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What physical properties has a dynamical system whose equation of motion are linear? When does it exist a change of coordinates which turn the equation of motions in a linear system?

My teacher says that Hamiltonian mechanics could answer this question. I need to see his argument made rigorous, because I am not too convinced. He says:

  1. If it exists a change of coordinates that makes the equation of motions linear, than the Hamiltonian $H$ is quadratic.

  2. Because $H$ is quadratic, it could be put in diagonal form; we can find coordinates such that $H = \sum{H_\lambda}$ and $H_\lambda = (q^\lambda)^2 + (p_\lambda)^2$.

  3. Every $H_\lambda$ is a constant of motion and also $\{H_\mu, H_\nu\} = 0$ because they depend on different degrees of freedom.

  4. We have $n$ independent and in involution constant of motion, so for Arnold-Liouville theorem, the system is completely integrable. There are $(\alpha^\lambda, I_\lambda$) action-angle variables.

  5. In these variables, we have $H_\lambda = \omega_\lambda I_\lambda$.

  6. By Hamilton equations we have (as always for action-angle variables) $\alpha_\lambda = \omega_\lambda t + \alpha_\lambda^0$. But $\omega_\lambda$ (the frequency on the invariant torus) does not depend by $I_\mu$ and so it does not depend on initial conditions.

Conclusion: the systems that can be made linear are only the ones which motion is isochronous in every degree of freedom ($\omega_\lambda$ is always the same for every initial condition).

My questions are

At step 2: ok, it could be put in diagonal form, but is this change of coordinates canonical? It is necessary, isn't it? If it is not canonical, we have different equation of motions, so we can't use action-angle, right?

Is this in general correct?

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Given a semipositive quadratic real Hamiltonian on a $2n$-dimensional phase space, one may show that there always exists a real & linear symplectic transformation that brings the Hamiltonian on diagonal form, cf. Ref. 1. See e.g. my related Phys.SE answer here.

So the problem can be reduced to $n$ decoupled simple harmonic oscillators (possibly degenerate). This is Liouville-integrable.

References:

  1. V.I. Arnold, Mathematical methods of Classical Mechanics, 2nd eds., 1989; Appendix 6.
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  • $\begingroup$ Thanks for the reply. So, if I have an Hamiltonian H quadratic semipositive, doest it exist a canonical transformation that 1. split the degrees of freedom, i.e $H = \sum{H_\lambda(q^\lambda, p_\lambda)}$ and 2. every $H_\lambda$ is of kind $H_\lambda = (p_\lambda)^2 + \omega_{(\lambda)}^2(q^\lambda)^2$? 3. I also suppose we cannot reduce every $\omega_{(\lambda)} = 1$, right? $\endgroup$
    – CNS709
    Commented Jan 31, 2019 at 22:03
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    $\begingroup$ 1. Yes. 2. In the non-degenerate case. 3. No, we cannot. $\endgroup$
    – Qmechanic ♦
    Commented Jan 31, 2019 at 22:10

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