@John Fredsted Your solution was the one I expected from my command. I'll use your method in the future.

@vv I didn't realize that parametric had to be stated when the equations for solve, as in your example, include an implied condition.

It is interesting that to obtain a correct solution one must solve for all "variables" even though one of them, y in your example, already has a stated value. 

@Carl Love 

@Kitonum Please explain how your commands below cause Maple to determine the period. I particularly do not understand how f-1 represents an equation.

f:=s->eval(y(t),sol(s)):
T:=fsolve(f-1, 2..2.5);  # The period

@Kitonum I believe I now fully understand your excellent graphical application of my original pair of equations.

@Kitonum How do you know that solving Sys for y = 1/2 gives you the point on the circle for the initial value of the parameter t=0? 

@Carl Love  I was not aware of command membertype which greatly increases the scope of select/remove.

@Rouben Rostamian   I made an attempt to adapt your solution to my original surface and start and end points, however the following command failed:

dsol := dsolve({de, bc}, numeric, method=bvp[middefer], maxmesh=500, abserr=1e-1);

The error message said that dsolve could not meet the abserr criteria and suggested increasing maxmesh and/or abserr. I tried both but dsolve continued to fail.

I suspected that the values of Pi in z(x,y) were causing the problems with dsolve and changed these values to 3.14 and your solution then displayed what appeared to be the correct fastest path, however the following failure occurred when attempting to calculate the time for this path:

dsol := dsolve({sys}, numeric, method=bvp[middefer],
  maxmesh=500, abserr=1e-1);
Error, (in fproc) unable to store 'HFloat(722917.2413392215)*I' when datatype=float[8]

Comments?

@Rouben Rostamian  

@vv I also isolated u^2 but attempts with commands minimize and Optimization[Minimize] failed, apparently due to the presence of tan(alpha) and cos(alpha). Your substitution of polynomials for these has enlightened me.

@Rouben Rostamian  Trial and error with your solution shows that a value on the range 0.65059998845 < omega < 0.6506  has the combined disks rolling until theta = PI/2 and then stopping in this position. Does this seem reasonable?

Why is your value for g = 1 rather than 9.8?

JohnS's question has anticipated my next question. When the combined disks rise from the x axis do they rotate about their combined centre of gravity?

@Rouben Rostamian  Sorry for the delayed response, I was out of town. I will spend some time absorbing your solution and reply again with any supplementary questions.

@Preben Alsholm 

@Preben Alsholm Thank you for an enlightening solution.

Can your method also find the value of DthetaZero which yields the simultaneous values theta(t)=Pi and diff(theta(t),t)=0?

@rlopez Your remarks help clarify this powerful coding.