Triangle mode movies

This script is is specialized for movie production.  It uses normal mode formulas copied from "Triangle Normal Modes", and it can imitate all possible vibrational motions of the equilateral triangle.  
						First run the movies of basis modes 1, 2, and 3.  Then as this afternoon's lab work, produce your own movies as linear combinations of the basis modes.  There are some profound and surprising things you can discover in these movies.  Hints are under foldout 4.

standard setup for all triangle motions

Off[General::spell, General::spell1,Graphics::gprim]

stdPlot={PlotRange->{{-1.4,1.4},{-1.4,1.4}},
AspectRatio->1};

restpoints = rp = 
			{{0, 0.866}, {-1, -0.866}, {1, -0.866}};

frame = Append[rp,rp[[1]]]

triangle = Show[Graphics[Line[frame]],stdPlot];

The unit of time is one period of the degenerate modes, times 2 Pi.

mode1 = {
{ 0.000,                         (*x1*)
 11.120 A1 Cos[p1 + 1.414 t]},   (*y1*)
 
{-9.730 A1 Cos[p1 + 1.414 t],    (*x2*)
 -5.595 A1 Cos[p1 + 1.414 t]},   (*y2*)
 
{ 9.730 A1 Cos[p1 + 1.414 t],    (*x3*)
 -5.595 A1 Cos[p1 + 1.414 t]}};  (*y3*)

mode2 = {
{-11.140 A2 Cos[p2 + t],        (*x1*)
   0.000              },        (*y1*)
 
{ 5.606 A2 Cos[p2 + t], 	    (*x2*)
  9.710 A2 Cos[p2 + t]},        (*y2*)
 
{  5.606 A2 Cos[p2 + t],        (*x3*)
 - 9.710 A2 Cos[p2 + t]}};      (*y3*)

mode3 = {
{ 0.000, 					    (*x1*)
 11.12 A3 Cos[p3 + t]},         (*y1*)
 
{ 9.67 A3 Cos[p3 + t],          (*x2*)
 -5.63 A3 Cos[p3 + t]},         (*y2*)
 
{-9.67 A3 Cos[p3 + t],          (*x3*)
 -5.63 A3 Cos[p3 + t]}};        (*y3*)

1.   the breathing mode

motion1 = restpoints + mode1 /.
				{A1->0.03, p1->0, t->phi/1.414}

mv1pics = Table[Show[triangle,
Graphics[{PointSize[.02],Map[Point,motion1]},stdPlot]],
{phi,0,2Pi(11/12),2Pi(1/12)}];

2.   the asymmetric stretch

motion2 = restpoints + mode2 /.{A2->0.03, p2->Pi/2}

mv2pics = Table[Show[triangle,
Graphics[{PointSize[.02],Map[Point,motion2]},stdPlot]],
{t,0,2Pi(11/12),2Pi(1/12)}];

3.  the bend

motion3 = restpoints + mode3 /. {A3->0.03, p3->0}

mv3pics = Table[Show[triangle,
Graphics[{PointSize[.02],
					Map[Point,motion3]},stdPlot]],
{t,0,2Pi(11/12),2Pi(1/12)}];

4.  Produce your own movies

Now you may pick your own values for amplitudes and phases, add them up to make the motion formula, and produce your own movie.  In the basis mode movies, there was no attempt to imitate the two different frequencies of motion.  However, when combining mode 1 with mode 2 and/or 3,  this becomes important.  We leave it to you to figure out how to do this, and to discover an important problem that arises in making movies of the general vibrational motion.
 
					Modes 2 and 3 have the same frequency, so they form a basis set for a function space containing a two-dimensional infinity of different motions, all with the same frequency.  The basis modes are special only in giving a symmetric role to atom 1, which was a tiny bit heavier than the other two. 
					
					 Try combining the degenerate modes 2 and 3 with various amplitudes and various phases.  Since the molecule is an equilateral triangle, there must be motions in which atom 2 plays the special symmetric role; and in others, atom 3.  Try using amplitudes of the form A2=r Cos[a] and  A3=r Sin[a].  These will take you all over the two dimensional space in a more intellegent way than just poking in random values of  A2 and A3.  Discover the separate roles of r and a.

						Keeping p2 and p3 equal and varying only a, see if you can make atom 2 run along a line that goes toward the triangle center. Then do it for atom 3.  Then try to generalize.   Remember, the matrix that performs rotations by a third of a turn contains the numbers +/-Sin[2 Pi/3] and Cos[2 Pi/3].

					Varying the relative values of p2 and p3 gives even  more  surprising motions.  Remember, circularly polarized light is made by combining a linear x-vibration with a linear y-vibration that is equal in amplitude but a quarter of a cycle out of phase.  Can vibrating atoms do something like this too?

This exercise is follows a long and distinguished tradition-  Hollywood also produces new movies as linear combinations of old ones.

The back of the book:  don't look too soon

a rotation of the bending mode

motion2p3 = restpoints +
((mode2 /. {A2->Sin[2Pi/3], p2->0})+
 (mode3 /. {A3->Cos[2Pi/3], p3->0}) )/50 // N

mv2p3pics = Table[Show[triangle,Graphics[
{PointSize[.02],Map[Point,motion2p3]},stdPlot]],
{t,0,2Pi(11/12),2Pi(1/12)}];

Note that this is a rotation of the bending mode, by 1/3 of a turn.

a vibrational motion that has angular momentum

Here we pick zero amplitude for mode 3, equal amplitudes for modes 2 and 1, and we set modes 1 and 2  out-of-phase with each other by 90 degrees.  Other initial conditions also produce interesting results.
				
				This movie runs in a smooth closed loop because modes 1 and 2 have the same frequency.  But generally, different modes have incommensurable frequencies, and smooth closed loop movies are not possible.

motion = restpoints +
(mode2 /. {A2->0.02,p2->0})+
(mode3 /. {A3->0.02,p3->Pi/2}) // N

mv12pics = Table[
Show[triangle,Graphics[
{PointSize[.02],Map[Point,motion]},stdPlot]],
{t,0,2Pi(23/24),2Pi(1/24)}];