A central force is a force that acts along the line connecting a particle to a fixed point (or the center of mass in a two-body system), with a magnitude depending only on the distance from that point. Examples include gravitational force in planetary orbits and the electrostatic force in atomic systems. When analyzing motion under a central force in a rotating frame, the dynamics are modified by fictitious forces, such as the centrifugal force and Coriolis force, which arise due to the frame’s constant angular velocity. This article explores the dynamics of central force motion in rotating frames, focusing on the interplay of real and fictitious forces, conservation laws, and applications in classical mechanics, astrophysics, and engineering.
Dynamics of Central Force Motion
\( \mathbf{F} = f(r)\,\hat{\mathbf{r}} \),
where \( r \) is the radial distance and \( \hat{\mathbf{r}} \) the radial unit vector. If the force is conservative it is derivable from a potential \( V(r) \):
\( \mathbf{F} = -\nabla V(r) \). Examples include the Newtonian gravitational force
\( f(r) = -\dfrac{G M m}{r^2} \) and other inverse-square laws.In a frame rotating with constant angular velocity \( \mathbf{\Omega} \) the equation of motion for a particle of mass \( m \) becomes
m \frac{d^2 \mathbf{r}}{dt^2}
= \mathbf{F}
– m\,\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})
– 2m\,\mathbf{\Omega}\times\mathbf{v}.
\]
Here the terms on the right are, respectively, the real central force \( \mathbf{F} \),
the centrifugal acceleration \( -\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r}) \),
and the Coriolis acceleration \( -2\mathbf{\Omega}\times\mathbf{v} \).
The centrifugal term acts radially outward with typical magnitude \( \Omega^2 r_\perp \) (where \( r_\perp \) is perpendicular distance to the rotation axis),
while the Coriolis term is orthogonal to both \( \mathbf{\Omega} \) and the particle velocity \( \mathbf{v} \).
Table of Contents
Effective Potential in Rotating Frames
When the central force is potential-derived, the rotating-frame dynamics are conveniently captured by an effective potential.
Taking the rotation axis along \( \hat{\mathbf{z}} \) and restricting motion to the \(xy\)-plane (so \( r \) is the planar radius), the centrifugal contribution can be represented as
V_{\text{centrifugal}}(r) = -\tfrac{1}{2}\,m(\mathbf{\Omega}\times\mathbf{r})^2 = -\tfrac{1}{2}\, m \Omega^2 r^2.
\]
The total effective potential in the rotating frame is therefore
V_{\text{eff}}(r) = V(r) \;-\; \tfrac{1}{2}\, m \Omega^2 r^2.
\]
The radial equation of motion (including the centrifugal contribution and the usual centripetal term arising from angular motion) may be written as
m\ddot r = -\frac{\partial V_{\text{eff}}}{\partial r} + m r \dot\theta^2,
\]
where \( \dot\theta \) denotes the particle’s angular speed relative to the rotating frame.
The Coriolis force affects tangential motion and deflects trajectories, but it does not contribute directly to \( V_{\text{eff}} \) because it is velocity-dependent and perpendicular to \( \mathbf{v} \).
Conservation Laws
In an inertial frame central forces conserve both angular momentum
\( \mathbf{L} = m\,\mathbf{r}\times\mathbf{v} \) and mechanical energy (kinetic + potential).
In a rotating frame these conservation statements are modified:
- Angular momentum: The z-component of angular momentum in the rotating frame generally is not strictly conserved because the Coriolis force can exert a torque (it depends on velocity and \( \mathbf{\Omega} \)).
For certain symmetric motions (e.g., steady circular orbits in the rotating frame) one can define effective conserved angular quantities by accounting for the frame rotation. - Energy / Jacobi integral: A useful conserved scalar in many rotating-frame problems is the Jacobi integral:
\[
E_J = \tfrac{1}{2} m v^2 + V_{\text{eff}}(r),
\]where \( v \) is the speed measured in the rotating frame. The Jacobi integral includes the centrifugal potential but does not include the Coriolis term since Coriolis is non-conservative (velocity dependent).
Applications
Astrophysics: Binary Systems & Accretion Disks
For a binary star system one typically works in a frame co-rotating with the binary orbital motion. The effective potential (the Roche potential) combines gravitational potentials of both components and the centrifugal term:
V_{\text{eff}}(\mathbf r)
= -\dfrac{G M_1}{|\mathbf r-\mathbf r_1|}
-\dfrac{G M_2}{|\mathbf r-\mathbf r_2|}
-\tfrac{1}{2}\Omega^2 r_\perp^2.
\]
Lagrangian points arise as equilibrium points of this potential; the Coriolis force governs dynamics near these points and affects mass-transfer streams and disk stability.
Classical Mechanics: Orbital Motion
For satellites or test particles, analysis in a rotating frame aligned with orbital motion simplifies perturbation theory.
The centrifugal force offsets part of gravity for circular motion, while the Coriolis term governs transverse deviations and stability of near-circular orbits.
Engineering: Rotating Machinery
In centrifuges, planetary gears, or rotating blades, centrifugal acceleration dominates radial loading and separation effects; Coriolis forces influence trajectories of internally moving parts and can create oscillatory instabilities that must be designed for.
Fluid Dynamics: Rotating Fluids
A rotating fluid in equilibrium often adopts a parabolic free-surface shape because of balance between gravity and the centrifugal potential. The Coriolis force drives large-scale swirling patterns (e.g., in laboratory rotating tanks or planetary atmospheres).
Challenges and Considerations
Two major complexities when working in rotating frames are:
- The Coriolis force is velocity-dependent and non-conservative, producing non-intuitive trajectory deflections that complicate analytical solutions.
- The centrifugal potential modifies the effective potential landscape and can create new equilibrium points (minima or maxima) that change stability properties versus the inertial frame.
Numerical simulations should include both centrifugal and Coriolis terms explicitly — especially when angular speeds are large or trajectories cross large radial distances — and coordinate transforms between frames must be handled carefully to preserve physical quantities consistently.
Conclusion
Motion under a central force in a rotating frame is a rich problem that unifies ideas across classical mechanics, astrophysics, and engineering.
The centrifugal term modifies the effective potential and simplifies equilibrium/stability analysis, while the Coriolis force produces velocity-dependent deflections that shape trajectories and flows.
Conserved quantities like the Jacobi integral provide useful handles for understanding motion in the rotating frame. As computational tools improve, rotating-frame analyses continue to deepen our understanding of natural and engineered rotating systems.