Linear momentum conservation for interaction between rigid body and point mass

Question

Let's say I have an interaction potential between a rigid body and a point mass expressed in spherical coordinate : $V=V(r,\theta, \phi)$. The coordinates $r, \theta$ and $\phi$ specify the position of the point mass and are defined w.r.t. the origin of the rigid body-fixed frame. I want to compute the trajectory of these two bodies in the space-fixed frame (SFF) using $F = -\nabla V$ and integrating the equation of motion.

Because of the nature of the two interacting objects, I'm confused about momenta conservation in the SFF. One object is a point mass, so has only translational degrees of freedom, the other one is a rigid body, so has both translational and rotational degrees of freedom. When computing $\nabla V$, the three components of the gradient will yield translation in the SFF for the point mass. However, for the rigid body, although the dimension of the components of $\nabla V$ in spherical coordinates are consistent with a force (and so translational motion), I feel like it makes physical sense that only $\nabla_r V$ generate motion of the center of mass (translation) whereas $\nabla_\theta V$ and $\nabla_\phi V$ should yield a torque and generate rotation of the rigid body in the SFF.

What am I missing? if some components of the gradient yield force for one object but torque for the other object, how can linear and angular momenta be conserved? Does it even make sense to try to compute the motion of the body-fixed frame from which the potential is defined? Does it make this frame non inertial and invalidate Newton's laws of motion?

Thanks as lot!

Answer 1

Point masses cannot rotate, but they still have (and can exchange) angular momentum. The angular momentum of a moving point mass is non-zero when calculated about an axis that is not on the line of motion.

That means forces on a point mass can cause a change in angular momentum, so there's no reason not to consider them a torque. It's just not a torque when considered about an axis that goes through the point mass.

how can linear momentum be conserved if some components of ∇V translate into orbital angular momentum for the point (c.om. moving) but into spin for the body (c.o.m. not moving) ?

Forces don't "split" into linear and angular momentum. All of the force (all of the $\Delta V$) goes into changing the linear momentum of both. In addition, if the force is off-axis, it goes into changing the angular momentum of both.