@C_R The closed loop is in the standard 3D inertial frame defined by the usual i,j,k orthogonal axes.

The beads are constrained to slide along the loop and cannot exchange forces via the loop.

The mass of the loop, if there is one, is not relevant to the question.

DirectSearch is a downloadable package with multiple problem solving capabilities include many for which Maple's solve command does not provide an answer. For example equations which include integrals.

A DirectSearch help file is also downloadable.

@Rouben Rostamian  

Thank you so much for providing a much better approach to the problem than the original in my worksheet, as corrected by other responders.

I have copied your code and adjusted the bead's starting position, its starting velocity and the limits of animation and the adjusted code works perfectly.

  Regards...Earl

@Carl Love 

Your guess in point 5 is correct. There is only one dependent function, phi(t).

I have corrected the ode and ics arrording to C_R 3962 and Reuben Rostamian's suggestions and the original error message has been replaced by the one in dharr's response.

I will try to manually convert to a first-order system.

@dharr 

I have corrected the ode and ics according to C_R 3962's, Carl Love's and Reuben Rostamian's suggestions and the original error has disappeared, however the new error is the one shown in dharr's reply. I will try manually to convert to a first order system according to your suggestion using the correct initial conditions.

@C_R I have corrected the ode and ics according to your and Reuben Rostamian's suggestions and the original error has disappeared, however the new error is the one shown in dharr's reply. I will try manually to convert to a first order system according to his (her?) suggestion

@Rouben Rostamian  You have enlightened me by your insight, as you have done so many times!!

@janhardo With this simple change Maple2020 executes your worksheet without a problem.

I like your display of the magnetic field induced in the wire. Also, the 3D display is more enlightening that the 2D display.

@janhardo Thanks for your quick reply.

Unfortunately when I executed your downloaded worksheet only the opening text displayed. The commands producing the animation did not execute. I presume Maple2020 could not handle your implementation written in a later Maple version.

@dharr Thank you for explaining my programming error.

I moved the ode definition inside the MoveRod procedure and the animation worked perfectly

@dharr The Maple2020 help has given me a good understanding of this form of dsolve. Thank you.

@Rouben Rostamian  

@Rouben Rostamian  I downloaded your worksheet, made changes to adapt it to Maple2020, and executed it.

It is a beutiful answer to my question!

Obviously something is wrong in my question's downloaded worksheet in my interpretation into Maple2020 of the youtube's differential equations. Maybe I can determine my errors.

I deeply appreciate all of the answers you have sent me over the past few years. Thanks again.

@C_R These are interesting. They also solve the problem with a completely different math than in the youtube video reverenced in my worksheet.

I especially enjoyed the history of the problem outlined in the video included in one of your referrals.

@acer Rouben Rostamian's solution to a similar problem is very different from that in the referenced youtube video, but interesting none the less.

Can any other shape pass through a cube while removing more of the cube's volume?