Week 5

Coupled Simple Harmonic Oscillators

We are now beginning our foray into systems with more than one degree of freedom.  We will soon see that the phenomena of oscillation in these systems is very rich, and exhibits the seed of what will grow into communication of information via waves.

Let us first look at the behavior of two oscillators when one of them is set into motion.  We are considering the case of two identical masses on springs with spring constant k connected to walls on the left and right with a weak spring connecting the two of them.

The potential energy function is given by:

U[x1_,x2_,k_,kc_]=1/2 k x1^2 + 1/2 k x2^2+1/2kc (x1-x2)^2;

m=1;

ω0[k_]=Sqrt[k/m];ωc[kc_]=Sqrt[kc/m];

ωp[k_,kc_]=ω0[k];ωm[k_,kc_]=Sqrt[ω0[k]^2 +2 ωc[kc]^2];

x1[t_,k_,kc_,Ap_,Am_,αp_,αm_]=1/2(Ap Cos[ωp[k,kc] t+αp]+Am Cos[ωm[k,kc] t+αm]);

x2[t_,k_,kc_,Ap_,Am_,αp_,αm_]=1/2(Ap Cos[ωp[k,kc] t+αp]-Am Cos[ωm[k,kc] t+αm]);

Manipulate[ParametricPlot[{x1[t,1,.05,1,1,0,0],x2[t,1,.05,1,1,0,0]},{t,0,tmax},PlotRange->{{-1,1},{-1,1}},AxesLabel->{x1,x2},AspectRatio->1],{tmax,0.1,50 π}]

Manipulate[ListPlot[{{x1[t,1,.05,1,1,0,0],x2[t,1,.05,1,1,0,0]}},PlotRange->{{-1,1},{-1,1}},AxesLabel->{x1,x2},AspectRatio->1],{t,0.1,500 π}]

Manipulate[ContourPlot[U[x1,x2,1,kc],{x1,-1,1},{x2,-1,1},Contours->{.05,.5,1,2}],{kc,0,5}]