It's never a good strategy to complexify unnecessarily an equation before trying to solve it.
This will only complicate Maple's task.

Download critical-point_mmcdara.mw

Download details.mw

Download test_mmcdara.mw

Here is an example that you can adapt to your own problem.

As I use Maple 2015, RandomTools[GenerateSimilar] is not supported, the end of the attached file presents a proposal based on Maple 2015 : it can be translated immediatly into more recent versions and easily customizable.

A_few_ideas.mw

Result:

What is equation (77)?

Your equation L3, which I guess is this unknown relation (77) after having substituted in it relations (79)  (which seems tocorrespondtowhat you name  L2, am I right?) contains far more monomials than only t^2, s*t, y*t... as said in the unusefull excerpt you provide.

Look to the attached file and please, keep in maind that if you need help you have to provide potential helpers all necessary informations they could need of.

Clarifications_needed.mw

Besides, I don't think that asking the same type of questions from different logins (proses, syntaxes and codes are too similar to be mere coincidence) will save you from all the inconvenience or disrespect you claimed to be the victim of.

With Maple 2015.2 : One_way.mw

(Insert content doesn't respond)

1. In procedure generate_ic, p2_sol is a sequence (at least with my Maple 2015 version) of two values containing p1.
   Inset this print line to see that:

   p2_sol := solve(H_eq, p2); 
   print('p2_sol'=p2_sol);
   

2. If p2_sol is indeed a sequence forn your Maple version, then this (simpler) command fails 

   select(x -> 0 < evalf(x), p2_sol);
   Error, invalid input: select expects 2 or more arguments, but received 1
   

   and you must write instead

   select(x -> 0 < evalf(x), [p2_sol]);  # note the square brackets
   

3. As p2_sol contains the name p1, no testing is applicable because Maple does not know anything about p1.
   To understand that run this simplified version of generate_ic:

   test_1 := proc (q1_val, ET_val, f_val) 
     local term, H_eq, p2_sol, p2_val; 
     term := sqrt(q1_val^2+1)-1+f_val; 
     H_eq := (1/2)*p1^2+(1/2)*p2^2-f_val+(1/2)*term^2 = ET_val; 
     p2_sol := [solve(H_eq, p2)]; 
     print('p2_sol'=p2_sol);
     select(x -> 0 < evalf(x), p2_sol)
   end proc:
   
   interface(displayprecision=5):
   
   test_1(.1, ET, f)
   
            p2_sol = [2.000*10^(-8)*sqrt(-2.500*10^15*p1^2+8.745*10^13), -2.000*10^(-8)*sqrt(-2.500*10^15*p1^2+8.745*10^13)]
   
   Error, (in test_1) selecting function must return true or false
   

Last but not least: can you provide a picture of your spring pendulum (is it a simple or double one or a mixed one?)
 

The worksheet is written with Maple 2015 and the output of solve(eqn1, allsolutions); (see @dharr ) differ from what you get in newer version.
This is why I explicitely split this output in two components named ans_2015_0 and ans_2015_1 wich corresponds to @dharr's ans1 and ans2 resoectively.

The plot serves as a guidance but is not at all mandatory: what matters is what comes next.

sets_mmcdara.mw

restart
solve(sqrt(2)*sin(2*x-(1/6)*Pi) = 1)
                             5    
                             -- Pi
                             24   
solve(sqrt(2)*sin(2*x-(1/6)*Pi) = 1, allsolutions=true)
                   5       3                
                   -- Pi - - Pi _B1 + Pi _Z1
                   24      4                
about(_B1)
Originally _B1, renamed _B1~:
  is assumed to be: OrProp(0,1)

about(_Z1)
Originally _Z1, renamed _Z1~:
  is assumed to be: integer

# BTW: you missed the Pi constant in your solution

eval(sqrt(2)*sin(2*x-(1/6)*Pi), x=5/24) = 1
                    (1/2)    /  5    1   \    
                  -2      sin|- -- + - Pi| = 1
                             \  12   6   /    
eval(sqrt(2)*sin(2*x-(1/6)*Pi), x=5/24*Pi) = 1
                             1 = 1

Download Allsolutions.mw

Simply use  package Student:-Statistics  (it's a shame that package Statistics forgot ExploreRV !)
 

with( Student:-Statistics ):
X := NormalRandomVariable(mu, sigma);
ExploreRV( X );


# Or, even more simply:
Student:-Statistics:-ExploreRV()

Raw result

By the time I was writting this reply Scot Gould was submitting his one.
Nevertheless I decided not to delete mine, even if there is likely nothing new compared to Scot's.

Customize_it.mw

You can animate wrt 1 or more parameters by selecting them.

BTW: You are Salim Barzani aren't you? 

I guess there is a conflct netween background in G1 and background in animate.
Here is a Maple 2015 Workaround.mw

Feel free to adjust the plotting ranges in G1.

Here is another alternative (more funny?)

f := unapply(PDF(Normal(mu, sigma), x), x):

G3 := proc(t)
  plots:-display(
    DensityPlot(X, color = blue, thickness = 5, range = mu-4*sigma-1..t)
    , plot(f(tau), tau = mu-4*sigma .. t, thickness = 5, color = red)
    , gridlines=true
    , view=[mu-4*sigma .. t, default]
  )
end proc:

plots:-animate(G3, [t], t = mu-4*sigma .. mu+4*sigma, background="AliceBlue") 

Download stop-working_mmcdara.mw

Point 1:
The signature of [2, 1, 4, 3] (your example p2) is +1 (even permutation) and not -1!

Point 2:
You can indeed ask Geminy for a code, but you can also ask Mapleprimers, or more simply search for parity in the help pages.
This is what an AIA (Astute IA, see @nm's reply) should answer you

Download PermParity.mw

@nm: It would be interesting to see what the IA you used in your reply answers you for this parity problem: does it know about GroupTheory package?

I suggest you to read the help pages to see what the correct syntaxes are

Download LA.mw