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freddy

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 massmis constrained to move on the surface of a sphere of radiusRby an applied forceF(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/dt2theta- 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 :)