Numerit[WIN32][1700][1703]d  qffffff)@j@fffffvq@?@ffffff9@?@ffffff9@ffffff)@ffffff)@         Times New RomanArialSymbol Courier NewVerdana??? G Ab@ffffffI@h1g1h2g2h3g3yc1xc1yc2xc2?@?@ ?@ ????>@ >@>@>@>@>@>@ >@ >@>@  A@ >==>>??-0.5210.010.01-3.43.40.50.010.01X ffffff)@@tX @@a3X @@a1X @@a2X @@e1X @@e2X @@e3G Ab@ C@e1x01e2x02e3x03>@>@ >@ >@>@>@>@ >@ >@ >@>@>@>@>@>@ >@  >==>>??01.10.50.010.01-3.43.410.010.01X /@UUUUUu@dirvvvpvvvpvvvpvvvpvvvpvvvpvvv0Coupled Pendulumsvvv.Yaron Silberbergvvv Xvvv This simulation demonstrates how the energy of the left pendulum can be transferred to the right pendulum through the middle one, yet without ever exciting it to a large amplitude. This transfer is accomplished by adiabatically modifying the coupling strength (symbolized in the drawing by the height of the connecting rod). This mechanical model is equivalent to the process known as STIRAP - for Stimulated Raman Adiabatic Passage - in atomic physics (see K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003, 1998). Note that to accomplish complete transfer, the stationary pendulum is initially coupled strongly to the central one, while the moving pendulum is not coupled. The strength of coupling is slowly changed, until at the end of the process the ratio is inverted. This process if often referred to as a 'counterintuitive scheme'. vvv As mentioned above, the coupling strength should be modified adiabatically. To see what happens when the coupling strength changes more rapidly, change the variable 'ptime' (process time) from 200 to 20 and run the program again (look at the amplitude of the middle pendulum).vvv C ,vvv3B $*Time:! p!Energy flow: 0) vvv  /vvv{BQB\B $*Amplitude: G # ! $ !" ,vvv{BQB\B $*Energy: Z % ! & !'  vvv{BQB\B $*( vvv vvv  q8ffffff)@j@fffffvq@?@ffffff9@?@ffffff9@ffffff)@ffffff)@         Times New RomanArialSymbol Courier New [+` This program simulates coupled pendulums.*` Press the Run button or F9 to start. The-` program runs indefinitely, so you must stop,` it manually (press the Stop button or F5).)` To speed up the simulation decrease the` delay factor (e.g., to 0.25)delay_factor = 0.5` position of pendulumsx01 = -2x02 = 0x03 = 2` pendulum lengthr = 2` initial amplitudea0 = 1#ptime = 200 ` process time (sec.)#dt = 0.04 ` time interval (sec.)direction = "--->","<---"idir = 10dir = direction[idir] ` direction of energy flow+delay = dt*delay_factor ` loop delay (sec.)t = 0 ` time#cc = 0 ` coupling coefficientx1 = a0x2 = 0x3 = 0reportloop ` couplingsc12 = 0.4*cc/ptimec23 = 0.4*(ptime-cc)/ptime ` amplitudesx1 = x1+dt*c12*x2x2 = x2-dt*(c12*x1+c23*x3)x3 = x3+dt*c23*x2co = cos(pi*t) xx1 = x1*co xx2 = x2*co xx3 = x3*co ` display ` pendulumsyy1 = sqrt(r^2-xx1^2)yy2 = sqrt(r^2-xx2^2)yy3 = sqrt(r^2-xx3^2)g1 = x01,x01+xx1 h1 = r,r-yy1g2 = x02,x02+xx2 h2 = r,r-yy2g3 = x03,x03+xx3 h3 = r,r-yy3` coupling rodsyc1 = 1.5*r*c12 + 0.2yc2 = 1.5*r*c23 + 0.2xc11 = yc1*xx1/yy1xc12 = yc1*xx2/yy2xc22 = yc2*xx2/yy2xc23 = yc2*xx3/yy3xc1 = x01+xc11,x02+xc12xc2 = x02+xc22,x03+xc23 yc1 = r - yc1 yc2 = r - yc2` amplitude & energy a1 = abs(x1) a2 = abs(x2) a3 = abs(x3)e1 = a1^2/a0^2e2 = a2^2/a0^2e3 = a3^2/a0^2refresh wait delaycc += dt*if cc < 0 or cc > ptime ` reverse process dt = -dt cc += 2*dt idir = 3-idirdir = direction[idir])t += abs(dt) ` absolute time for displayd:\program files\numerit\ . delay_factorx01 x02 x03 ra0ptimedt directionidirdir delayt ccx1x2x3c12c23coxx1xx2xx3yy1yy2yy3g1 h1 g2 h2 g3 h3 yc1 yc2 xc11xc12xc22xc23xc1 xc2 a1 a2 a3 e1 e2 e3  e  G        5   4   B        !"$  B C%   AB C(   B B@)    B  B@BA*   B B@+ B,  B-  B.  B2  E  EA3  E  EA4  E  EA5   @56   A57   @58   A59   @5:   A5= B B @ > B B @!? B C"@ B C#A ! B C$B ! B C%C  "@  #@5&D  $@  %@5'E  A F  !A!I (J )K *L ( E  EC+M ) E  EC,N * E  EC-PQ S N T Y Xb7U GV  BN W A X 4 Y N 6d=^ 0.5?2@01?200i@0.04{Gz?---><---0.4?1.5?0.2?3@