How to find the zeroes of the function S, below?want to find the zeroes of S = 4 ψ 2 Where ψ ˙ 1 = 2 ψ 2 ψ ˙ 2 = − 2 ψ 1

Question

How to find the zeroes of the function   S, below?want to find the zeroes of   S  =  4      ψ    2  Where                    ψ        ˙              1    =  2      ψ    2                      ψ        ˙              2    =  −  2      ψ    1

wlanauee · Accepted Answer

When you have                    ψ        ˙              1    =  2      ψ    2  and                    ψ        ˙              2    =  −  2      ψ    1    ,you can re-write this as      (                                                                      ψ                ˙                                      1                                                                                          ψ                ˙                                      2                                )    =      (                            0                          2                                      −          2                          0                      )        (                                        ψ            1                                                            ψ            2                                )    .Call the matrix above   A.   A takes the general form of a matrix   M defined as  M  =      (                            σ                          ω                                      −          ω                          σ                      )    ,and the exponential for such matrices takes the form  exp  ⁡  (  M  t  )  =      e          σ      t            (                            cos          ⁡          (          ω          t          )                          sin          ⁡          (          ω          t          )                                      −          sin          ⁡          (          ω          t          )                          cos          ⁡          (          ω          t          )                      )    .Using the values of   σ  =  0 and   ω  =  2 we have in   A above, we get  exp  ⁡  (  A  t  )  =      (                            cos          ⁡          (          2          t          )                          sin          ⁡          (          2          t          )                                      −          sin          ⁡          (          2          t          )                          cos          ⁡          (          2          t          )                      )    .Then solving for       ψ    1   and       ψ    2   gives      (                                        ψ            1                    (          t          )                                                  ψ            2                    (          t          )                      )    =      (                            cos          ⁡          (          2          t          )                          sin          ⁡          (          2          t          )                                      −          sin          ⁡          (          2          t          )                          cos          ⁡          (          2          t          )                      )        (                                        ψ            1                    (                      t            0                    )                                                  ψ            2                    (                      t            0                    )                      )    .From this we get that  S  =  −  4  sin  ⁡  (  2  t  )      ψ    1    (      t    0    )  +  4  cos  ⁡  (  2  t  )      ψ    2    (      t    0    )  .You can plug in any initial condition,             (            ψ    1    (      t    0    )  ,      ψ    2    (      t    0    )                    )              T   of interest and solve for the zeros of   S from there.

How to find the zeroes of the function S, below? want to find the zeroes of S=4psi_2 Where psi_1=2 psi_2, psi_2=−2 psi_1

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