Motion of a mass point on a spring of relaxed length L and spring constant k in the x-y plane. All units in SI

m=0.25

k=10

omega0=Sqrt[k/m]

t0=2*Pi/omega0

L=1

Initial conditions :

rinit=L

vinit=0

ptheta=2

Angular momentum = constant of motion

s=NDSolve[{r''[t]==ptheta^2/mass^2/r[t]^3-omega0^2*(r[t]-L),r[0]==rinit,r'[0]==vinit},r,{t,0,100,0.001}]

Plot[r[t]/.s,{t,0,5}]

Plot[ptheta/m/(r[t]^2)/.s,{t,0,5},PlotRange->{0,10}]

theta[tx_]:=NIntegrate[ptheta/m/(r[t]^2)/.s,{t,0,tx}]

Plot[theta[tx],{tx,0,5}]

a=10

Total Energy

b=k/2

1/2 Spring Constant

c=ptheta^2/2/m

Angular Momentum Squared/ 2 m

Integrate[1/Sqrt[1-x^2],x]

r0=Solve[b*(rr-L)*rr^2-c==0,rr]

Minimum of “quasi-potential” (r0 for equilibrium)

omegasmall=omega*Sqrt[3*c/b/(rr)^4+1/.r0]

Frequency of small oscillations around equilibrium

tt=N[2*Pi/8.209444528299498`]

Time period for one full oscillation (small oscillation) around equilibrium

Solve[a*rr^2-b*(rr-L)^2*rr^2==c,rr]

Turning points for periodic motion

Plot[c/rr^2+b*(rr-L)^2,{rr,0,3},PlotRange->{0,50}]

Pseudopotential for given parameters

tpred=2/Sqrt[2/m]*NIntegrate[1/Sqrt[a-c/rr^2-b*(rr-L)^2],{rr,0.8968174011656064`,2.3032222479710054`}]

Time period of one full oscillation between r = rmin and the other turning point and back