@Kitonum Your procedure is the best way to generate any similar chain of tangent circles. Thank you for this.

@Rouben Rostamian  Thank you, Mr. Rostamian

@Kitonum Your answer implies that the formula in the cited book of the relationship between circle diameters as the key to forming the sequence of smaller, tangent circles can be replaced by your formula for the parabola, once the diameter of the original large circle is set. A wonderful discovery!

@Kitonum Thank you for discovering my error. I will try your finding with a few very different sized large circles.

@Rouben Rostamian  Once I feel familiar with the techniques you describe for the movable outer circle, I will have a go at solving the fixed outer circle solution.

@Rouben Rostamian  I will spend some time studying your replies in detail and will undoubtedly learn a great deal from you as always.

By the way. The first of the two problems, 12.25, stated that the larger circle is fixed to the line (or plane) on which it sits. Can your reply accommodate this fact?

@dharr Your answer is a step forward.

I also have animated the birds' flight, and have run into a puzzle.

My worksheet below displays the birds' spiral flight using spacecurve which functions as expected for t values from zero to one even thought the ode halts functioning at a t value just below 0.85.

The vectors showing the instantaneous directions of flight display with the plots,arrow command which halts functioning at the same t value as the ode.

What explains this contradictory behaviour between these two commands?

Birds_flight.mw

@Rouben Rostamian  I am again most grateful for you for your rapid response to my question. I will research the Googol websites on variational calculus and will acquire the reference therein which you mention.  The calculus of variations is a fascinating topic and I am keen to know more of it and become more proficient in using it in Maple.

@Rouben Rostamian  

Please tell me if any questions I want to ask you regarding your reply should be expressed in a further reply such as this one, in a comment, or in the option to contact you.

Here is my first question:

I am familiar with basic use of EulerLagrange techniques but do not know why your Lagrangian includes this element in it:

+ lambda*(f(x(t),y(t))-z(t));

Please explain this to me.

@mmcdara 

Thanks for pointing out all the possible subtleties in this scenario. I will appreciate if you can point me towards any Maple worksheets that delve into these ( I don't know Maple Sim)

@Rouben Rostamian  

I much appreciate your quick response! I will study it and learn from it and likely have a few questions.

@acer Thanks for your quick response.

Much of your code is unfamiliar to me, but I'm going to work at understanding it over the next day or two.

Could complex techniques provide an answer?

@acer Between this reply of yours and the previous one, as a blind stab, I replaced the faulty libname command with one from an operating worksheet and the problem disappeared.

I admire your perceptiveness in finding my problem!

All that's left is to wonder at the resulting error messages, which give no hint of their cause.

Thank you for your timely help!

@acer 

Thank you for your quick reply.

### I just replaced the libname command in the "troubled" worksheet with apparently the same command from a previously working worksheet and the problem no longer occurs. ###

### Somehow I corrupted the libname command in the problem worksheet and replacing it fixed the problem. ##

I have many operating worksheets beginning with the same libname command so I don't understand what you mean by "augment libname" .

Does your phrase "commands out of order" mean that all procedures do not precede the commands invoking them?

This is because I have modified large sections of the worksheet several times and have not bothered to make it neater.

I usually do that after I manage to get a significant part of a worksheet to work correctly.

That is the case here. I have just debugged the commands preceding: 

display(DispLC, DispUC, DispTL1, DispTL2, DispInvCircleRight, DispInvCircleLeft, scaling = constrained, axes = boxed). 

Commands following this one (including the final plot command which you mention) are as yet untested and mostly likely will be discarded.

Since my worksheet is only partially debugged I expect further "normal" error messages but the strange one I quoted appeared without my altering the worksheet at all.

Should I completely re-write the worksheet's working portion in a new worksheet, or is my Maple 2020 itself corrupted and needing to be replaced?