Cyclic variables for Hamiltonian

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Main Question or Discussion Point

A single particle Hamitonian $H=\frac{m\dot{x}^{2}}{2}+\frac{m\dot{y}^{2}}{2}+\frac{x^{2}+y^{2}}{2}$ can be expressed as: $H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{x^{2}+y^{2}}{2}$ or even: $H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{\dot{p_{x}}^{2}+\dot{p_{x}}^{2}}{4}$. Does it mean that the system has no cyclic coordinates? Since all relevant coordinates appear explicitly in $H$. In that case, there are no constants of motion?

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Orodruin
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First of all, the Hamiltonian is a function of the generalised coordinates and the corresponding canonical momenta (or more generally, of coordinates on phase space). You should not write the Hamiltonian as a function of derivatives of the momenta. Second, it is unclear to me what "it" that is supposed to mean that the system has no cyclic coordinates is. What property of the expression are you referring to? Clearly, none of the coordinates that you have are cyclic, but that does not mean that there are necessarily other coordinates you could use where one of them would be cyclic (as in fact there is in this case).

First of all, the Hamiltonian is a function of the generalised coordinates and the corresponding canonical momenta (or more generally, of coordinates on phase space). You should not write the Hamiltonian as a function of derivatives of the momenta. Second, it is unclear to me what "it" that is supposed to mean that the system has no cyclic coordinates is. What property of the expression are you referring to? Clearly, none of the coordinates that you have are cyclic, but that does not mean that there are necessarily other coordinates you could use where one of them would be cyclic (as in fact there is in this case).
What other coordinate is that?

Orodruin
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What other coordinate is that?
What symmetries does the Hamiltonian have? In what coordinates would the Hamiltonian therefore not depend on one of the coordinates?
Hint: What you have is quite clearly a 2-dimensional harmonic oscillator.

What symmetries does the Hamiltonian have? In what coordinates would the Hamiltonian therefore not depend on one of the coordinates?
Hint: What you have is quite clearly a 2-dimensional harmonic oscillator.
Yes, the frequency is $\frac{1}{\sqrt{m}}$. Should I put everything in polar coordinates?

Orodruin
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