Coupled Harmonic OscillatorsPeyam TabrizianFriday, November 18th, 2011This handout is meant to summarize everything you need to know about thecoupled harmonic oscillators for the final exam.Note: In what follows we will assume that all masses m = 1 and all springconstants k = 1.1 Case N = 2 (two harmonic oscillators)Question: Find the proper frequencies and eigenvectors / proper modes of twocoupled harmonic oscillators.Equation:x00= Ax, where:x(t) =x1(t)x2(t), A =−2 11 −2Proper frequencies:Find the eigenvalues of the matrix A: λ = −1, −31Fact: The proper frequencies are ±√λ.Hence the proper frequencies are: ±√−1 = ±i, ±√−3 = ±√3i.Proper modes:To find the modes, use the following trick: Since N = 2, N + 1 = 3, henceall the modes will involve sinqπ3, where q is some number. Then:v1=sin1π3sin2π3, v2=sin2π3sin4π3Note: The way you get the other values is by using multiples, i.e. the multi-ples of 1 are 1 and 2, the multiples of 2 are 2 and 4.Hence the proper modes are:v1="√32√32#, v2="√32−√32#Note: If you ever get00, then you probably used N = 2 instead of N +1 = 3.22 Case N = 3 (three harmonic oscillators)Question: Find the proper frequencies and eigenvectors / proper modes of threecoupled harmonic oscillators.Equation:x00= Ax, where:x(t) =x1(t)x2(t)x3(t), A =−2 1 01 −2 10 1 −2Proper frequencies:Find the eigenvalues of the matrix A: λ = −2, −2 −√2, −2 +√2Fact: The proper frequencies are ±√λ.Hence the proper frequencies are:±√−2 = ±√2i, ±q−2 −√2 = ±q2 +√2i, ±q−2 +√2 = ±q2 −√2iProper modes:To find the modes, use the following trick: Since N = 3, N + 1 = 4, henceall the modes will involve sinqπ4, where q is some number. Then:v1=sin1π4sin2π4sin3π4, v2=sin2π4sin4π4sin6π4, v3=sin3π4sin6π4sin9π4Note: The way you get the other values is by using multiples, i.e. the multi-3ples of 1 are 1, 2 and 3, the multiples of 2 are 2, 4 and 6, the multiples of 3 are 3,6, and 9.Hence the proper modes are:v1=√221√22, v2=10−1, v3=√22−1√22Note: If you get000, then you probably used N = 3 instead of N + 1 = 4.3 General caseEquation: x00= AxA =−2 1 0 ··· 01 −2 1 0 ··· 00 1 −2 1 0 ···..................0 0 ··· 1 −2 10 0 0 0 1 −2Proper frequencies: ±2i sinkπ2(N +1), k = 1, 2, ···NProper modes: vk=sinkπN +1sin2kπN +1...sinN kπN
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