## Force on a particle constrained to move on the surface of a sphere

 Quote by Vanadium 50 PF has a problem where people try and make things more complicated than they have to be. This is Chapter 2 in Marion, for heaven's sake, and it should be answered at a level appropriate to Chapter 2 in Marion. In this formulation, the constraining force is not part of the problem, and there is no net force in the r-direction, because r is constant. Yes, one can formulate this problem in other ways, some of which can provide additional answers - but the question is about understanding things at the level of the beginning of Marion.
Agree with the first paragraph, disagree with the second. As noted before, there is a radial component of the acceleration vector, equal to $-r \left(\theta '^2+\sin ^2(\theta ) \phi '^2\right)$ and so there is a radial component to the net force on the particle. Motion on the sphere implies a radial component of the force on the particle.

So what is the answer to the OP? The force on the particle is:$$F_r(\theta,\phi)=-mr \left(\theta '^2+\sin ^2(\theta ) \phi '^2\right)$$$$F_\theta(\theta,\phi)=mr(-\cos(\theta)\sin(\theta)\phi'^2+\theta'')$$$$F_\phi(\theta,\phi)=mr(2\cos(\theta)\theta'\phi'+\sin(\theta)\phi'')$$This is three equations in three unknowns ($F_r,\theta,\phi$), so solve the last two for $\theta$ and $\phi$ (given initial conditions) to get the equations of motion, then solve the first for $F_r$, if you feel like it.

That's a big job, but the above is brute force. Maybe there is a more elegant way? I think solving it by the Lagrangian method will be tricky, since $F_\theta$ and $F_\phi$ might not be described by a potential function.

Maybe the bottom line is that it's a much more complicated question than Thornton and Marion thought.

Recognitions:
 Quote by Vanadium 50 But this is not orbital mechanics. This is Chapter 2 in Marion and it should be answered at a level appropriate to Chapter 2 in Marion. We should try and make this concept clear, and not drown the OP in advanced "yeahbuts".
That doesn't mean you can claim that radial acceleration is zero when it's not. It means you can claim that radial acceleration is irrelevant.

Recognitions:
Homework Help
 Quote by Vanadium 50 In this formulation, the constraining force is not part of the problem, and there is no net force in the r-direction, because r is constant.
K^2 is right. There is a net force in the radial direction, but it is not relevant to the problem. Saying there is no net force is not making things easier for beginners - it just creates confusion.

 Quote by K^2 That doesn't mean you can claim that radial acceleration is zero when it's not. It means you can claim that radial acceleration is irrelevant.
 Quote by haruspex K^2 is right. There is a net force in the radial direction, but it is not relevant to the problem. Saying there is no net force is not making things easier for beginners - it just creates confusion.
Yes.

 Tags classical dynamics, force, marion, thornton