Pendulum Oscillation
The differential equation to describe oscillation of a pendulum of length l is
This is a nonlinear equation. For small oscillation amplitude,it can be linearized asThe oscillation frequency is thus given byAs the amplitude increases, the oscillation frequency becomes smaller. The frequency corrected for a finite amplitude is approximately given byAnimation shows oscillations with small and large amplitude, with(plots):
k1:=1/sqrt(2.):
k2:=0.1:
a:=[seq(plot([[sin(2*arcsin(k1*JacobiSN(1.8+0.25*(i-1),k1))),-cos(2*arcsin(k1*JacobiSN(0.25*(i-1)+1.8,k1)))]],style=point,symbol=CIRCLE,color=black),i=1..30)]:
b:=[seq(line([0,0],[sin(2*arcsin(k1*JacobiSN(1.8+0.25*(i-1),k1))),-cos(2*arcsin(k1*JacobiSN(0.25*(i-1)+1.8,k1)))],color=red),i=1..30)]:
aa:=display(a,insequence=true,axes=normal,view=[-1..1,-1..0]):
bb:=display(b,insequence=true,axes=normal,view=[-1..1,-1..0]):
display({aa,bb});
Amplitude = 0.1
with(plots):
k1:=1/sqrt(2.):
k2:=0.1:
c:=[seq(plot([[sin(2*arcsin(k2*JacobiSN(0.25*(i-1)+Pi/2,k2))),cos(2*arcsin(k2*JacobiSN(Pi/2+0.25*(i-1),k2)))]],style=point,symbol=CIRCLE,color=black),i=1..27)]:
d:=[seq(line([0,0],[sin(2*arcsin(k2*JacobiSN(0.25*(i-1)+Pi/2,k2))),-cos(2*arcsin(k2*JacobiSN(Pi/2+0.25*(i-1),k2)))],color=red),i=1..27)]:
cc:=display(c,insequence=true,axes=normal,view=[-1..1,-1..0]):
dd:=display(d,insequence=true,axes=normal,view=[-1..1,-1..0]):
display({cc,dd});
