rho$ is the distance to the axis.

Application to the CR3BP Synodic Frame

The standard derivation of the circular restricted three-body problem is a direct application of the Coriolis theorem (Szebehely 1967, §1.5; Vallado 2022, §1.5). In the synodic frame the two primaries are fixed and the frame rotates with their orbital angular velocity $\vec\omega=n\hat z$ . Moving Newton's gravity $\vec a_I=-\mu_1\vec r_1/r_1^3-\mu_2\vec r_2/r_2^3$ into the rotating frame gives

\ddot{\vec r}+2\vec\omega\times\dot{\vec r}+\vec\omega\times(\vec\omega\times\vec r) = -\frac{\mu_1}{r_1^3}\vec r_1-\frac{\mu_2}{r_2^3}\vec r_2,

or equivalently

\ddot{\vec r} = -\nabla\Omega - 2\vec\omega\times\dot{\vec r},\qquad \Omega = \frac{\mu_1}{r_1}+\frac{\mu_2}{r_2}+\tfrac{1}{2}\omega^2(x^2+y^2).

Once $\Omega$ is written as a single scalar "two-body gravity + centrifugal" potential, the Coriolis term $-2\vec\omega\times\dot{\vec r}$ is a gyroscopic force (does no work, perpendicular to velocity), from which the Jacobi integral follows. This is the mechanical origin of the "Coriolis + centrifugal" terms in the synodic-frame equations.

Application to Body-Fixed Frames

Lunar soft landing, Earth re-entry and similar missions require dynamics in body-fixed frames co-rotating with the body. For the Moon-fixed frame (Zhou & Zhou 2007):

\vec V_I = \vec V_L + \vec\omega_M\times\vec R,

with $\vec V_I$ the inertial velocity, $\vec V_L$ the Moon-fixed velocity, $\vec\omega_M$ the lunar rotation rate, and $\vec R$ the position vector. Differentiating and applying the Coriolis theorem gives the relative equations of motion containing Coriolis $2\vec\omega_M\times\vec V_L$ and centrifugal $\vec\omega_M\times(\vec\omega_M\times\vec R)$ terms; adding lunar non-spherical gravity and control forces yields the precision dynamics model for lunar soft landing. This is the basis of engineering-level landing guidance laws.

Common Pitfalls

"Does the Coriolis force do work?" No. The term $2\vec\omega\times\vec v_R$ is always perpendicular to $\vec v_R$ and contributes nothing to the energy integral — one of the reasons the Jacobi integral is conserved.
"Is Coriolis acceleration the 'cost' of relative velocity?" Yes. It is the inertial observer's view of relative motion being "dragged along" by the rotating frame.
Everyday "Coriolis force": rivers scouring one bank, Foucault pendulum precession, etc. These are the result of multiplying the acceleration term by mass and treating it as a "pseudo-force"; strictly it is a kinematic effect, not a physical force.
Frame of $\vec\omega$ : in the Coriolis theorem, $\vec\omega$ is the angular velocity of $\mathcal R$ relative to $\mathcal I$ , not of the body relative to $\mathcal R$ .

References

Szebehely, V. (1967). Theory of Orbits: The Restricted Problem of Three Bodies, §1.5. Academic Press.
Vallado, D. A. (2022). Fundamentals of Astrodynamics and Applications, 5th ed., §1.5 (rotating frames and the transport theorem).
Goldstein, H., Poole, C. P., & Safko, J. L. (2002). Classical Mechanics, 3rd ed., Chapter 4. Addison-Wesley.
周净扬, 周荻 (2007). 月球探测器软着陆精确建模及最优轨道设计. 宇航学报.