Two integration schemes were used, fairly similar, but of different order of accuracy
Standard initial conditions of the pendulum subject to Hooke's law tension with 
neutral length l0, and Earth's gravity are as follows.
 
computation no.1: standard case
	T_end = 1000d0 ! s
	l0 = 0.1 	! m 
	k_m = 33 	! Hz**2
	g = 9.81  	! m/s**2
	x = -0.1;    y = 0.2;   z =  0.1 
	vx = 0;     vy = 0;    vz = -0.1
computation no. 2 has dt = 2 times shorter timestep, to see if the integration is accurate
computation no. 3 has x = -0.1; y = 0.21; z = 0.1 to see the sensitivity to initial position,
	therefore, rather than init l0 =  0.244949, has  l0 =  0.2531798
computation no. 4 has a different spring constant k_m = 40 (not 33) just to see..
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Euler scheme, 1st order, dt=1e-4	 (fin-prob1-eu-2024.f90)
 
 #steps.    final:  x       y        z        l       dE/E  
10000000 i,r,l  -0.0492   0.0985   0.0754   0.1334 -0.133E-03 <- standard init cond. k/m=33 
20000000 i,r,l  -0.0494   0.0988   0.0752   0.1336 -0.672E-04 <- 2x longer timestep, k/m=33
10000000 i,r,l   0.0915  -0.1921  -0.6594   0.6928  0.599E-03 <- y0 variation, no.3, k/m=33
10000000 i,r,l   0.0973  -0.1946  -0.6076   0.6456 -0.403E-03 <- computation no. 4,  k/m=40

Leapfrog scheme, 2nd order, dt-1e-3	 (fin-prob1-lf-2024.f90)

 #steps.    final:  x       y        z        l       dE/E  
 1000000 i,r,l  -0.0501   0.1003   0.0742   0.1340  0.344E-05 <- standard init cond.    
 2000000 i,r,l  -0.0497   0.0995   0.0747   0.1338  0.861E-06 <- 2x longer timestep     
 1000000 i,r,l   0.0915  -0.1921  -0.6548   0.6904  0.708E-05 <- y0 different by 1 cm

Conclusions: 

(i) The system can be successfully (accurately) simulated by both schemes for 1000 seconds. 
Euler being lower order needs a 10x shorter time step for that (despite the much
longer runtime, it conserved the specific energy E worse, at the level of < 1e-3, 
while leapfrog achieves energy conservation to < 1e-5 level). 
Results of both schemes almost coincide, and timestep is adequate for both 
schemes - improvement due to 2x shorter timestep is unimportant.

(ii) Results of all schemes show chaos-like, large sensitivity to initial position: 
 1 cm difference in one of the coordinates causes ~50 cm difference in final 
 coordinates as well as length. (For comparison: the length of pendulum varies in 
 time between ~0.1 and ~1 m)