To plot graphs for possible motions for 2 (frictionless) carts and 2 walls,

with three equivalent springs connecting them all,

 phys. 304, Spring, 200 9

We measure the coordinates relative to some amplitude, which we take, initially, to be just 1,

and also take the original frequency as 1, i.e., measure time in terms of it.

>	x1:=cos(t); v1:=-sin(t);

>	plot(x1,t=0..30,title=`x1=x2\n first normal mode`, labels=[t,`x1=x2`]);

>	plot([x1,v1,t=0..30], title=`x1=x2\n phase plane for \n first normal mode`, labels=[`x1=x2`, `v1=v2`]);

Now we go on to the case where both normal modes are involved,

remembering that the ratio of the two frequencies, for this problem, is the square root of 3.

In what may seem to be a strange approach, I first write down the velocities, and then the

corresponding displacements.

>	w1:=-(1/2)*(sin(t)+sqrt(3)*sin(sqrt(3)*t)); w2:=-(sin(t)-sqrt(3)*sin(sqrt(3)*t))/2;

>	q1:=(cos(t)+cos(sqrt(3)*t))/2; q2:=(cos(t)-cos(sqrt(3)*t))/2;

>	plot([q1,q2],t=0..30,title=`x1 and x2\n both modes`, labels=[t,x],linestyle=[SOLID,DASHDOT], thickness=[2,2],legend=["x1","x2"]);

>	plot([[q1,w1,t=0..30],[q2,w2,t=0..30]],title=`x1 and x2 vs. v1 and v2\n both modes`, labels=[`x`,`v`], linestyle=[SOLID,DASHDOT],thickness=[2,2],legend=["cart 1", "cart 2"]);

>	plot([q1,w1,t=0..30],title=`cart 1 phase plane`,labels=[x,v],thickness=2,legend="cart 1");

>	plot([q2,w2,t=0..30],title=`cart 2 phase plane`,labels=[x,v], color=green,thickness=2,legend="cart 2");

>	plot([[q1,w1,t=0..10],[q2,w2,t=0..10]],title=`both carts, phase plane; t=0 to 10`,labels=[x,v], color=[red,green],thickness=2,legend=["cart1","cart 2"]);

>	plot([[q1,w1,t=0..15],[q2,w2,t=0..15]],title=`both carts, phase plane; t=0 to 15`,labels=[x,v], color=[red,green],thickness=2,legend=["cart1","cart 2"]);

>	plot([q2,w2,t=0..20],title=`cart 2 phase plane; t=0 to 20`,labels=[x,v], color=green,thickness=2,legend="cart 2");

>	plot([q2,w2,t=0..60],title=`cart 2 phase plane; t=0 to 60`,labels=[x,v], color=green,thickness=2,legend="cart 2");

>	plot([q2,w2,t=0..200],title=`cart 2 phase plane; t=0 to 200`,labels=[x,v], color=green,thickness=2,legend="cart 2");

>	plot([q2,w2,t=0..1000],title=`cart 2 phase plane; t=0 to 1000`,labels=[x,v], color=green,thickness=2,legend="cart 2");

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