# Acceleration on the surface of a sphere

freddy
Hello everyone!

I need some help with problem 2.2 out of Classical Dynamics of Particles and Systems by Thornton & Marion. I can't believe I got stuck on the second problem The text of the problem is:

A particle of mass m is constrained to move on the surface of a sphere of radius R by an applied force F(theta, phi). Write the equation of motion.

Seemed simple to me at first, F_theta = m * d2/dt2 theta and I'm done. I figure since the surface of a sphere is a two dimensional surface and thetahat and phihat are perpindicular to eachother I should be able to just apply forces in either thetahat or phihat and expect the particle to stay on the surface of the sphere. Looking at the answer in the back of the book I can see that I'm very wrong:

F_theta = m*R*(d2/dt2 theta - d/dt(phi^2)*sin(theta)*cos(theta))
F_phi = m*R*(2*d/dt(theta) * d/dt(phi) * cos(theta) + d2/dt2(phi) * sin(theta)

After having given it some thought I think I understand why the two accelerations must be dependent upon eachother, but I still don't know how I'm supposed to work it mathematically.

The accelerations must depend on one another because of how phi and theta are set up, almost anywhere on the sphere if I want to travel solely through either phi or theta I'm not actually traveling in a straight line; a straight line on a sphere is a line that bisects the sphere. So, in general, if I have a particle on the surface of a sphere and want it to travel in only one of theta or phi I must exert a force in both in order to keep it on track.

Is my reasoning correct? If not, what's wrong with it, and if so, how the heck am I supposed to get there mathematically?

Thanks in advance for any help :)

arildno
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You need a vector representation of your surface in order to manage this.
In addition, you'll need to know the unit vectors in spherical coordinates.

Any point on the sphere with radius R may be given the vector representation:
$$R(\sin\gamma\cos\psi\vec{i}+\sin\gamma\sin\psi\vec{j}+\cos\gamma\vec{k})$$
$$0\leq\psi\leq2\pi,0\leq\gamma\leq\pi$$
(You may find out how $$\gamma,\psi$$ are related to the used angle variables $$\theta,\phi$$)

The unit vectors are:
$$\vec{i}_{r}= (\sin\gamma\cos\psi\vec{i}+\sin\gamma\sin\psi\vec{j}+\cos\gamma\vec{k})$$
$$\vec{i}_{\gamma}=\frac{\partial\vec{i}_{r}}{\partial\gamma}=(\cos\gamma\cos\psi\vec{i}+\sin\gamma\sin\psi\vec{j}-\sin\gamma\vec{k})$$
$$\vec{i}_{\psi}=\frac{1}{\sin\gamma}\frac{\partial\vec{i}_{r}}{\partial\psi}=-\sin\psi\vec{i}+\cos\psi\vec{j}$$

Clearly, the position of the particle with respect to time can be written as:
$$\vec{r}(t)=R\vec{i}_{r}(t)$$
with angular functions of time $$\gamma(t),\psi(t)$$
Use the chain rule to determine the acceleration components.