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I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics":

Another method of construction the same symplectic structure on complex projective space consists of the following. Consider small oscillations of a mathematical pendulum with an $n+1$-dimensional configuration space. We make use of the integral of energy to decrease by 1 the degree of freedom of the system, The phase space obtained after this operation is $\mathbb C P^n$ and the symplectic structure on it agrees with the form $\Omega$ described above by a factor.

Could anybody clarify what is written here and how to get the projective space from a pendulum?

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    $\begingroup$ Consider on $\mathbb{R}^{2(n+1)}$ the harmonic oscillator Hamiltonian $H(x,\xi) = \frac{1}{2}(|x|^2 + |\xi|^2)$. Its flow on $\mathbb{S}^{2n+1} \cong \{ H(x,\xi) = 1\}$ is periodic and the quotient map is exactly the (complex) Hopf fibration, hence the quotient is $\mathbb{C}P^n$. $\endgroup$
    – mcd
    Commented Oct 19, 2020 at 12:58
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    $\begingroup$ @mcd Wait. You gave a way to obtain CP^n but it has nothing in common with the Arnold writing. He definitely did not mean factorisation by trajectories. There should be factorisation by a first integral. $\endgroup$
    – Nikita Kalinin
    Commented Oct 20, 2020 at 7:09

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