(a) In the problem of small oscillations about steady motion, show that at the point of st...

(a) In the problem of small oscillations about steady motion, show that at the point of steady motion all the Hamiltonian variables η are constant. If the values for steady motion are η0 so that η = η0 + ξ, show that to the lowest nonvanishing approximation the effective Hamiltonian for small oscillation can be expressed as

 

where S is a square matrix with components that are functions of η0 only.

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(b) Assuming all frequencies of small oscillation are distinct, let M be a square 2n × 2n matrix formed by the components of a possible set of eigenvectors (for both positive and negative frequencies). Only the directions of the eigenvectors are fixed, not their magnitudes. Show that it is possible to apply conditions to the eigenvectors (in effect fixing their magnitudes) that make M the Jacobian matrix of a canonical transformation.

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(c) Show that the canonical transformation so found transforms the effective Hamiltonian to the form

 

where ωj is the magnitude of the normal frequencies. What are the equations of motion in this set of canonical coordinates?

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(d) Finally, show that

 

leads to a canonical transformation that decomposes H into the Hamiltonians for a set of uncoupled linear harmonic oscillators that oscillate in the normal modes.