@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?

@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.