# Linear momentum conservation for interaction between rigid body and point mass

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!

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.
• Thanks for the replies. I agree point masses can have angular momentum, orbital angular momentum in this case. As you said, it depends on the choice of axis. But regardless of this choice, the center of mass of the point is moving, right? This is not the case with intrinsic or spin angular momentum, which makes the body rotates around its center of mass. If this is right, my question would be : how can linear momentum be conserved if some components of $\nabla V$ translate into orbital angular momentum for the point (c.om. moving) but into spin for the body (c.o.m. not moving) ? Thanks! Aug 18 '18 at 16:08