@Rouben Rostamian  This is great! Your worksheet above has enabled me to display Slinky extended by the centrifugal force of rotation.

Please check the code I have added to yours in the worksheet below. I will appreciate any comments you care to make.

Slinky2.mw 

If the above is accurate, I will proceed to animate the rotation.

@Rouben Rostamian  Thank you greatly for the lesson on elasticity.

I was prompted to ask this question by the book "200 More Puzzling Physics Problems" by Gnadig, Honyek and Vigh, problem 61, which describes a Slinky anchored at a point in a horizontal tube which rotates horizontally about one end at a fixed angular velocity.

The book provides the following ODE solution:

diff(r(m),t,t) = -omega*sqrt(M/k)*r(m).

M is the mass of the Slinky, k is its spring constant, omega is the tube's angular velocity, and r is the length from the tube's rotation axis to any point on the Slinky as a function of the Slinky's mass within this length.

I would like to animate the Slinky as the tube containing it rotates but I don't know how to obtain the value of m which itself is a function of r. Can you refer me to a source which describes a mathematically accurate profile of a Slinky in this scenario?

@Kitonum In my Maple2016 this procedure does not produce the above results and I can't find relevant help text.

Is there Maple2016 code that can produce these results? Is there help text explaining this interleave capability?

@tomleslie Thank you for your answer.

I recently found the technique below in one of Dr. Lopez's applications.

Please comment on its validity.

Efield_contours.mw

Sorry, I forgot to eliminate the reference to my document library.

@Rouben Rostamian  You have provided me with many answers and I am grateful for each of them.

I will take the time to digest your worksheet and communicate any questions which arise.

For your interest, my question is an expansion of problem 13 in "200 Puzzling Physics Problems" by Peter Gnadig, Gyula Honyek and Ken Riley, Cambridge University Press.

@Rouben Rostamian  

@Rouben Rostamian  I thought I recognized the law of cosines but was thrown off by seeing derivatives multiplied within one of its terms. 

Thank you for your clear explanation.

Later I discovered that the statement below also yields the pearl's kinetic energy.

simplify((1/2)*m*VectorNorm(v, 2, conjugate = false)^2)

@Rouben Rostamian  Lovely code.

I am going to try using your technique to flatten the more complex dodecahedron and icosahedron.

@Kitonum Further changes to Contourplot_with_Labels corrects the problems you identified.
 

Contourplot_with_labels.mw

@Kitonum Originally I only intended for this modification to handle curves defined as [f(x,y),g(x,y),x or y] . 

Taking your review as motivation, I will try to modify the modification to work with examples such as you have given, namely

[f(x,y),g(x,y),h(x,y)].

I admire your work and appreciate your interest.

Thank you for this clear insight.

@Joe Riel Are my comments below correct? Does the discrete event below have to contain the discrete variable dv(t)? 

DE1 := diff(x(t), t, t) = -x(t)-(diff(x(t), t))+y(t):

DE2 := diff(y(t), t, t) = y(t)-(diff(y(t), t))+x(t):

ICs := x(0) = 0, (D(x))(0) = 1, y(0) = 0, (D(y))(0) = 1, dv(0) = 0:

sol1 := dsolve({DE1, DE2, ICs}, {x(t), y(t)}, numeric, discrete_variables = [dv(t)], events = [[x(t)-1, dv(t) = 10], [[0, dv(t) > 1], dv(t) = -dv(t)]]):

odeplot(sol1, [t, dv(t)], -1 .. 3, title = "Plot of dv(t)", titlefont = [Courier, bold, 14]);

The second event is a discrete event with a conditional trigger and only occurs when a prior event (here the first event) has earlier occurred i.e. if the first event is omitted the second does not occur, even if option event_initial is true. If the action on the second event is halt, the halt occurs but the odeplot of dv(t) shows its value as zero. Why?

@Preben Alsholm Your suggestion works, but I would still like to know all the capabilities of the conditional trigger feature.

@Rouben Rostamian  Thank you Rouben.

Your use of the Lagrangian probably fits more diverse motion scenarios than mine.

In addition your second order ODEs are directly solvable by Maple whereas I had to manually reduce mine to first order when Maple's dsolve, numeric sent me a message that Maple could not perform this reduction. 

@acer You have supplied an example of what I meant i.e. a procedure invoked by a plot command to tailor some aspect of the plotted display, in this case its color.

Thank you for your effort and I apologize for this delayed reply, I was tied up in family business.