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In Classical Dynamics by José & Saletan [section 4.2.2] they give the example of a 2D Harmonic Oscillator whose equations of motion are

\begin{equation} \ddot{x}_i+\omega_i^2x_i=0 \ \ \ \ \ \text{for} \ \ \ \ i=1,2\tag{3.38} \end{equation}

This system has two obvious conserved quantities

\begin{equation} E_i = \frac{1}{2}\dot{x}_i^2+\frac{1}{2}\omega_i^2x^2 \tag{3.39} \end{equation} which are just the energies of each independent oscillator. The motion is obviously integrable and everything works out. However, in their explanation on section 4.2.2 they use this example to show that if the two frequencies are incommensurate

\begin{equation} \frac{\omega_1} {\omega_2 } \notin \mathbb{Q} \end{equation}

then the motion is not periodic since the trajectory $(x_1(t),x_2(t))$ will never return to its initial position again. Because of this, solutions densely populate the phase space of the system and any conserved quantity defined as

\begin{equation} \Gamma (x_1,x_2,\dot{x}_1,\dot{x}_2)=C \end{equation}

will be pathological discontinous. This is because for any initial condition $\chi_0=(x_1,x_2,\dot{x}_1,\dot{x}_2)$ there's another point arbitrarily close that belongs to a trajectory with an arbitrary different value of $\Gamma$. I think I understand the explanation. However, he claims that when we have this pathological we can't define conserved quantities other than $E_1$ and $E_2$. This, to me, sounds like it implies the system is not integrable, due to a lack of constants of motion. But I already know the system is fully integrable given it's just two copies of an harmonic oscillator. So my main questions are:

  1. Why are they saying that we can't define conserved quantities other than $E_1$ and $E_2$? What's special about those? They are also constants of motion defined as functions of $x_i$ and $\dot{x}_i$.

  2. What is the relation between incommensurate frequencies, the lack of conserved quantities and integrability?

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  • $\begingroup$ You have a system with 4 degrees of freedom and a set of 2 independent first integrals. That's all you need to conclude that your system is integrable. Why would you want more first integrals? $\endgroup$
    – Phoenix87
    Apr 20, 2020 at 13:23
  • $\begingroup$ I know that two independent first integrals are enough. What I don't understand is then why is he giving this example to show systems where there's a lack of constants of motion. $\endgroup$ Apr 20, 2020 at 17:05

1 Answer 1

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  1. OP has already noted that the 2D harmonic oscillator is completely Liouville-integrable with 2 globally defined, Poisson-commuting, real integrals of motion $H_1$ and $H_2$.

  2. Since the phase space has 4 real dimensions, there can at most be 3 independent real integrals of motion, and 4 independent real constants of motion. By definition an integral of motion cannot depend explicitly on time $t$ while a constant of motion can, cf. e.g. this Phys.SE post.

  3. We can rewrite the 2D harmonic oscillator $$\begin{align}H~:=~&H_1+H_2, \cr H_j~:=~&\frac{p_j^2}{2}+\frac{\omega_j^2q_j^2}{2}~=~\omega_jz_j^{\ast}z_j,\qquad j~\in~\{1,2\},\end{align}\tag{A}$$ in complex notation $$\begin{align}z_j~:=~&\sqrt{\frac{\omega_j}{2}}q_j + \frac{ip_j}{\sqrt{2\omega_j}}, \cr \{z^{\ast}_j, z_k\}_{PB}~=~&i\delta_{j,k},\qquad j,k~\in~\{1,2\}.\end{align}\tag{B}$$ For technical reasons we exclude the singular zero-leaf, i.e. the phase-space becomes $M=(\mathbb{C}^{\times})^2$, where $\mathbb{C}^{\times}:=\mathbb{C}\backslash\{0\}$. The phase-space $M$ has 2 complex dimensions. We can easily find 2 independent, globally defined, complex constants of motion $$F_j~:=~z_je^{i\omega_j t}, \qquad j~\in~\{1,2\},\tag{C}$$ which is the maximal number. The two Hamiltonians $H_j=\omega_j|F_j|^2$ depend on their absolute values.

  4. On one hand, if $$\frac{\omega_1}{\omega_2}~=~\frac{n_1}{n_2}~\in~\mathbb{Q}\tag{D}$$ are commensurate frequencies, then we can construct a globally defined, complex integral of motion $$ \frac{z_1^{n_2}}{z_2^{n_1}}.\tag{E} $$ Its argument is independent of $H_1$ and $H_2$, which shows that the system is maximally superintegrable.

  5. On the other hand, if the frequencies are incommensurate, then we can only define a 3rd independent integral of motion $${\rm Im}\left(\frac{{\rm Ln}(z_1)}{\omega_1}-\frac{{\rm Ln}(z_2)}{\omega_2}\right)\tag{F}$$ locally, because of the branch-cut of the complex logarithm ${\rm Ln}$.

References:

  1. J.V. Jose & E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998; Subsection 4.2.2 p. 183-185.
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