r[x_,y_]={x,y}
norm[rr_]:=Sqrt[rr[[1]]^2+rr[[2]]^2]
r1=r[Indexed[q,1],Indexed[q,2]]
r2=r[Indexed[q,3]+a,Indexed[q,4]]
r3=r[Indexed[q,5]+a Cos[Pi/3],Indexed[q,6]+a Sin[Pi/3]]
r12=norm[r1-r2]
r13=norm[r1-r3]
r23=norm[r2-r3]
U= k/2 ( (r12-a)^2 + (r13-a)^2 + (r23-a)^2)
Uij=Table[D[U,Indexed[q,i],Indexed[q,j]],{i,6},{j,6}]
Uij /. {Indexed[q,1]->0,Indexed[q,2]->0,Indexed[q,3]->0,Indexed[q,4]->0,Indexed[q,5]->0,Indexed[q,6]->0}
Uij0=Simplify[%,a>0]/k
{values,vectors}=Eigensystem[Uij0]
 values                                                                 

             3  3
Out[71]= {3, -, -, 0, 0, 0}
             2  2

breathing, pull3, odd, odd, x, rotation about #3
to show: l.c. 2&3 are pull1 & pull2
	 l.c. 4&6 are y or CM rotation

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[1]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[1]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[1]][[5;;6]]}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[2]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[2]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[2]][[5;;6]]}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[3]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[3]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[3]][[5;;6]]}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[4]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[4]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[4]][[5;;6]]}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[5]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[5]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[5]][[5;;6]]}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[6]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[6]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[6]][[5;;6]]}]}]]

Solve[{a1,a2,a3}.{vectors[[4]],vectors[[5]],vectors[[6]]}=={0,1,0,1,0,1},{a1,a2,a3}]
Out[42]= {{a1 -> 1, a2 -> 0, a3 -> 1}}

Solve[{a1,a2,a3}.{vectors[[4]],vectors[[5]],vectors[[6]]}=={Cos[Pi/3],-Sin[Pi/3],Cos[Pi/3],Sin[Pi/3],-1,0},{a1,a2,a3}]
                                    Sqrt[3]
Out[46]= {{a1 -> 0, a2 -> -1, a3 -> -------}}
                                       2
v={Cos[Pi/3],-Sin[Pi/3],Cos[Pi/3],Sin[Pi/3],-1,0}
Show[Graphics[{Arrow[{{0,0},{0,0}+v[[1;;2]]}],Arrow[{{10,0},{10,0}+v[[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+v[[5;;6]]}]}]]

v1=RotationMatrix[2 Pi/3].vectors[[2]][[1;;2]]
v2=RotationMatrix[2 Pi/3].vectors[[2]][[3;;4]]
v3=RotationMatrix[2 Pi/3].vectors[[2]][[5;;6]]
v=Flatten[{v3,v1,v2}]

Solve[{a1,a2}.{vectors[[2]],vectors[[3]]}==v,{a1,a2}]
                   1         Sqrt[3]
Out[62]= {{a1 -> -(-), a2 -> -------}}
                   2            2

v1=RotationMatrix[-2 Pi/3].vectors[[2]][[1;;2]]
v2=RotationMatrix[-2 Pi/3].vectors[[2]][[3;;4]]
v3=RotationMatrix[-2 Pi/3].vectors[[2]][[5;;6]]
v=Flatten[{v2,v3,v1}]

Solve[{a1,a2}.{vectors[[2]],vectors[[3]]}==v,{a1,a2}]

GraphicsGrid[{{%77,%78,%79},{%80,%81,%82}}]
Export["H3+eigen.eps",%83]


Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[1]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[1]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[1]][[5;;6]]}],
Line[{{0,0},{10,0},{10 Cos[Pi/3],10 Sin[Pi/3]},{0,0}}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[2]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[2]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[2]][[5;;6]]}],
Line[{{0,0},{10,0},{10 Cos[Pi/3],10 Sin[Pi/3]},{0,0}}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[3]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[3]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[3]][[5;;6]]}],
Line[{{0,0},{10,0},{10 Cos[Pi/3],10 Sin[Pi/3]},{0,0}}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[4]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[4]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[4]][[5;;6]]}],
Line[{{0,0},{10,0},{10 Cos[Pi/3],10 Sin[Pi/3]},{0,0}}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[5]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[5]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[5]][[5;;6]]}],
Line[{{0,0},{10,0},{10 Cos[Pi/3],10 Sin[Pi/3]},{0,0}}]}]]

Show[Graphics[{Arrow[{{0,0},{0,0}+vectors[[6]][[1;;2]]}],Arrow[{{10,0},{10,0}+vectors[[6]][[3;;4]]}],
Arrow[{{10 Cos[Pi/3],10 Sin[Pi/3]},{10 Cos[Pi/3],10 Sin[Pi/3]}+vectors[[6]][[5;;6]]}],
Line[{{0,0},{10,0},{10 Cos[Pi/3],10 Sin[Pi/3]},{0,0}}]}]]

GraphicsGrid[{{%87,%88,%89},{%90,%91,%92}}]
Export["H3+eigen2.eps",%93]

vectors[[2]][[1;;2]]+vectors[[2]][[3;;4]]+vectors[[2]][[5;;6]]
