Here is the proper way of defining your Lagrangian:

L := J/2*(diff(x(t),t)/R)^2
   + m__r/2*diff(x(t),t)^2
   + m__p/2*((diff(x(t),t) + l*diff(phi(t),t)*cos(phi(t)))^2 + (l*diff(phi(t),t)*sin(phi(t)))^2)
   + m__p*g*l*cos(phi(t));

but be sure to verify that I have not introduced errors in transcribing the image that you have provided.

If you are interested in the final form of the Euler-Lagrange equations and don't care about the intermediate steps, then Maple's EulerLagrange function in the VariationalCalculus package will produce the result directly, as in Method 1 below.  Otherwise look at Method 2.

Method 1:

tmp1 := VariationalCalculus:-EulerLagrange(L, t, [x(t),phi(t)]);

tmp1 consists of a set of four expressions—two equations of the form something=K1 and something=K2, which are the integrals of motion (conservation of energy and momentum).  The other two are the differential equations of motion.  In the next step I delete the integrals of motion, and keep the differential equations:

tmp2 := remove(type, tmp1, equation);

tmp2 consists of two differential equations as I noted above.  To verify that we indeed have two items, we do:

nops(tmp2);

and indeed it returns "2".

Next, we name those equations de1 and de2:

de1 := tmp2[1] = 0;
de2 := tmp2[2] = 0;

Then you do whatever you need to do with de1 and de2

Method 2:

If you insist on doing the calculations one-step-at-a-time as you have outlined in your statement, you need to load the Physics package.  The package redefines Maple's "diff" operator which then enables you to calculate derivatives with respect to derivatives.

with(Physics):
dL_dxdot := diff(L, diff(x(t),t));         # derivative of L with respect to xdot
dL_dphidot := diff(L, diff(phi(t),t));     # derivative of L with respect to phidot
dL_dphi := diff(L, phi(t));                # derivative of L with respect to phi
diff(dL_dxdot, t);

p1 := plot(x^2 - 1, x = -2 .. 2, color=blue):
p2 := plots:-pointplot([3/2,5/4], symbol=solidcircle, symbolsize=20, color=white):
p3 := plots:-pointplot([3/2,5/4], symbol=     circle, symbolsize=20, color=black):
plots[display](p1, p2, p3);

In the attached worksheet I show how to generalize the definition of the pedal curve to pedal surface and calculate the equation of the pedal surface in the general case.  Then I illustrate the result by calculating the pedal surface of an ellipsoid with respect to its center.  Here is what it looks like.

Download pedal-surface.mw

No need for extra packages.  Just do this:

restart;
k := x -> piecewise(x < -6, 1/2*x + 10, x < -3, 1, x < 6*Pi, sin(1/2*x) + 2, x < 25, x - 15);
plot3d(k(x), t=0..2*Pi, x=-20..20, coords=cylindrical);

The parametrization of a surface (any surface) is not unque.  The way a surface is parametrized can have a drastic effect in how it is rendered on screen.

The standard parametrization of the oloid does not produce good computer-generated graphics as you have observed. But an oloid has a great deal of symmetries.  It is not difficult to see that it may be decomposed into four congruent patches, each of which has a quite "tame" parametrization for which Maple's default plotting grid is more than adequate.  Here I construct the oloid by putting together the four patches. 

restart;
alpha[1] := sin(t):
alpha[2] := -1/2 - cos(t):
alpha[3] := 0:
beta[1] := 0:
beta[2] := 1/2 - cos(t)/(1+cos(t)):
beta[3] := sqrt(1 + 2*cos(t))/(1+cos(t)):
x := (1-m)*alpha[1] + m*beta[1];
y := (1-m)*alpha[2] + m*beta[2];
z := (1-m)*alpha[3] + m*beta[3];
plot3d([[x,y,z], [x,y,-z], [z,-y,x], [-z,-y,x]], m=0..1, t=-Pi/2..Pi/2,
    scaling=constrained, color=[red,green,blue,yellow]);

I have colored the patches in four different colors in order to delineate them.  Change surface style and color as it suits your needs.

T := t -> (t+459.47) * 5/9;
seq(printf("%10d %10.2f\n", t, T(t)), t=100..110);
       100     310.82
       101     311.37
       102     311.93
       103     312.48
       104     313.04
       105     313.59
       106     314.15
       107     314.71
       108     315.26
       109     315.82
       110     316.37

Inversion in a sphere is much simpler than what you have in your worksheet.  Here is how I would do it.

Download inversion-in-sphere.mw

There are mnay ways of doing it, depending on what you want to do with the results.  Also it depends on how you represent quaternions.  You say that you want to represent a quaternion as a vector of length 4, but in your example your write qaa and qab as lists.  Lists and vectors are different types of objects in Maple.

Let's say you represent a quaternion as a list of length 4.  Then this will do what you want:

That said, I much prefer the [scalar, vector] pair representation of quaternions (see the Wikipedia article) rather than lists of length 4.  But that again depends on what you want to do with them.

de := a*diff(y(x),x,x) + b*diff(y(x),x) + c*y(x) = 0;
dsolve(de) assuming b^2 < 4*a*c;

Download mw.mw

Okay, your hand-written calculations are good.  Here I will do those in Maple.

Download mw.mw

How do you like this?

restart;

# macro emulates Matlab's "jet" colorscheme:
macro(jet=["zgradient", ["DarkBlue", "Blue", "DeepSkyBlue", "Cyan", "LawnGreen", "Yellow", "Orange", "Red", "DarkRed"]]):

plot3d((1-xi^2)*(1+tau), tau=-1..1, xi=-1..1,
    colorscheme=jet, orientation=[-120,72,0], axes=framed);

plot3d((1-xi^2)*(1+tau), tau=-1..1, xi=-1..1,
    colorscheme=jet, gridstyle=triangular, orientation=[-120,72,0]);

As Kitonum has noted, this is not an infinite fraction.  The integrand is z = ..  Then we see that z^2 = x*z, and therefore z = x.  Then the integral of z is x^2/2 + constant.  QED.

Forget about solve.  After you define psi, just do:

plots:-contourplot(psi, x=-1..1, eta=-2..2);

Try this:

restart;
plot(x^2, x=-1..1, labels=[typeset(NFE), typeset(log__10(`Max Err`))]);